No title

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

Quo Vadis

History of Mathematics? Detlef D. Spalt

History of mathematics seems to have little prestige in mathematics as a whole. In The Mathematical Intelligencer, vol. 15, no. 4 (1993), pp. 4-6, the prominent historian Professor Grattan-Guinness describes history of mathematics as a "residual category," and his essay seems to blame this on the neighbouring subjects, especially mathematics and education, because of their poor support. I do not feel this kind of lament to be very helpful. In my opinion it is the task of the historians of mathematics themselves to till their field. In fact, history of mathematics is a fundamental category, and it is a sound subject on its o w n - - or at least it could be one. Let us briefly examine the present state of affairs and then see what to do.

ological standards. It is a do-it-yourself s u b j e c t - - y o u just start working, without any training. The only standard you have to meet is the support of a somehowestablished m e n t o r - - h e might not even be a historian. To get such support is easier of course if you follow his lines of thinking and if your research helps to elevate your mentor's achievements. What generally in science counts as the heart of innovation, namely the criticism of established ideas, is completely undesired in the history of mathematics today. At

Situation To start at the beginning, we have to realise that history of mathematics today is not a sound subject on its own. This can be seen in two ways. First, history of mathematics today has no topics and no methodological standards of its own. Instead, it is a field where anything goes, where everybody arbitrarily follows his or her personal interests. In history of mathematics today there does not exist any well-defined research project comprising more than one person at more than one locality (publishing of collected works aside). Nor are there puzzling open problems that historians of mathematics all over the world impatiently hope to solve. Rather one finds an absolute indifference concernin$ topics. At any conference on the history of mathematics you will meet an indefinite variety of topics, most of them dealing with the plain stating of mere facts (from lives, institutions, statements) such as any undergraduate can compile in a few weeks. Secondly, history of mathematics today lacks methodTHE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3 (~ 1994 Springer-Verlag New York

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least in Germany a novice may develop new and critical ideas but the establishment will not show the slightest interest in discussing them with him within a dozen years. As far as the rational development of knowledge stems from the criticism of established ideas, in history of mathematics today no such development is possible.

Consequences for the Learning Historian? One of the most striking consequences is the lack of a standard of competence within the community of historians of mathematics. You can have a career without having any sound training even in mathematics. Let us suppose that right at the beginning of your career an outstanding mathematician proves publicly in detail that one of your first essays is mathematically completely unsound; let us further suppose that he tries to point out your mathematical errors to you prior to publication. Even if you ignore this and publish the essay all the same, this may not stop you from achieving an academic position. On the other hand, it seems self-evident to some that any mathematician is able to do history without any special training and without knowing much about the historical context. The mathematician usually sees his task in doing history as telling us what our mathematical ancestors ought to have said but did not really say because they were not up to date with us. It is proof of the deplorable condition of the history of mathematics that this silly approach is well accepted nearly everywhere. If you are introduced to a historian of mathematics you have to be alert to the possibility that basic insights may simply not be there. He or she might not know what a mathematical proof is, or that Descartes couldn't have cartesian coordinates-- and worse, it might happen that he or she is not even able to grasp such matters. Because of this depressing condition of the subject, it is absolutely justified that the field per se does not get any respect in the scientific community. It is not the colleagues of the neighbouring subjects who have to account for the marginal esteem of the "residual category." It is the historians themselves, most of whose publications do not deserve respect. (Let me say aside that there do exist some very respectable historians of mathematics, although none of them is referred to in Professor Grattan-Guinness's essay.) Reflection If the historians of mathematics are dissatisfied with this state of their a r t - - and I think they should b e - - then they could start reflecting on their task. Historians in general are first of all the chroniclers of the transitoriness of things, and historians of mathematics are (or should be!) the chroniclers of the transitoriness of mathematical things. If well done, history of mathe4 T,E MATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994

matics first of all focusses on the changes of mathematical knowledge. At its best, it guides us to a former way of mathematical t h i n k i n g - - a n d to an understanding of this thinking in such a way that we are able to practise it ourselves. This former way of mathematical thinking is different from ours today but valid in its own concepts and categories. Only by grasping the difference between this sleeping art and our current way of handling the subjects are we able to realise the true development of mathematical thinking.

Essentials But this task demands a sound training in mathematics. Only by knowing the present state of mathematical affairs very well can the historian disclose a different state of affairs. This does not mean that only the most prominent researchers of present-day mathematics are able to do historical work. In fact, experience has shown that this does not really work. Sometimes the outstanding representatives of today's mathematics are unable to grasp the concepts of their forefathers insofar as these concepts fundamentally differ from their own. Mathematics is an intellectual endeavour, only one expression of the totality of human culture. That is why historians not trained in the search for former states of intellectual life are condemned to produce results of very limited significance. Since ancient times, mathematics is known to be deeply rooted in philosophy. So for the historian of mathematics to re-create past thinking, a prerequisite would be some substantial training in history of philosophy. If these two essentials for the art of doing history of mathematics were m e t - - a sound training in actual mathematics as well as in the history of philosophy-then the products of this subject would be much more substantial. Then it really could be hoped that historians of mathematics would establish a discourse of research focussed on programs (instead of personal interests), on criticism (instead of complacency), and on the change of knowledge. And then history of mathematics would be able to get rid of its deplorable reputation as a scrap heap of wrecked or retired mathematicians.

The Challenge of History We should not downplay the provocative essence of history of mathematics. Well done, history undermines the common ethos of mathematics in a two-fold way. It destroys the concept of eternal mathematics to be understood in absolute perfection. Instead of this, mathematics is shown as being dependent on the general changes in the human mind, as are all the other human sciences. History of mathematics questions the exceptional position of mathematics as untouched by the zeitgeist. The second way in which the history of mathematics undermines the common ethos of mathematics is this:

It is the only part of mathematics which cannot work within the Church of Platonism. Maybe this statement sounds a bit harsh, but I think it is valid. Let me give you the example of the sacrilege of non-Euclidean geometry during the last century. NonEuclidean geometry changed the status of the age-old theorem "The angles of the triangle add to pi" from being an indubitable truth to being a proposition only, which might be true or false, depending on the mode of geometrical thinking. A comparable example arose in our century with the emergence of non-standard (or if you like non-Cauchy) analysis. This pariah field shows that such esteemed theorems as "There does not exist a 6function" or "The real numbers are uncountable" are true only in a certain meaning (or system of axioms) but false

in another one. History of mathematics is antithetical to absolute truths. It is my experience that history of mathematics can play an important role in the training of mathematical students: It strengthens the ability to see mathematics as an essential cultural force; it opens up the thinking of students to different mathematical models. Mathematicians are blind to its potential use, it is true. But before reproaching them too harshly, historians of mathematics should set their own house in order. Fachreferat Mathematik und Informatik Niedersffchsische Staats- und Universita'tsbibliothek D-37070 G6ttingen Germany

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994 5

Reminiscences about the 1930s Ralph Phillips

Saunders Mac Lane has recently lectured on American mathematics, giving a bird's-eye view from the vantage point of a well-established mathematician. I would like to present a worm's-eye view of American mathematics in the 1930s from the perspective of a beginning mathematician. For me it was not a pretty picture. I entered UCLA in 1931, about 3 years after its metamorphosis from a teachers' college in central LA to a university in Westwood. The faculty was essentially the same with the addition of two or three new mathematicians, principally Earl Hedrick and William Whyburn. Hedrick was an imposing and charismatic figure, an excellent administrator and a flamboyant teacher; in time he became president of the American Mathematical Society (AMS). Among mathematicians he is best known for his translation of Goursat's Cours d'analyse. Whyburn was a student of R. L. Moore, an eccentric Texas mathematician who, together with his students, dominated the field of point-set topology for many years. He popularized the idea of laying out a course in the form of progressively inclusive lemmas and theorems and having his students prove the material. The pace is fairly slow, but such a course is great for developing a student's ability to do mathematics. I had the good fortune to take courses of this kind from three of Moore's students: Whyburn at UCLA and later Ayres and Wilder at Michigan. After finishing a year of graduate studies at UCLA, majoring in both mathematics and physics, I went on to the University of Michigan as a graduate student in physics in 1936. I immediately attached myself to Samuel Goudsmit, who soon afterward encouraged me to get out of physics and into mathematics. At that time, the graduate work in mathematics was essentially in the hands of W. L. Ayres, T. H. Hildebrandt, G. Y. Rainich, and R. L. Wilder. Ayres made a name for himself in point-set topology but gave up on research in mid-career to become chairman and later dean at Purdue. Hildebrandt was a well-educated mathematician, a student of E. H. Moore at Chicago; his research started out very well but 6

slowed down when he became chairman of the department, a position he retained for 23 years. Rainich was a very nice man who was especially kind to me. He specialized in relativity theory and was pleasantly surprised when, toward the end of his life, his work received a lot of attention. Wilder worked in topology; of the four, I believe that he left the most substantial mathematical legacy. Both Hildebrandt and Wilder were presidents of the AMS. I wrote my thesis under Hildebrandt in the (at that time) rapidly developing field of functional analysis. Hildebrandt was socially awkward, an organist in his church, a man of strict moral judgments and of many prejudices. Along with most other mathematics chairmen of that period he would not hire a black and believed that there should be a quota on the number of Jews in a department. However, his was a generous quota: there

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3 (~) 1994 Springer-Verlag New York

were at least three Jews in the mathematics department at that time. By contrast, it should be noted that in many mathematics departments the quota on Jews was at most one (see [3], p. 182). To this day I am sure that Hildebrandt liked me personally and thought well of m y mathematical abilities. Yet he felt obliged to warn schools where I applied for a position that I was Jewish, knowing that it would probably eliminate me as a candidate. I suppose that had he acted otherwise, many of his fellow chairmen would have felt betrayed. I had a Rackham Fellowship during the 2 years I was in the Mathematics Department at Michigan and for the following year (1939-1940), which I spent at the Institute of Advanced Study. I had written to von Neumann for permission to spend the year at the Institute and he had replied that he looked forward to collaborating with me. I suppose it was naive of me to take his reply literally, but I did and was disappointed when I was unable to see much of him. I have recently learned (see [2], pp. 194197) that 1939 was a troubling time for von Neumann, both for personal reasons and because he was already deeply involved in war work. Nevertheless, my stay at the Institute was very rewarding. I took courses from von Neumann and Hermann Weyl and wrote three papers, one in collaboration with Salomon Bochner. I remember Bochner as a kind and friendly man, still troubled by scars inflicted by Nazi anti-Semitism. There was a very good group of young mathematicians at the Institute that year: Warren Ambrose, Hugh Dowker, Paul Erd6s, Paul Halmos, Walter Strodt, and Henry Wallman. Among our other activities, we organized a seminar on almost periodic functions. I can report that Erd6s was just as eccentric then as he is now. At the end of the year, Oswald Veblen (a professor at the Institute) asked me to go to the University of Washington to replace Abe Taub, who was to visit him the next year. I accepted with the understanding that I would have a fellowship waiting for me at the Institute the following year. The reason I accepted was that I was becoming anxious about obtaining a permanent job. This was at the height of a 10-year depression, and there were almost no academic positions available for a n y o n e - - especially for Jews and foreigners. Talented mathematicians from Europe like Carl Loewner and Fritz John were fortunate to obtain jobs in universities that did not emphasize research. Witold Hurewicz, who arrived in the United States in 1938 with a well-established international reputation, spent a year looking for a job; he had given up and was on his way to China when at the last moment he got an offer from Chapel Hill. The job situation for young Jewish American Ph.D.'s was equally bad. Many of them with Jewish-sounding names changed their names to more acceptable ones, even at the cost of forgoing credit for their earlier publications. In the summer of 19401 visited Stanford, hoping to get a job. J. D. Tamarkin was there at the time and recom-

mended me to Gabor Szeg6, who was the chairman of the Mathematics Department. Szeg6, who was Jewish, later told Peter Lax that he tried to hire me but that the appointment was blocked by Professor Manning because I was Jewish. At the University of Washington I was one of three new instructors, one of whom was clearly not a research mathematician. In addition to the required 15-hour-aweek load, I taught a beginning topology class. At the end of the year I was the one instructor who was not kept. What made it all worthwhile for me was getting to know my future wife, Jean, who was a teaching assistant at the time. In the summer of 1941 Jean and I set out for the Institute, stopping off at Ann Arbor on the way. Hildebrandt almost offered me a job but abruptly stopped the negotiations when I informed him of my living arrangements. I do not attribute this to anti-Semitism. However, while we were in Ann Arbor I got a telegram from H. B. Phillips, the chairman at MIT, asking me if I would accept a position there; in those days this was tantamount to an offer. This came at the urging of Ted Martin, who had heard about me from Bochner. I, of course, telegraphed back my willingness to come to MIT, but my reply was never acknowledged. Ted Martin later told me that Phillips had in the meantime been informed by Hildebrandt that I was Jewish and that this ended the offer. I should mention that this was not an isolated event at MIT. A couple of years earlier, Norman Levinson came up against the same prejudice, and it was only because G. H. Hardy intervened on his behalf that he was eventually appointed. Hardy is said to have confronted the provost of MIT, asking him whether he was running a scientific or a theological institute [1]. At the end of the summer, we arrived at Princeton, looking forward to a year at the Institute. But this was not to be. The country was just beginning to prepare for the war in Europe, and the universities were being asked to train officers for the military. This happened in the summer and there was suddenly a shortage of qualified mathematicians. I was approached by Solomon Lefschetz who was empowered to offer me an instructorship at Harvard. "Approached" is probably not the right word, for he made it clear that if I did not accept, my future in mathematics would be very bleak; so I ended up at Harvard in 1941. Incidentally it should be noted that although Lefschetz was Jewish, he was not above engaging in a mild form of anti-Semitism. He told Henry Wallman that he was the last Jewish graduate student that would be admitted to Princeton because Jews could not get a job anyway and so why bother [1]. For the most part, the already established Jewish mathematicians whom I came in contact with (Bochner, Rainich, Szeg6, Tamarkin) were very supportive but were limited in what they could do by the prevailing culture. During my year at Harvard the younger analysts organized a seminar on functional analysis. Gelfand's paper THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

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on n o r m e d rings h a d just a p p e a r e d a n d I presented it at the seminar. Word got around, a n d I was asked to repeat it for s o m e of the older m e m b e r s of the faculty. At the second showing, Garrett Birkhoff attended and w a s so i m p r e s s e d that he asked m e to repeat it again for his father. M y only personal contact with G. D. Birkhoff w a s at a colloquium tea. While w e w e r e talking I m u s t h a v e been s o m e w h a t flustered, because m y t e a s p o o n fell off m y saucer; Birkhoff graciously picked it u p for me. At the time, G. D. Birkhoff w a s one of the m o s t eminent a n d politically p o w e r f u l m e m b e r s of the A m e r i c a n Mathematical Society. Because of this, his prejudices h a d an u n d u e influence on the m a t h e m a t i c a l c o m m u n i t y and, unfortunately, his anti-Semitic v i e w s were widely k n o w n [even to Einstein (see [3], p. 184)]. During his reign at H a r v a r d there were no tenured Jews in the Mathematics D e p a r t m e n t . His true feelings on the subject are e x p o s e d in a p r i v a t e letter that he w r o t e to R. G. D. Richardson a b o u t the n o m i n a t i o n of Lefschetz to be president of the A m e r i c a n Mathematical Society in 1934 (see [3], p. 183). H e said, "I h a v e a feeling that Lefschetz will be likely to be less pleasant than he has been, in that f r o m n o w on he will try to w o r k strongly a n d positively for his o w n race. T h e y are exceedingly confident of their o w n p o w e r and influence in the good old USA. The real h o p e in o u r m a t h e m a t i c a l situation is that w e will be able to be fair to o u r o w n kind." A n d he w e n t on, " H e will get v e r y cocky, v e r y racial and use the Annals as a g o o d deal of racial perquisite. The racial interests will get deeper, as Einstein's and all of t h e m do." In all fairness it should be noted that, in spite of his stated position on refugees ([3], pp. 196, 199), Birkhoff did help s o m e (Jewish) refugees get positions in less prestigious schools. In 1941 the United States entered the war, a n d I started w o r k i n g at the Radiation L a b o r a t o r y at MIT, at first part-time but, after the school y e a r ended, full-time until 1946. F r o m that point on, the job situation greatly i m p r o v e d a n d anti-Semitism m a r k e d l y diminished, alt h o u g h traces of anti-Semitism were still evident as late as 1960. I realize that the job m a r k e t in m a t h e m a t i c s is v e r y tight right n o w but I expect it will i m p r o v e in a y e a r or two. In a n y case it is not nearly as b a d as in the 1930s. References

1. Zipporah Levinson, personal communication. 2. Norman Macrae, John von Neumann, New York: Pantheon Books, 1992. 3. Nathan Reingold, Refugee mathematicians in the United States of America, 1933-1941: Reception and reaction, A Century of Mathematics in America, Part I (Peter Duren, ed.) Providence, Rh American Mathematical Society, (1988), 175-200.

Department of Mathematics Stanford University Stanford, CA 94305-2125 USA 8

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

Jobs in George

1930s and the Views of Birkhoff Saunders Mac Lane

Ralph Phillips has written eloquently of the difficulties of getting a job in the 1930s and early 1940s. Then antiSemitism was present in many American universities. There were quotas for Jewish students at a number of eastern universities (e.g., Yale, Harvard). In many mathematics departments, it was often felt that mathematicians who were Jewish were too contentious, and were likely to urge the appointment of others like them. These opinions, then or later, are clearly wrong. But they were not limited to mathematics or to universities, but were present in the larger society, for example in my uncle's prosperous New York law firm. I saw prejudice against blacks (then called Negroes) when I was in the third grade in Jamaica Plains, Boston. I heard of the Ku Klux Klan hate for Catholics when I was in High School in a small town (Leominster, Mass.); my 80-year-old grandfather defended the Catholics in a two-hour lecture in his Congregational church. I was then unaware of anti-Semitism. My high school chum Jack Cohen was easily accepted at Harvard, while family tradition and my uncle's money brought me to Yale. It was at once clear to me that Yale was in part a "finishing school for young men" (in a telling phrase later used by Kingman Brewster), but I did not recognize the corollary: the quota for Jewish s t u d e n t s - - although I had several friends who came from the New York ghettos. I simply did not observe that the Yale faculty involved no Jewish professors. There are persistent statements that George Birkhoff was notably anti-Semitic. This essay is to record such information as I may have on this question. Early in the century, it was common to "recognize" in each country the "best" mathematician: In France, Poincar6; in Germany, Hilbert; in Great Britain, G.H. Hardy; in the USA, E.H. Moore. George Birkhoff, a Ph.D. student of Moore's, taught at Princeton, but in 1912 left his position as Professor at Princeton to accept an Assistant Professorship at Harvard (Harvard was then thought to have a stronger mathematics department than Princeton). Poincar6 died in 1912, and in the same year published a paper on the 3-body problem: the existence of periodic solutions of this problem could be deduced from a conjectured "geometric theorem" Poincar6 was able to prove

only special cases of. Within a year, Birkhoff found a simple proof of "Poincar6's Geometric Theorem." This made Birkhoff famous. Until his death in 1944, he was generally regarded as the "best American mathematician." This ranking even appeared in stories: "George, who is the second best mathematician?" Why, probably Wiener. "And who is the third best?" John von N e u m a n n - - and so on to the twentieth best, listing no Birkhoff. This recognition meant that Birkhoff was prominent and a natural target for criticism. He was very influential in the American Mathematical Society; a former secretary once told me that when Birkhoff's term as member of the AMS Council came to an end, he simply continued to attend all Council meetings. Birkhoff had served as President, and so was always in touch with his close friend Dean R.G.D. Richardson (Brown University), a long-term Secretary of the AMS. Ralph Phillips writes about Birkhoff's eminence and then refers to an article by Nathan Reingold to show that GDB was well known (even by Einstein) to be antiSemitic ("the leading academic anti-Semite"). This reference is worthless: Einstein did not then carefully follow the American academic scene, Birkhoff had a (then well-

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3 (~) 1994 Springer-Verlag New York 9

known) competing theory of relativity, and this was at a time when there were highly active mathematical antiSemites in Berlin (a quotation from them: "Princeton ist ein kleines Negerdorf"). Reingold also quotes a private letter from Birkhoff to his good friend Richardson about the proposed nomination of Solomon Lefschetz to be President of the AMS (it appears that GDB had a different candidate). This letter is cited as evidence of anti-Semitism. The opinions there cited were common among mathematicians of Birkhoff's generation. The public fact is that Lefschetz was at exactly that time chosen as president of the AMS. One common concern was worry about obstreperous people. At Princeton, there were ditties about professors: Here's to Lefschetz, Solomon L. Irrepressible as hell When he's at last beneath the sod He'll then begin to heckle God In my own experience, Lefschetz was both obstreperous and enthusiastic-- about research in mathematics. In the 1930's there was a great tension in the American mathematics community: to help emigr4s or to support young Americans. Oswald Veblen and others at Princeton were vigorously active in finding positions for many emigr6s. George Birkhoff, to my clear recollection, was equally emphatic in wanting to support promising young Americans. It is hard to tell now whether Birkhoff's emphatic views on the point were also partly anti-Semitic. His views did allow him to reach across the Atlantic to invite Lars Ahlfors from Finland. In the event, most mathematicians fleeing Europe were helped to some sort of position in the United States-in some cases, a position not up to their known merits. I recall two cases of failures: Max Dehn, noted for work in topology, got only a weak position. It is shocking that Andr6 Weil, coming to the USA in 1941, found only a junior position at Lehigh University. During the depression, some young American Ph.D.s also didn't get academic positions; for example, my Yale classmate Eugene Northrop (Ph.D., 1933) chose to accept a position at a preparatory school. Getting jobs was hard for everybody. One cannot know what Northrop (and others) might have accomplished with better opportunities. It is my view that the 1930s tension between placing refugees and helping young Americans came to a reasonable balance of those interests--and that the differing views of Veblen and Birkhoff served to help the balance. My view on this issue cannot be objective. In 1934, just married and finishing a post-doc, my only job prospect was as a master at Exeter (a famous preparatory school). Then Harvard offered me a two-year Benjamin Pierce Instructorship (I suppose, by recommendation from Birkhoff and Marshall Stone). I spent no time thinking that Harvard might have chosen any one of a half-dozen well-qualified emigr6s. By then I was well aware of the extent of academic anti-Semitism, but I did 10 THEMATHEMATICALINTELLIGENCERVOL.16, NO. 3,1994

not attach these views to particular individuals. Birkhoff never took steps to appoint Norbert Wiener to a Professorship at Harvard. Wiener's father had been a Professor of Slavic Literature at Harvard: I knew Norbert well in 1934-36, and it was clear to me (and many others) that he strongly hoped to get an offer from Harvard. This never transpired. It is my impression that friends of Wiener, as perhaps represented by Reingold's article, hold this against Birkhoff, as due to anti-Semitism. In this c a s e - - and in that of Andr6 Weil-- it is not easy to disentangle and judge the various reasons behind employment actions taken in the distant past. Wiener attended Harvard's colloquia; Birkhoff clearly listened to Norbert's ideas. There could be many reasons w h y Birkhoff did not take steps to appoint him (say in the 1920s). Great achievement does not entitle one to demand appointment at a specific university. I have a careful list of all new faculty (= tenure track) in mathematics appointed at Harvard 1922-1944. There are eleven. I know directly that George paid careful attention to such matters. Nine of the eleven held Harvard degrees; the exceptions are Ahlfors (Assistant Professor 1935; Fields medal 1936) and Mac Lane (Assistant Professor 1938). All except Van Vleck were initially appointed without tenure. This pattern is clear; GDB thought that he could recognize young talent early, and considered candidates whom he knew personally. As is often emphasized, none of these eleven was a Jew. But this is just not the real question. There was then no affirmative action, no propaganda for underrepresented groups, and no talk of diversity. The central question was and is: Did Birkhoff's actions keep Harvard a top department? The answer must be "yes." After the war, there were many changes. Harvard had to reach out for the best in newly active fields (algebraic geometry, Oscar Zariski, and group theory, Richard Brauer). 1 What matters is not their origins, but their achievements. And let us now join in emphasizing tolerance, with less concern for past errors. I was a regular member of the Harvard Department of 1938-47; I knew George Birkhoff well. I do not now recall any explicit statement of his about Jews or about the ethnic origins of mathematicians; I have dependable evidence that he thought that Jewish mathematicians stopped doing research early. I recall no discussion of racial issues at Department meetings (which were frequent, friendly, and usually at the Chairman's home). It seems likely that Birkhoff shared the somewhat diffuse and varied versions of anti-Semitism held by many (most?) of his contemporaries. He should not now, fifty years later, be made the whipping boy for those regrettable views. Department of Mathematics The University of Chicago Chicago, IL 60637 USA 1 Editor's Note: On the hiring of Zariski, see C. Parikh, The Unreal Life

of Oscar Zariski (Boston: Academic, 1991).

Hyperbolic Geometry and Spaces of Riemann Surfaces Linda Keen

Introduction Classifying Riemann surfaces is a problem that has fascinated mathematicians for more than a century. Real analytic, complex analytic, and geometric solutions have been found using a variety of techniques. In this article I shall examine several approaches; I shall restrict myself to the situation where the surface is a torus or a punctured torus and make the description very explicit.

Moduli Spaces for Riemann Surfaces A Riemann surface is a topological surface with a complex analytic structure on it; that is, the surface is covered by a set of charts so that the relation between the maps defined on overlapping neighborhoods is complex analytic. If $1 and $2 are two Riemann surfaces, it can happen that there exist homeomorphisms from $1 to $2, and yet none of these homeomorphisms is complex analytic. In other words, $1 and $2 have the same underlying topological surface but are distinct as Riemann surfaces. It turns out that, unless the underlying surface is the 2-sphere or the 2-sphere minus 1, 2, or 3 points, there is a continuum of distinct Riemann surfaces with the same underlying surface. How then might we characterize the set of all distinct Riemann surfaces for a given topological surface S? This set is known as the moduli space of the surface and is denoted Mod(S). To put the characterization problem more concretely:

This article looks at several examples that illustrate what the problem is about and some of the methods that have been used to attack it. The geometric key is that a complex analytic homeomorphism is conformal; that is, it preserves angles locally. It is an easy exercise in calculus to show that if a map is complex analytic and invertible at a point, then the angle between two curves intersecting transversally at that point, measured as the angle between their tangents, is equal to the angle between the image curves at the image point. Maps that distort angles cannot be complex analytic.

First Simple Example Let )~ be a real number in the unit interval / = {0 < ,~ < 1}, and consider the cyclic group G~ = {g~ : z ~ ~ z , n 9 Z}

9 Can we realize Mod(S) as some natural geometric object (e.g., as a real analytic manifold, or perhaps even as a complex analytic manifold)? 9 Can we find parameters (these are the "moduli'), at least for some large open subset of this manifold, so that as we vary the parameters there is some aspect of the complex structure of the corresponding Riemann surfaces that is visibly varying with the parameters?

Research partially s u p p o r t e d by NSF g r a n t DMS-8902881. THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 3 ~)1994 Springer-Verlag New York

11

of conformal homeomorphisms of the punctured plane, C* = C - {0}, to itself. The natural m a p C* --+ C * / G ~ -~ S~ maps the half-open annulus A~ = {IAJ < Jz I < 1} oneto-one onto the quotient S;~. Because ~]1 : z ----+ ,~Z maps the unit circle one-to-one onto the inner b o u n d a r y of A~, we see that S;~ is topologically a torus: The image a of the unit circle and the image fl of the real axis are a pair of generators for its homology. Projecting the complex structure from C onto S;~ makes it a Riemann surface. When A is close to 1, fl is very short, so we get a very "skinny" torus, and as ), decreases, the torus gets "fatter." Because complex analytic maps preserve conformal geometry, this distortion is reflected in the complex structure, so it is plausible that we get a whole continuum of different complex structures as A varies in the interval. Now suppose that A is no longer real but is a complex number re i~ in the punctured unit disk, D* = {z : 0 < ]z I < 1}. Define the group G~, the quotient C * / G ~ ~- S;~, and the annulus A~ as above. The element g~ E G~ still identifies the inner b o u n d a r y of A~ with the outer boundary, but n o w the inner circle is twisted by the angle/9 before it is glued. This twisting distorts the complex structure of the quotient, so for fixed r and varying/9, there is another whole continuum of different structures. In fact, classical theorems from elliptic function theory tell us that every possible complex structure on the torus is obtained from some ), E D*. Thus, D* is a good candidate for our natural realization of Mod(S) and A is a natural parameter. However, D* is not quite Mod(S) because m a n y different k's m a y give rise to the same structure. We shall return to this question after the next section. We shall see that the parameter space D* is, in fact, a covering space of Mod(S). It is typical that the moduli space is difficult to find; one often has to settle for a covering space.

Second Simple Example

A more usual representation of the torus is obtained by considering the group Gr = {gm,n (Z) = z + m + n r : m , n E Z}, where r is in the upper half-plane U, and forming the quotient C ---, C / G r -~ Sr. The parallelogram Pr spanned by 1 and r, with its opposite sides glued, is the analogue of the annulus. The complex structure on the torus is inherited from C and depends on G~-. As uniform stretching doesn't change the complex structure, the quotient of C by the group {z -+ z + r m + r n r : m , n E Z}, where r is any positive scalar, determines a torus equivalent to Sr. The space of moduli is the collection of groups G~-, r E U. The parameter space U is, therefore, another covering space of Mod(S) that is easy to find and to work with. To find Mod(S) one has to see h o w different choices of generators for the groups G~ are related. For these groups one knows h o w to do this: The classical m o d u l a r group P S L ( 2 , Z) relates pairs of generators. The plane C is simply connected and is the universal cover of the torus. The exponential m a p s the parallelogram P~ onto the annulus A;~ for A = exp(2rrir). Because 12 THEMATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

the parameters r and - 1 / r give rise to equivalent tori, so do the parameters k = exp(2rcir) and k' = e x p ( - - 2 r r i / r ) . If r = it, for t > 0 real and large, A is real and very s m a l l so the underlying torus is fat. On the other hand, k' is real and close to 1, so the underlying torus is skinny. Thus, a torus that is fat from one perspective is skinny from another. The relation between r and A also shows that the parameter space D* is an intermediate covering space between U and Mod(S).

Boundary Behavior In our examples, the plane domains in which the parameter spaces are embedded have boundaries. This means that if our parameter reaches the boundary, something has h a p p e n e d - - t h e construction of the torus no longer works. Let us look at what happens w h e n A = re i~ tends to the b o u n d a r y of D*. The absolute value of the parameter, r = ]AI, is measuring the size of the annulus. The open arc from 1 to A projects to a closed curve fl on the torus S;~. If 8 = 0, so that A = r, the length of fl on S;~ is Jlog rI; as r ---+0, it becomes infinite. On the other hand, suppose /9 = 2rcip/q for p / q rational, and consider the collection of arcs {(r, 1), (re i~ ei~

(re 2~~ e2i~

(rg (q-1)i8, e(q-1)i8)}.

They project to a closed curve on S;~ that I again call ft. N o w if r --+ 1, the arcs get short, fl becomes "pinched," and its length on S;~ goes to zero. In either case, there is no longer a torus; it has become a doubly infinite cylinder. N o w let us look at the parameter space U. What happens as r approaches the rational points on the boundary of U? Suppose r = p / q + it, p, q E Z , t E R +. Draw the parallelogram P~ spanned by 1 and r; it contains the vertical line joining the origin and - p + qv = qit, which projects to a closed curve fl on & . As t --+ 0, fl is "pinched," and w h e n t = 0, &- has degenerated to a doubly infinite cylinder again. It is m u c h more difficult to describe what is degenerating on &- as we approach the irrational points on the boundary. If we write r = r + it, where r is irrational and t > 0, the projection fl of the vertical line in the parallelogram joining 0 and - r + r = it never closes up on &and, hence, is an open curve of infinite length. If we call a the projection of the generator joining 0 and 1, we see that w h a t is getting shorter is the length of the segment of fl between its successive intersections with c~.

Third Example: A First Taste of Hyperbolic Geometry In our first example, we can think of the annulus A;~ as the torus cut open along a curve. The domain C* is "tiled" by more annuli An = gn(Ax); the annuli An don't overlap and together fill out all of C*. The group G;~ is a discrete group of conformal self-maps of C*. In our second example, we can think of the parallelogram Pr as the torus cut along a pair of simple curves that intersect exactly once. The group of trans-

and let G = (g, h} be the group they generate. We easily compute g(-1)=cc,

-i

-0.5

0.5

Figure 1. The hyperbolic quadrilateral.

lations G; determines a collection of parallelogram tiles, Pm,n = gm,n(P~), that do not overlap and fill out all of C. Again G~ is a discrete group of conformal self-maps of C. We can apply this idea of "tiling" to obtain techniques that work not only on tori but also on more complicated Riemann surfaces. We cut the surface up to obtain a piece P of complex plane; we then try to find a group G of conformal maps to obtain a collection {gP} of images of P that fill up some simply connected domain f~ in the plane without overlap; each element of G should be a conformal self-map of fL I illustrate again with a simple example. Start with a torus, and take a pair of simple closed curves on it intersecting in the point ~7. N o w remove the point ~7to obtain a punctured torus. Cutting along the curves gives us a quadrilateral with its corners removed. As we try to make our tiling, we see that because f~ is simply connected, the removed corners will have to be on the boundary of the domain f~ we are tiling. It follows that f~ will have to have at least four boundary points, and hence, by the Riemann mapping theorem, cannot be complex analytically equivalent to either C or C*. Let us get very specific. Suppose that P (see Fig. 1) is the region inside the upper half-plane U bounded by the semicircles C, = {]z + 1/21 = 1/2} A U and C2 = { I z - 1/21 = 1/2} A U and by the semi-infinite vertical lines h={{Rz=l}nU

and

/2={~z=-I}NU.

N o w consider the linear fractional transformations 2z+1 z+l h(z) z+l ' z+2'

g(0)=l

and

g((-1+i)/2)=1+i,

so g maps the semicircle C1 onto the vertical line h. Moreover, it maps P onto a quadrilateral gP adjacent to P along h. It does not overlap P, and its vertices are again on R. Similarly we see that h maps the vertical line I2 to the semicircle C2; and that hP is a quadrilateral with vertices on R, not overlapping P, and adjacent to P along the semicircle. The group G is a discrete group of conformal self-maps of U, and because P has zero angles, one may show that the images of P under G do, in fact, tile U. This example gives us a once-punctured torus, and it has a complex structure inherited from U. N o w we can puncture any torus (the torus being homogeneous, it doesn't matter where we puncture it), so there is again a whole family of possible complex structures for the punctured torus. H o w can we introduce parameters into the group we just constructed to vary it and obtain these other punctured tori? In the late nineteenth century, Poincar6 [1] discovered a technique which he, Fricke [2], and others used on this problem. It was used again by a number of people in the mid-1960s, including Ahlfors [3], Bers [4], Fenchel [5], Maskit [6], and the author [7-9]; in the 1970s, it was enlarged and developed further by Thurston, Sullivan, and Gromov [10]. What Poincar6 remarked on was that the group of linear transformations

(az+b)/(cz+d),

a,b,c, d E R , a d - b c > O ,

are not only conformal homeomorphisms of U but are also isometries with respect to the hyperbolic metric on U. The hyperbolic metric is defined by ds = Idzl/,~z. Geodesics are circles orthogonal to the real axis (and vertical lines). The distance from any point inside U to a point on R tO {oo} is infinite. In our second example, where we tile the plane by parallelograms, we may convince ourselves that we can choose the basic parallelogram in any shape by choosing the lengths of the sides and the angle between them. These lengths and the angle determine generating translations for the group. Because rescaling doesn't change the complex structure of the quotient, we may always assume one of the lengths is 1. Then, as the angles of a parallelogram add up to 2rr, four copies fit around each corner, and we can tile the plane. In our punctured torus example, the quadrilateral P is bounded by hyperbolic geodesics, but they have infinite length. Moreover, they meet at 0 angles at the boundary. Are there hyperbolic geometric invariants sitting inside P somewhere? Does it have a "hyperbolic shape"? The answer is yes! THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

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The hyperbolic isometry g(z) fixes exactly two points on R and leaves the hyperbolic geodesic A a joining them invariant. This geodesic is called the axis of g. Unlike the Euclidean case, the hyperbolic distance between z and g(z), du(z,g(z)), is not the same for all z E U. This distance is minimal for any z E As; the minimum distance Ig is called the translation length of g. Similarly the isometry h has an axis An and a translation length lh; o n e sees that A a and Ah intersect in exactly one point. Suppose now that we try to construct an arbitrary hyperbolic quadrilateral P with four infinite sides, meeting in vertices on the real axis, and such that there are hyperbolic isometries g and h identifying the pairs of opposite sides. It is a theorem, certainly known to Fricke and Fenchel, but first published by the author [8], that 9 the "shape" of such a hyperbolic P is determined by the translation length of either isometry, lg or lh, and the angle 0 between the axes Aa and Ah, and 9 there is a P and a group for any given shape. Only one length is necessary in this case because there are no isometries that change scale. In sum, we have constructed a simply connected covering of the moduli space of a punctured toms parametrized by two real variables, {(lg, 0) E R + x (0, 7r)}. These parameters have a geometric interpretation on the surface. There is also a simple way to write these parameters as real analytic functions of the coefficients of the generators of the group. An important point here is that the methods of Examples I and 2 do not generalize to surfaces of higher genus, but these methods do.

Complex M o d u l i Spaces

For # C U, consider the group G a = (g, ha) where

g(z)=z+2,

14 THEMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994

z

Using techniques originated by Maskit (e.g., [6], VII), one can show that for appropriately chosen #, there is a simply connected domain f~(Ga) such that the group G a is a discrete group of conformal automorphisms and f~(Ga)/G a is a punctured torus. To get an indication of how this works, choose # = 3i and let P be the region (see Fig. 2) 9 between the vertical lines ~z = -1, ~z = 1 and the circles IzI = 1 and Iz - 3il = 1, 9 with vertices -1, 1, 1 + 3i and -1 + 3i. The map g(z) takes the left side of P to the right side and maps P to a translate adjacent along the right side. The map h(z) takes the lower semicircular boundary onto the upper one and maps P to a quadrilateral adjacent along the upper semicircle. This is the start of our tiling. It is not obvious, but it follows from Maskit's theory that the images of P under G3i do not overlap. As we generate these images of P, they fill out some domain f~(G3i) in C, which, by construction, is invariant under G3i. Unlike Example 3, where the images of the tile P filled out the recognizable upper half-plane, the domain f~(G3i) is not easily described; in fact, f~(Ga) is different for different choices of #. To get some idea of what the domains ft(Ga) can look like, in Figures 3, 4, and 5 I show the computer pictures made by Ian Redfern at Warwick University for the groups G a w i t h # = 3i, # = 0.0533 + 1.9i, and # = 0.5001 + 1.667i. The domain f~(Ga) is the complement of the closed circles; its boundary is quite intricate.

The parameters for conformal structures on Riemann surfaces that we found above using hyperbolic geometric methods have many desirable properties. They are intrinsically defined; we can explicitly compute the polygonal tile and, hence, the group they determine; they work for arbitrary Riemann surfaces. In the first two examples, we see how the complex structure on the torus depends on the parameter as a complex variable, so these parameter spaces have a rich structure. The methods, however, depend on elliptic function theory and work only for tori. In the third example, where the methods do generalize, the complex structure on the punctured torus depends on the parameters as independent real variables, so the parameter space has less structure. Ideally one would like to find a method for constructing parameter spaces for general Riemann surfaces so that the parameters are complex and the dependence of the geometry of the Riemann surface on these complex variables can be understood. The Punctured Torus Revisited Here is another complex representation of the moduli space of a punctured torus that will generalize.

1 ha(z)= -+#.

y 1.5

,

-1

,

,

,

,

,

. .

-0.5

Figure 2. The tile P for C.~.

0.5

1

1.5

2

Figure 3. ~ ( ~ ) .

Figure 4. ~(Go.os.~lm). THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 3,1994 1 5

Figure

5. ~'~(~.50o1+1.667i).

Figure 6. T h e M a s k i t e m b e d d i n g w i t h p l e a t i n g coordinates. 16 THEMATHEMATICALINTELLIGENCERVOL.16,NO. 3,1994

The Maskit Parameter Space. To show that the specific group G3i gives us another way to represent a particular punctured torus by forming the quotient ~ (G3i)/G3i requires Maskit's combinatorial theory for groups that represent Riemann surfaces. To show that every punctured torus can be represented by some group in the family {Gu} takes a different set of techniques, from partial differential equations and the theory of quasi-conformal mappings developed by Ahlfors and Bers in the early 1960s (see, e.g., [3, 4, 11]). I will not explain these theories, but just report that for the punctured torus they tell us that: 9 there is a simply connected domain A4 c U such that for # c M there is a domain fl(Ga) on which the group G~ acts as a group of conformal homeomorphisms and such that ~(G~)/G~ represents a punctured toms, and 9 every punctured torus is represented by some #EM. In the first example, the cyclic group depending on a parameter ~ E D*, we saw that the boundary of the disk was a natural boundary because the tori had degenerated. Similarly, the Maskit parameter space A,~ is emb e d d e d as a domain inside the upper half-plane and the tori must degenerate as we approach its boundary. H o w can we find or describe this boundary? Using Maskit's techniques one can prove that the half-plane ~# > 2 is contained in de/. Therefore, the boundary OA/I in C is in the horizontal strip 0 < ~# _< 2. Wright [12] used experimental techniques to compute this boundary and came up with the picture in Figure 6. This picture was the jumping-off point for the author's ongoing collaboration with Caroline Series on complex moduli spaces [13-17].

Hyperbolic Geometry Again-- This Time in Three Dimensions When Poincar6 realized that linear fractional transformations with real coefficients were isometries of the hyperbolic plane, he also realized that linear fractional transformations with complex coefficients were isometries of hyperbolic 3-space. Hyperbolic 3-space can be modeled by the upper half-space H 3 = {(z, t) : z c C, t c R+}. The hyperbolic metric there is given by ds = v/idz] 2 + dt2/t. Geodesics are circles orthogonal to the base C, and hyperbolic planes are hemispheres orthogonal to the base C. Linear fractional transformations map circles and straight lines in C onto circles and straight lines; in fact, each is a product of an even number of reflections in lines and inversions in circles. Given a circle in C, we can view it as the equator of a sphere in R 3. An inversion in the circle extends naturally to an inversion in the sphere. Similarly, reflections in lines in C extend to reflections in the planes through them orthogonal to C. The isometry of H 3 corresponding to a linear transformation is a product of these extended inversions and reflections. One checks that since there are an even number, the upper half-space

is preserved. It is an exercise to check that the metric is preserved. Fenchel [5] and Greenberg [18], in the early 1960s, began to use techniques of 3-dimensional hyperbolic geometry to study groups representing Riemann surfaces. The idea was that because the group was discrete, one could look at the quotient 3-manifold, HB/G. It is a manifold with boundary, and the Riemann surfaces represented by the group are the boundary components. Let us see how this works in our third example, the group G = (g, h). The action of G on U can be extended to H 3 and to the lower half-plane L. Reflect the circles Ci and the lines Ii, i = 1, 2, in the real axis. These reflections determine a region/~ in L, and L/G is again a punctured toms. Note that the surfaces U/G and L/G are antiholomorphically equivalent, for the maps on local neighborhoods are given by complex conjugation. The hemispheres over the circles Ci and the vertical halfplanes over the lines Ii, i = 1, 2, bound a region R C H3; and g identifies the hemisphere over C1 with the plane over h , while h identifies the plane over/2 with the hemisphere over C2. The polyhedron R is a tile for the group G acting on H 3. The quotient (U U L UH 3)/G is a 3-manifold whose boundary consists of a pair of antiholomorphically equivalent punctured tori. As we shall see, not all groups of linear fractional transformations acting on H 3 act so symmetrically with respect to the real line, nor are the relations among their boundary surfaces so easy to determine. In the early 1970s, Marden [19] studied the relationship between groups of linear fractional transformations acting on H 3 and the topological properties of their quotient 3-manifolds, and in the late 1970s, Thurston [10] introduced revolutionary new techniques involving this hyperbolic geometry to attack classification problems for both Riemann surfaces and 3-manifolds.

Convex Hulls and Pleated Surfaces Let us return to the family of groups {G~} for # E .Ad. For each group we have an open plane domain f~(Ga) invariant under Ga. The boundary, A(Ga) = 0f~(Ga), is a closed Ga-invariant set called the limit set of Ga. Let us turn our attention to it. One of Thurston's ideas was to consider the (hyperbolic) convex hull C in H 3 of the set A(G~) and to study its boundary. This boundary is also G~-invariant, and one can prove that there is a G~-invariant component of this boundary, OC(G~), that is homeomorphic to f~(G~). The quotient, OC(G~)/G~, is, therefore, again a punctured toms. Thurston saw that OChad certain geometric properties that were very useful. It is a surface in H 3 made of pieces of hyperbolic plane joined along geodesic curves that, because of the convexity, can only meet on (~ = OH3 in points of A(G~). The quotient surface S~ = OC(G~)/G~ is, therefore, also made up of pieces of hyperbolic plane joined along nonintersecting geodesics. Thurston called THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

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such surfaces pleated, and the geodesics along which they are pleated, the pleating locus. Before we see how to extract information about the groups G~ from these ideas, let us see w h y they don't give any new information about the groups of Example 3. There, the set 12is always U and the limit set A is the real line. The convex hull of A is the vertical plane above A and so is equal to its boundary. There is only one hyperbolic plane and so no geodesic where two planes are joined; the pleating locus is, therefore, empty. Figure 4 is a picture of A(G0.0533+l.9i). This limit set is very intricate and its convex hull C has interior. To get a sense of what C looks like, note that by definition the hyperbolic geodesic joining any pair of points in A belongs to C, as does the hyperbolic triangle spanned by any three points in A. If four or more points of A lie in a circle C, the convex polygon they span is in a plane and in C. If, moreover, there are no points of A inside C, then the hyperbolic plane spanned by C intersects OC in a hyperbolic convex polygon. If we look carefully at Figure 3, 4, or 5, we see a pattern of closed circles and other overlapping circles with missing boundary arcs. The interiors of these circles contain no points in A. The boundary of the convex hull in H 3 consists of the intersection of the hyperbolic planes spanned by all the circles that we see. The full planes spanned by the closed circles belong to cqC. Over the other circles, the piece of the plane belonging to c9C is an infinite-sided convex polygon. When a pair of circles intersect, the planes spanning them intersect in a circular arc that is a boundary curve of the polygon on each plane. It is a geodesic with its endpoints in the limit set; c9C is "bent" along this geodesic at an angle equal to the angle between the circles. The set of geodesics formed by the intersecting planes is the pleating locus. In computer pictures for various groups Gg E .&l made by Wright and Redfern, and particularly for those groups near the boundary, one could see patterns of circles in the limit sets A(Gg), and we thought that there should be meaning to the patterns. For example, note that the patterns in Figures 3, 4, and 5 are decidedly different. What Series and I realized is that whenever a pattern of circles appears in A(G~), the quotient S~ is pleated along some simple closed curve and the curve is determined by the pattern!

Enumerating Simple Closed Curves on the Punctured Torus Consider the unpunctured torus with simple closed geodesics a and fl intersecting once. We may assume a is the projection of the line in C joining 0 and 1, and fl is the projection of the line joining 0 and T. Every simple closed geodesic on the torus then has the form pa + qfl and is the projection of a line joining 0 to p + q~for relatively prime integers p and q. For each such pair, (p, q), we have a family of parallel lines projecting onto a family of parallel geodesics. 18

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

The fundamental group of the punctured toms, zrl (S), is also generated by a pair of simple closed geodesics intersecting once, but it is a free group not an abelian group. We know from Maskit's theory that for # E .At, the domain f~(Ga) is simply connected; it follows that G~ is isomorphic to 7rl (S). The "forgetful map" from S into the unpunctured toms, defined by forgetting the puncture, shows that each simple closed curve on S is also a simple closed curve on the unpunctured toms. It induces a projection on fundamental groups zrl (S) ~ Z + Z,

from which we see that there are many elements in 7rl (S) that project to (pa + qfl). It is an interesting fact, proved by Series [20], that there is a unique simple closed geodesic 7v/q in the inverse image of (pa + qfl). Moreover, there is a unique conjugacy class in G~ containing a shortest cyclically reduced representative for that geodesic. Hence, there is a canonical word Wv/q in Gg associated to each simple closed geodesic on the punctured toms. These words may be enumerated recursively using continued fractions.

Pleating Curves and Moduli Given a linear fractional transformation, (az + b)/(cz + d), we may assume without loss of generality that ad - bc -- 1. The trace of the transformation, a + d, is then well defined, and conjugate transformations have the same trace. If we look at words in Gg, they are compositions of the maps g and h a, so their coefficients and their traces are polynomials in # with integral coefficients. The crucial observation [13] that relates the complex parameter # to the geometry of the hyperbolic 3manifold H3/Ga is THEOREM 1. Whenever the quotient of the convex hull boundary Sa is pleated along the curve %/q, the trace polynomial Tr Wp / q ( # ) is real-valued. We also prove THEOREM 2. For any pair of relatively prime integers, (p, q), there is some # E .M such that S~ is pleated along 7p/q" These theorems together give this picture of the parameter space: THEOREM 3. The space A4 is foliated by real analytic curves 7Jr, r E R, a dense subfamily of which is defined by properly chosen branches of the curves defined by

.~ Tr Wp/q(#) = 0 (the "vertical" curves in Fig. 6). These curves meet the boundary of .M at points where the torus has degenerated because the curve %/q has been pinched. The final piece of the relationship between the complex and geometric parameters is given by

T H E O R E M 4. There is a family, {lr(#)}~cR, of analytic maps from Ad to C that vary continuously with r. The value l~ (#), for r rational and # E 7~, equals the appropriately normalized length of the pleating locus. The pairs (r, l~ (#)) define a new set of coordinates for 34. The level curves lr(#) = const are the "horizontal" curves in Figure 6. We have generalized these techniques to twicep u n c t u r e d tori and expect them to generalize to arbitrary surfaces [15, 17]. References 1. H. Poincar6, Papers on Fuchsian Functions, translated by J. Stillwell, New York: Springer-Verlag (1985). 2. R. Fricke and E Klein, Vorlesungen fiber die Theorie der automorphen Funktionen, New York: Johnson Reprint (1965), Vol. 2. 3. L. Ahlfors, Lectureson Quasiconformal Mappings, New York: Van Nostrand (1966). 4. L. Bers, Finite dimensional Teichmiiller spaces and generalizations, Bull. AMS (2) 5 (1972), 257-300. 5. W. Fenchel and J. Nielsen, Discrete groups, unpublished manuscript. 6. B. Maskit, Kleinian Groups, New York: Springer-Verlag (1987). 7. L. Keen, Canonical polygons for finitely generated Fuchsian groups, Acta Math. 115 (1966), 1-16. 8. L. Keen, Intrinsic moduli on Riemann surfaces, Ann. Math. 84(3) (1966), 404-420. 9. L. Keen, On Fricke moduli, Advances in the Theory of Riemann Surfaces, Princeton, NJ: Princeton University Press (1971), 205-224. 10. W. E Thurston, The geometry and topology of threemanifolds, unpublished manuscript. 11. E Gardiner, Teichm~iller Theory and Quadratic Differentials, New York: Wiley (1987). 12. D. J. Wright, The shape of the boundary of Maskit's embedding of the Teichmfiller space of once punctured tori, preprint. 13. L. Keen and C. Series, Pleating coordinates for the Maskit embedding of the TeicmiJller space of punctured tori, Topology 32(4) (1993), 719-749. 14. L. Keen and C. Series, Pleating coordinates for the Teichmffiler space of punctured tori, Bull. AMS. (2) 26 (1992), 141-146. 15. L. Keen and C. Series, The Riley slice of Schottky space (to appear). 16. L. Keen, B. Maskit, and C. Series, Geometric finiteness and uniqueness for Kleinian groups with circle-packing limit sets, J. Reine angew. Math. 436 (1993), 209-219. 17. L. Keen, J. Parker, and C. Series, Pleating coordinates for the twice-punctured toms. In preparation. 18. L. Greenberg, Fundamental polyhedra for Kleinian groups. Ann. Math. 84(2) (1966), 433-441. 19. A. Marden, The geometry of finitely generated Kleinian groups, Ann. Math. 99 (1974), 383-462. 20. C. Series, The geometry of Markoff numbers, Math. Intelligencer 7(3) (1985), 20-29.

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What Is the Philosophy of Mathematics, and What Should It Be?* Bonnie Gold

"Oh, n o - - h o w tedious. Another article on the philosop h y of mathematics. I think I'll skip it: some weird axiom system, or a bizarre model of set t h e o r y - - j u s t tell me if it has some applications to m y work." WAIT! That's exactly the purpose of this article: to discuss what there is to think about in the philosophy of mathematics other than foundations, and to convince you that some of the questions are of at least passing interest to you. As Stewart Shapiro [1] says, the philosophy of a subject should be useful to the practitioner by giving an orientation to "his work by providing a clear account of what he is trying to accomplish and h o w his practice contributes to this." However, in the first part of this century, urgent foundational questions diverted virtually all the work in the philosophy of mathematics into foundations, and most of those writing on nonfoundational questions in the philosophy of mathematics in the second half of the century (there are a few notable exceptions) have some mathematical training but are not mathematicians. Thus, very little of w h a t has been written has been of much interest to mathematicians. Philosophers have been discussing with m u c h fervor questions (such as "What is a number?") which strike mathematicians as silly and irrelevant; and the philosophical questions mathematicians would like to learn more about are rarely addressed. During the last 2 years I have been reading books and articles both by philosophers of mathematics and by mathematicians writing in a philosophical vein, motivated by the question, "What is mathematics?" (see IB). While reading, I have been noting what questions are discussed and also w h a t other questions seem to me to be slighted.

Let me offer my list of questions. I have not seen a similar list in the literature in a reasonably compact form (The Mathematical Experience [2] asks m a n y of them, but it takes 460 pages to do so). By making more mathematicians and philosophers aware of some questions which still need addressing, I hope to encourage work on these questions. I will indicate at the end which of these questions have been extensively worked on, and which have not. I have collected an extensive bibliography of articles on these topics, which I would be happy to send to anyone interested, but it's too long to include here. I have not classified these questions in the traditional philosophical categories of epistemological, ontological, moral, and so on because m a n y of them require consideration of several of these aspects to answer adequately. I must warn you that, as with m a n y mathematicians, I am at heart a platonist. I don't k n o w whether we invent or

*This articlewas begun while the author was on sabbatical at the Universit4 Catholique de Louvain and was partially supported by Lilly FacultyOpen Fellowshipgrant number 910043.A preliminary version w a s presented in Belgiumto the logic seminar,and the author thanks the participantsin that seminar,especiallyThierry Lucas,as well as Jan Denef for their suggestions. 20

THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 3 (~)1994Springer-VerlagNew York

discover mathematics, but I do believe that, at least once a mathematical object has been invented/discovered, it is independent of us: It is objective. This belief colors what I say, but I have also tried to list questions philosophers are currently asking which come from points of view different from, and often antagonistic to, this platonistic position.

I. General Philosophical Questions m The Nature of Mathematics A. What is the nature of mathematical objects? For example, the numbers 2, 45, 7r, and i; the set of integers, of real numbers; the group $5; the idea of a topological space; the exact sequence 0 ~ 2E -* G --* G2 ---+ 0 - do they belong to the physical world, to the mind of each individual mathematician, to the communal mind (whatever that might be) of the community of mathematicians, to a nonmaterial realm? After all, although mathematical objects seem very concrete to mathematicians, they seem unlike tables and mountains in that we cannot see them with our eyes or smell them or hit them with our hands. Are they sets, are they fictions which don't exist but are nonetheless useful? (If they belong to the physical world, how does one resolve the paradox that there seem to be infinitely many mathematical objects but perhaps only finitely many physical objects?) B. What is common to those subjects (e.g., algebra, analysis, topology, geometry, combinatorics, category theory) that are classified as mathematics which causes us to classify them, but not other subjects, as mathematics? C. Are there mathematical truths? If so, are they facts about the physical world, are they simply definitions, are they nothing but theorems of logic, "if-then" statements? Are they simply conventions? Are they a priori or a posteriori, analytic or synthetic, falsifiable or independent of the world of experience? D. Is there such a thing as mathematical knowledge, and if there is, how do we acquire it? If the objects of mathematics belong to a nonphysical world, how is it possible for us to make contact with this other realm, let alone have knowledge of it? 1. Is mathematical knowledge more sure than other forms of knowledge? It seems so (it was, over many centuries, taken as the one form of absolute knowledge), but is that simply an illusion? 2. Do there exist any absolute limits to mathematical knowledge? Are there theorems whose proofs are so long or so difficult that we will never be able to discover or comprehend them? Are there mathematical structures which are too subtle for'people to comprehend? E. What is the relationship between mathematics and the physical world? Certainly the sciences, and, more and more, the social sciences, use mathematics, and increasingly complicated and apparently abstract mathematics. If mathematics is just a fiction, or if it is only

in the minds of mathematicians, why is mathematics so useful in the world? Do the little doodlings and notes we make when we are thinking about mathematics somehow help bridge the gap between us, physical beings, and the mathematical realm? How? In another direction, what role do the questions which come from users of mathematics play in the development of mathematics? Describe the dialectic between theory and application. F. Does mathematics belong with the sciences or with the humanities? It resembles the humanities in not needing complicated instruments or (physical) tools, usually not even computers. It resembles the sciences in that a mathematical statement is either true or false, and there is a parallel with scientific experiments when we use examples in the process of trying to make an idea or a conjecture precise, or in trying to find a proof.

II. Formal Methodology: Proofs A. How do we know the fundamental facts of mathematics, such as that 2 + 3 = 5, that 321 + 123 = 444, that the union of two sets forms a set, the axioms of logic? 1. How do we initially discover them? If they come from experience, how do we get from the fact that two apples plus three apples make five apples to the fact that 2 + 3 = 5 for all conceivable cases? Could experiences in the future conceivably falsify these facts? If, on the other hand, they come from our intuition, what is this intuition? How does it work? Or, looking at a third possibility, if these are just definitions, w h y do they seem to be verifiable facts? 2. Is this method of initial discovery of mathematical facts sufficient to give us knowledge? a. If so, does our knowledge come about (i) in a manner similar to our knowledge of physical facts or (ii) in a different manner, and then why do we think it is knowledge? b. If our method of initial discovery is not adequate to give knowledge, why not, and when and how does it become knowledge? B. How does mathematical proof confer knowledge? Thanks to the logicians, we now know how to determine when something is really a proof, but if proof is a way to obtain knowledge, it's a completely different way from the one for knowledge of physical objects. And if a proof is so long that it is not surveyable, or can't all be kept in the mind at once, does it still confer knowledge? Hardy [3] said that proofs are just the gas of mathematics, they don't give knowledge. It's perceiving the truth that gives knowledge, and the purpose of a proof is simply to lead us to this perception, as if someone is looking at a series of mountain peaks and tries to explain to someone else how to find a particular peak. He points out a peak that the other can clearly see, and then the route to follow to see the peak he is trying to point out. From this perspective, rigor is of no importance; only understanding the ideas counts as knowledge. THEMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994 21

C. What is the importance of definitions and of symbols for mathematics? From the point of view of logic, they have no importance; one can always eliminate them. All they do is artificially make proofs shorter. But it appears that we as human mathematicians really do need them; one can even say that many good theorems had to wait for the proper notation. Wilder [4] has suggested, for example, that if the ancient Greeks had had our algebraic notation, they might have invented analytic geometry or even calculus. Can we go on indefinitely inventing better notations or definitions handling arbitrarily intricate concepts? Certainly we already have mathematical proofs which, if they were written out in the predicate calculus without any abbreviations, would be too long for anyone to read in his lifetime. Is there any limit on how far one could overcome these limitations of length or complication of proof? D. What is the importance of abbreviations, such as lemmas, in proofs? This may not be exactly the same question as C: Lemmas are simply a formal device, whereas definitions involve new ideas which may not be completely formalizable and, thus, may introduce genuinely new proofs. E. Mathematical errors. 1. Analyze the role of changes in the level of rigor expected in mathematical communication. If one doesn't know a mathematical fact until one has a rigorous proof, then the ancient mathematicians knew nothing. That's clearly absurd, but, on the other hand, the gaps which Hilbert found in Euclid have some meaning. Are we safe from future gaps in our logic, perhaps even some which we unwittingly would program our computers not to see? 2. Are errors in mathematics different from mistakes in other disciplines (sciences, humanities)? Are errors rarer in mathematics? Do they have the same effects? Do they come about in the same way? If a result (of sufficient interest that it has been read by several people) remains unchallenged in the mathematical literature for 20 or 30 years, one can be effectively certain that it is true. Is this different from the other sciences? Could we in principle (say, by writing our papers in the predicate calculus) eliminate errors, or, at least, some sorts of errors? 3. Why does it so often occur that someone gives a false proof, but of a theorem which in fact is true and is later p r o v e d - - o r is later made provable by adding a hypothesis? This seems inexplicable from any formalist standpoint. However, if one uses the idea of mathematical intuition to explain it, how does this intuition work? 4. Does mathematics normally proceed, as Lakatos [5] claims, by a process of proofs and refutations? Lakatos gives the example of the development of Euler's formula, but are there many other such examples? As one tries to discover mathematics for oneself, some such process seems to be at work, a parallel to the scientific method: observing some examples, making a conjecture to ex22

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

plain those examples, testing this conjecture on other examples, trying to develop a proof, and where this runs into a wall, looking for counterexamples (refutations) and for modified conjectures. 5. Are the facts of mathematics eternal, unchanging? Did they exist prior to their discovery by mathematicians? Even if not, having been discovered, will they remain true, or will we perhaps later find that they were completely wrong? In one sense, when we look at the differences in rigor between the time of Newton and Leibnitz and our own time, there have been vast changes. But in another sense, certainly the facts of mathematics endure longer than those of any other subject: The mathematical discoveries of the ancient Greeks are still true today. These two senses need to be made more precise.

III. I n f o r m a l

Methodology

A. Are there routes to mathematical knowledge besides that of proof? 1. Describe the relationship between mathematics as it is written down and mathematics (mathematical ideas) itself. Of course, from a strictly formalist point of view, these are the same, but to us actually doing mathematics they are not. For example, what is the role of heuristic arguments in mathematics? As no one actually writes mathematical proofs in the predicate calculus, one could say that all our proofs are heuristic, but mathematicians mean something else by a heuristic proof; the notion and its uses need to be better understood. Are there things that heuristic arguments do that proofs can't; for example, to motivate or aid in the finding of a real proof, or to expose the deep idea hidden in a technical proof? Or does the latter always mean that one hasn't yet found the "true proof"? 2. Analyze the various forms of mathematical intuition. It seems to me that there are different sorts, but that when philosophers talk about mathematical intuition, they aren't aware of the distinction, leading to confusions. The most elementary is our perception of mathematical objects, for example, numbers, groups, theories. When we speak of 7 or Z or $5, something appears in our heads: for some of us, pictures, for others, something more difficult to describe, but in any case, something very concrete, which enables us to answer questions we have not thought of before, as well as ones we have thought about. When we think of the set I~ of positive integers, of which one cannot give a complete first-order recursive axiomatization, do we all have the same underlying idea? How could we tell? Next is what mathematicians call mathematical intuition, that which gives us the ability to make true conjectures before finding their proofs. This is more than just perception because one must also have an idea of the relations between mathematical objects to make conjectures.

Children develop the first kind of intuition for numbers before or soon after they start school, but this second type of intuition is not always present in undergraduate majors. Perhaps there is yet a third sort of mathematical intuition? B. Explain the processes by which mathematics develops. It does not develop in the linear fashion of proofs. As with other sciences, one can try to understand that development and the forces which help or slow it. 1. What are the motivations which give mathematical research its direction? What is it in mathematics that interests mathematicians so strongly? a. Where do mathematical questions come from? What makes a question a good question? For example, for some, the motivation comes in trying to find the deep reasons behind surprising connections. b. Where do new mathematical subjects come from, and how? Some come from outside mathematics, some from mathematics itself, but what is the process? 2. What is the role of proof in mathematical discovery? Attempting to find a proof of a conjecture helps one understand better the ideas involved, and often leads to possible counterexamples, which then help in the revision of the conjecture. But I think there is much more to say about the role of proof in the development of informal mathematics. Why is the dialectic of examples/theorems (proofs) (see IIE4) so pervasive in mathematical discovery? 3. When a new mathematical idea begins to develop, how does it become precise? 4. What role does esthetics play in mathematical development? Is it a sense which helps one to find new mathematical ideas, or which enables one to judge if one has found the true solution, or is it simply something which allows one to appreciate ideas after the work is done? Or perhaps it's merely a fiction of mathematicians. What is meant by "good" mathematics, by "A's work is better than B's," by "this is a better proof of the theorem than that"? Are such judgments purely subjective? Why in mathematics are there fewer problems making such judgments, and achieving a high degree of community agreement, than in other subjects (philosophy or sociology, for example, where it seems one group regards X's work as of no value, whereas another views it as the most important of the century)? 5. To what extent is the development of mathematics a matter of free choice, to what extent is it necessary? Is mathematical discovery like discovering new landscape, where one can choose which direction to go, but not what one will see there? Or is it more like creating a sculpture, completely free except for the limits imposed by the material with which one is working? Peter Hilton once said that when he worked, it seemed to him a free creation, but when he read the works of others, it seemed they had made necessary discoveries. If it is free, why does one have the frequent event of simultaneous discovery? If it is necessary, w h y does one have the sense of being

free? 6. Does the surrounding culture have an effect on (a) what mathematics is discovered, (b) what the answer to a mathematical question turns out to be, and (c) the way mathematics is written? It seems to me that the answer to (c) is definitely yes, to (b) definitely no, and to (a) probably; but if so, how does it have this effect. For example, are there national differences in mathematics? If so, why is it so easy to translate mathematics from one language to another, unambiguously? Is there such a thing as "women's mathematics," and so on? 7. What is the role of generalization in mathematical development? On the one hand, one can often find a good explanation of a phenomenon by generalizing; on the other hand, not all generalizations are fruitful. Can one describe what sorts of generalizations have been successful, and how? P61ya has written extensively on part of this: inductive generalization from examples in the process of mathematical discovery. 8. What is the significance of the relationships between different mathematical subjects? Why does one find these relations between subjects with such different origins? C. Why do we find results which astonish us in a subject which it appears we make up? For example: (1) Resuits which don't seem to have been present in our definitions or axioms, but which follow from them, such as the fact that a finite division ring is, in fact, a field; (2) results which link subjects which seem to be very different and distant from one another, as complex analysis with all its continuity and differentiability yields results in number theory, which is so discrete. D. The sociology of mathematics 1. Why is collaboration among mathematicians so often effective? One thinks of Erd6s, but there are many other examples. Why is collaboration sometimes so difficult? 2. Why does one so often find simultaneous discoveries of the most important theorems or theories, often by people completely isolated from each other, often in different countries? Does this simultaneity imply some sort of necessity in the development of mathematics, at least under certain circumstances? This is certainly not a complete list of questions. When one starts to work on a subject, one always finds new questions. I would certainly be happy to hear from those who have questions to add. As I mentioned at the beginning, some of these questions have been worked on a lot, others hardly at all. (When a question has many subquestions, these usually either appear in the literature or occurred to me as I read a discussion of the question in the literature.) Due to various "crises" (the discovery of nonEuclidean geometries and the gaps in Euclid's geometry, the problems with the foundations of calculus, the paradoxes of set theory), the philosophy of mathematics in the THEMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994 23

first half of this century was preoccupied with founda- References tionalist schools, intuitionism, logicism, and formalism, and by careful attention to logic itself, to the exclusion 1. Stewart Shapiro, Mathematics and reality, Phil. Sci. 50 (1983), 525. of other sorts of philosophical questions. (See Ref. 6 for 2. Philip Davis and Reuben Hersh, The Mathematical Experian introduction to these schools as well as some articles ence, Boston: Birkh/iuser (1981). at the base of the current discussion by philosophers of 3. G. H. Hardy, Mathematical proof, Mind 38 (1929), 18. mathematics.) Then it came to seem that none of these 4. R. L. Wilder, The origin and growth of mathematical confoundationalist schools could provide the philosophical cepts, Bull. AMS 59 (1953), 428. 5. Imre Lakatos, Proofs and Refutations (J. Worral and E. Zahar, justification of mathematics for which they aimed. The eds.), Cambridge: Cambridge University Press (1976). questions which they had been developed to answer (the 6. Paul Benacerraf and Hilary Putnam (eds.), Philosophy of status of mathematical objects, truth and k n o w l e d g e - Mathematics: SelectedReadings, 2nd ed., Minneapolis: Cammathematical ontology and epistemology) were asked bridge University Press (1983). again, and variants of some of these schools, as well as 7. William Aspray and Philip Kitcher (eds.), History and Philosophy of Modern Mathematics, Milwaukee: University of of the platonism which preceded them, have been atMinnesota Press (1988). tempted. Among those attracting attention currently are 8. A. D. Irvine (ed.), Physicalism in Mathematics, Boston: versions of structuralism, which considers mathematics Kluwer (1990). to be the study of abstract structure, and attempts to 9. Thomas Tymoczko (ed.), New Directions in the Philosophy of deal with the problem of what mathematical objects are Mathematics, Boston: Birkh/iuser (1986). by eliminating the need for them; and a version of con- 10. Saunders Mac Lane, Mathematical logic is neither foundation nor philosophy, Philosophia Mathematica (2) 1 (1986). ventionalism which attempts to incorporate the fact that 11. Reuben Hersh, Some proposals for reviving the philosomathematics changes over time, by viewing mathematphy of mathematics, Adv. Math. 31 (1979), 31-50. ical truth as simply the communal agreement of mathematicians. Thus, among the questions which have received attention by philosophers in the last 30 years are Department of Mathematics and Computer Science questions IACD, IIA, and a bit of IIE; many (leaving out Wabash College the "philosophers of mathematics" who are simply logi- P.O. Box 352 Crawfordsville, IN 47933-0352 USA cians) write basically only about these questions. Which other questions have received any philosophical consideration? Several philosophers have directed their attention to the relationship between mathematics and the physical world (IE). Several authors, including some mathematicians, have discussed whether there are alternatives to proofs which yield mathematical knowledge (IliA). Several mathematicians and a few philosophers are interested in the development of mathematics (IIIB). References 7-9 are collections which give a good introduction to issues currently being considered in the philosophy of mathematics; all three have articles by both mathematicians and philosophers. The questions which interest mathematicians and those which interest philosophers, as I've mentioned, are somewhat disjoint. But when mathematicians write on philosophical questions, they aren't as careful as when they are doing their own mathematical research: They have a tendency to simply state that this is how it is, without much effort at justification, in the tone of an after-dinner conversation. In any case, I've found almost nothing written about questions IB, ID2, IF, IIB-IID, IIIC, IIID, and much work still needs to be done on IE, IIE, IIIA, and IIIB. I'm certainly not the only one to urge a renewed interest in the philosophy of mathematics. Both Saunders Mac Lane [10] and Reuben Hersh [11] have written articles which encourage further thought about it. I hope this list has included at least some questions of interest, and that some of you" will consider looking seriously at the philosophy of mathematics. 24

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

Symmetrical Combinations of Three or Four Hollow Triangles H. S. M. Coxeter

A hollow triangle is the planar region bounded by two homothetic and concentric equilateral triangles, that is, a flat triangular "ring." The edges of the inner triangle are half as long as those of the outer triangle. We will look at two striking constructions in 3-space made from such figures. The Australian sculptor John Robinson assembled three such triangular rings to form a structure (entitled intuition) in which certain points on two outer edges of each ring fit into two inner corners of the next, in cyclic order (see Figure 6). The topology of the assembly is that of the "Borromean rings," and its symmetry group is C3, cyclically permuting the three hollow triangles. Quite independently, the American artist George Odom assembled four such triangular rings to form a rigid structure in which the midpoints of the three outer edges of each ring fit into inner corners of the three remaining rings. The four rings are mutually interlocked, and the symmetry group is the octahedral group O or $4 : 24 rotations permuting the 4 hollow triangles in all the 4! possible ways. The positions of the 24 vertices of the 4 pairs of homothetic equilateral triangles are most naturally indicated by means of 4-dimensional Cartesian coordinates involving the integers

reciprocal tetrahedra (Klein's "tetrahedron and countertetrahedron') whose 4 + 4 vertices coincide with the 8 vertices of the cube. In the case of the icosahedral group A5, the five objects which he evenly permuted were the five triads of mutually perpendicular lines (like Cartesian coordinate axes) which join the midpoints of pairs of opposite edges of the icosahedron (or of its reciprocal, the pentagonal dodecahedron {5, 3}); (compare [3], pp. 273-275). By holding up a model of the icosahedron, with two opposite edges in one's two hands, one soon recognizes these edges as belonging to a golden rectangle whose two

0, 4-1, +2.

Felix Klein's Polyhedral Groups In the opening chapter of his beautiful Lectures on the Icosahedron [5], pp. 14-20, Klein identified the rotation groups of the regular tetrahedron {3, 3}, octahedron {3, 4}, and icosahedron {3, 5}, with the alternating and symmetric groups A4, $4, As. For the tetrahedral and octahedral groups, the four objects which he permuted were the four diameters ("diagonals") that join pairs of opposite vertices of the cube {4, 3}, or of Kepler's stella octangula consisting of two THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3 @1994 Springer-Verlag New York

25

square so that the two congruent diagonals A B and B C are perpendicular. The angle 2r = BOC, between the diagonals of the rotated rectangle, is given by tan 2r = BC/ 89A B = 2. Thus, the acute angle between the diagonals of a golden rectangle is arctan 2. This perspicuous proof of a familiar result was kindly supplied by Jan van de Craats. In ,terms of the icosahedron, 2r is the angle between two adjacent diameters ([3], p. 156). These diameters of the icosahedron are perpendicular to two planes containing adjacent faces of the reciprocal dodecahedron, providing an easy proof for a still more familiar fact: The

dihedral angle of the dodecahedron is ~r - arctan 2.

M a x Briickner's F i v e Tetrahedra The 4 triangles are permuted by 4! rotations (see cover). longer sides are diagonals of pentagons. In such a rectangle [6], p. 12, the ratio of lengths of the sides (longer to shorter) is the "golden ratio" 7 = 2 cos(Tr/5) = 1 + 1/r = 1 + 1/1 + 1/1 + . . . 1.6180339887 [3], pp. 140-143. The convention used here is that each solidus dominates whatever follows: 1/1 + 1 means 1/2. This convention facilitates the printing of continued fractions. The equation T = 1 + 1 / r shows that a golden rectangle with sides r and 1 can be dissected into a square with side 1 and a smaller golden rectangle with sides 1 and T-1. In Figure 1, the smaller rectangle with diagonal B C has been rotated by a quarter-turn and placed inside the

For the icosahedral group (5, 3, 2) ~ As,

(1)

Klein's five Cartesian frames may conveniently be replaced by five tetrahedra whose 5 x 4 vertices coincide with the 20 vertices of the dodecahedron {5, 3} ([2], Plate IX, Figure 11). A simple way to see how this compound arises is to observe the square that lies behind any edge of the dodecahedron. This square and the antipodal square are two opposite faces of a cube whose eight vertices belong to the dodecahedron. Either of the two tetrahedra inscribed in the cube can serve as one of the five tetrahedra inscribed in the dodecahedron; the other four can be obtained by rotation about any pentagonal axis. Because the 5 x 4 faces of the 5 tetrahedra lie in the 20 face-planes of an icosahedron, this compound is one of the 58 stellations of the icosahedron, namely, Ell or E l l

Figure 1. The diagonals of golden rectangles.

J,

C ! !

A

,r-t

t 26

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

B

"r-'

[4], pp. 5, 25-26 and Plate XVI. A solid model, with five colours for the five tetrahedra, provides a perspicuous demonstration of the isomorphism (1): Each rotation of the model yields an even permutation of the five colours. The problem of stellating the icosahedron inspired George Odom's discovery of a new construction for the golden ratio ([1], p. 23; [7]): Let A and B be the midpoints of the sides E F and E D of an equilateral triangle D E F . Extend A B to meet the circumcircle (of D E F) at C. Then B divides A C according to the golden section.

Symmetrical Lattices When circular discs are closely packed in the Euclidean plane, so that each is surrounded by six others, their lattice of centres is most conveniently coordinatized by working in space and taking the plane to have the equation

011

110

Xl q - X 2 A - X 3 = 0.

For then, if the discs have diameter v~, the centres are simply all the points (xl, x2, x3) whose coordinates are integers satisfying that equation. In particular, the origin (0, 0, 0) is surrounded by a regular hexagon given by the six permutations of (1, 0, -1). In Figure 2, for convenience, (1, 0, - 1 ) is contracted to 101. Note that when the vertices of the hexagon are interpreted as vectors, each is the sum of its two neighbours. This fact is not surprising when we recall that the set of vectors belonging to any lattice is closed with respect to the operation of subtraction. Analogously in 4-dimensional space, the points (Xl, X2, X3, X4) whose coordinates are integers satisfying the equation X 1 ~'- X 2 + X3 + X4 = 0 (2)

Figure 2. A regular hexagon.

o ,oi i10ff'

X2 = 0 ,

X3=0,

001i /

/

f

Tolo

form the lattice of centres of congruent balls (of diameter v'2) in their most symmetrical close-packing. N o w the centres nearest to the origin (0, 0, 0, 0) are the 12 vertices of a cuboctahedron, given by the 12 permutations of (1, 0, 0, -1) or 1001. As we see in Figure 3, its edge-length is v~. The four planes Xl = 0 ,

100i

X4=0

(by which we mean "sections of four hyperplanes by the special hyperplane (2),") each contain a 2-dimensional lattice of the kind we considered earlier.

George Odom's Four Hollow Triangles Doubling the coordinates, we obtain the 12 vertices of a larger cuboctahedrori: the permutations of (2, 0, 0, -2) or 2002. The 24 vertices of the two homothetic cuboctahedra can be differently joined by new edges (12 of length v ~ and 12 of length 2v~) so as to form 4 pairs of homothetic triangles, as in Figure 4. The 12 vertices of the four

Figure 3. A cuboctahedron. inner triangles are easily seen to coincide with the midpoints of the 12 sides of the 4 outer triangles. It follows that when the space between each pair of homothetic triangles is filled in with a cardboard (or steel or wooden) lamina, the four interlocked "hollow triangles" form a rigid structure. Such a model, with four colours for the four hollow triangles, was made by George Odom as a piece of abstract sculpture (see the cover). THEMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994 27

zioo

2_0~0

0i"01

00~_ ~

072 0

O~O2

o

o~o i

~a~o

~eo2

Figure 4. Coordinates for the four hollow triangles. As the outer vertices are given by the permutations of (2, 0, 0, -2), the whole sculpture can be enclosed by the cube of edge 4 whose faces lie in the three pairs of parallel hyperplanes Xv + X 4

=

4-2 (v = 1,2,3).

the model in the box provides a different permutation of the four hollow triangles, that is, of the four colours. The sculpture thus yields a strikingly perspicuous demonstration of the isomorphism (4, 3, 2) _~ 84.

For instance, the two parallel planes Xl + X2 =

:F2,

X3 + X4 =

+2

contain the squares (-2, 0, 2, 0) ( - 2 , 0 , 0 , 2 )

(0,-2,0,2)

(0,-2,2,0)

and (0, 2, 0, -2) (0, 2, -2, 0) (2, 0, -2, 0) (2, 0, 0, -2), whose distance apart is 4. In other words, a model will fit neatly into a cubical box of edge 4. Because of its symmetry, it can be placed in the box in 24 ways, 1 for each element of the octahedral group (4, 3, 2), which is the rotation group of the cube {4, 3} (and of the octahedron {3, 4} and of the cuboctahedron {34} )" Each way of placing 28

THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 3, 1994

John R o b i n s o n ' s Three H o l l o w Triangles It is interesting to compare Figures 5 and 6, bearing in mind that Odom (in the United States) and Robinson (in England) had no communication, though both used hollow triangles of almost exactly the same shape. The differences are as striking as the resemblances. Every two of Odom's four hollow triangles are interlocked, and their assembly is inherently rigid. No two of Robinson's three hollow triangles are interlocked, but paradoxically the whole assembly is inseparable in the manner of "Borromean rings" ([8], pp. 66, 67, 119). When a model is made, using thin laminae of cardboard, and the lowest vertices of the three outer triangles are allowed to slide freely on a tabletop, the as-

sembly collapses under its own weight so as to form the planar pattern of Figure 7. The blue hollow triangle lies over the yellow, the yellow over the red, and the red over the blue. Conversely, when the outermost edges are lifted while trigonal symmetry is preserved, a complicated twisting motion takes place. This continues smoothly for a while and then stops abruptly. It remains true that certain points on two outer edges of each ring fit "into two inner corners of the next, in cyclic order," but these "certain points" slide along those "outer edges" and do not remain midpoints. Robinson fixed his rings at the stage when the lifting motion abruptly stops.

At Marjorie Senechal's "Regional Geometry Institute" in Smith College (funded by NSF), a workshop organized by Doris Schattschneider discovered that the "twisting motion" is an illusion, caused by the modelling material's flexibility. In fact, Robinson's hollow triangles are slightly narrower than Odom's: If the inner equilateral triangle has side 1, the side of the outer triangle is not 2 but (2v~ + 1)/3 ~ 1.9663265. The details are as follows. When the Borromean assembly in Figure 7 is turned over, it looks like Figure 8. An upside-down version of Robinson's sculpture is obtained by lifting the corner A of one hollow triangle A B C D E F and the corresponding

Figure 7. The collapsed "Intuition." Figure 5. Odom's sculpture.

Figure 6. Robinson's sculpture.

Figure 8. The other side of Figure 7.

THEMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994 29

it follows that A

t = 1+

R Z

2 sin fl(v~ cos fl + sin fl) cos fl + v ~ sin fl + 1

(4)

and v ~ cos fl + sin fl x = v~(cos fl + v ~ sin fl + 1)"

(5)

In the collapsed arrangement (Figs. 7 and 8) we have fl = 60 ~ yielding t=2,

x= 89

y=l,

z=l.

In John Robinson's sculpture itself, Elizabeth Whitcomb observed that D Q appears to be one edge of a regular tetrahedron with centre O, so that the angle Q O D is arcsec(-3) and fl = arcsec 3 ~ 70o31'44 ''. Setting cos fl = 89and sin fl = 2 v ~ in Eqs. (3)-(5) we obtain

Figure 9. The intersection P Q of two hollow triangles.

1

t = 3 ( 2 v ~ + 1) ~ 1.9663265, corners (one initially at E) of the other two. The point (9 on the inner side D F of the first hollow triangle is seen to coincide with analogous points on the other two hollow triangles. The trigonal symmetry ensures that the OQ and OD of Figure 9, being two of three congruent segments, have the same length, say x. P Q is an inner edge of the third hollow triangle. If P Q = D E = 1, the length of the outer edge A B = t can vary from 2 (in the collapsed position) to a slightly smaller value. Extensions of E D and F D form, with A, a rhomb of side (t - 1)/3, whereas OD and OQ span another rhomb, of side x. Let fl denote its angle at Q and D. In terms of fl and t, we can calculate O D = x, A Q = y, and A P = z by observing that

Y-

AQ PQ-

RO PO-

x + (t - 1)/3 1-x

whereas, from the same triangle A P Q , y sin(120 ~ - fl)

z sin fl

1 sin 60 ~

Thus, y = cos

+

2 sin fl z- - -

sin fl

and, since (by similar triangles) t- 1

( t - 1)/3

x

y - ( t - 1)/3

2v/3 sin fl

z

1

y+ 1

cos fl + (1/v~) sin fl - (t - 1)/3 cos fl + (1/v~) sin fl + 1

30

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

(3)

x -

1

v~

2

12

1

2~-6

~ 0.2958809,

y = -~ + - -

z -

9

~ 0.8776780,

~ 1.0886893,

in good agreement with Robinson's artistic "Intuition."

References 1. Albert Beutelspacher, Der Goldene Schnitt, Wissenschaftsver-

lag, Mannheim, 1988. 2. Max Brfickner, Vielecke und Vielflache, Leipzig, 1900. 3. H. S. M. Coxeter, Introduction to Geometry (2nd ed.), Wiley, New York, 1969. 4. H. S. M. Coxeter, Patrick Du Val, H. T. Flather, and J. E Petrie, The Fifty-nine Icosahedra (2nd ed.), Springer-Verlag, New York, 1982. 5. Felix Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree (2nd ed.), Kegan Paul, Trench Trfibner, London, 1913. 6. George Markowsky, Misconceptions about the Golden Ratio, College Math. J. 23 (1992), 2-19. 7. George Odom and Jan van de Craats, Problem E 3007, Amer. Math. Monthly 90 (1983), 482. 8. John Robinson, Symbolic Sculpture, Edition Limit6e, Carouge-Geneva, 1992.

Department of Mathematics University of Toronto Toronto, Ontario, M5S 1A1 Canada

David Gale* For the general philosophy of this section see Vol. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.

When I first was given the job of editing this column I tried to define entertainment in a way that would allow publication of just about anything mathematical that I thought would appeal to the readers. I wrote at that time, "The term entertainment is sufficiently vague to allow a wide variety of material, provided only that it should not require technical expertise in any particular area." In this issue, the notion of entertainment is being stretched to a greater extent than in any of the previous columns to accommodate an exposition by Hugh Woodin of some very recent work related to the Continuum Hypothesis. As will be evident, Woodin's write-up would have been suitable for a separate expository article, and in fact, the only reason it appears here is because it was at my urging that he agreed to write up this material. If readers end up being educated and edified as well as entertained, then so much the better.

Large Cardinal Axioms and Independence: The Continuum Problem Revisited W. Hugh Woodin How many angels can dance on the head of a pin? For many of us this is one of 'the first questions we encounter which cannot be solved on the basis of the axioms. * Column editor's address: Department of Mathematics, U n i v e r s i t y o f California, Berkeley, C A 9 4 7 2 0 U S A .

Write down some axioms to describe your car. These for example could be the content of the manual for your car. Very likely the color of your car cannot be deduced from your axioms. Even more likely, the number of times your car has been started is not deducible from your axioms. These are lowbrow examples of independence. G6del proved in the 1930s that for any nontrivial system of axioms that can be enumerated, there must exist propositions that cannot be settled on the basis of the axioms. This first became a practical concern 30 years later when Cohen showed that the Continuum Hypothesis of set theory cannot be proved from the axioms of set theory. We take for the axioms of set theory, the standard Zermelo-Fraenkel axioms together with the Axiom of Choice (ZFC). G6del had proved 20 years earlier that the Continuum Hypothesis cannot be refuted from these axioms. The Continuum Hypothesis is just as intractable given only the axioms of set theory as the c~)lor of your car is from the mechanical description of your car. G6del, anticipating that independence was inevitable, proposed before Cohen's work that large cardinal axioms might resolve questions in set theory that otherwise could not be solved. Unfortunately the method that Cohen discovered was so powerful that it could be used to show that even large cardinal axioms could not help to resolve the Continuum Hypothesis. This was first argued independently by Levy and Solovay shortly after Cohen's work. Is this the end of the matter? The answer now seems clearly no. In fact, it is only the beginning.

THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 3 ~)1994 Springer-Veflag New York

31

There are many formulations of the Continuum Hypothesis. Here is one. For every infinite set A C I~, either there is a bijection of A with H, the set of natural numbers, or there is a bijection of A with ~, the set of real numbers. Consider the following effective versions of the Continu u m Hypothesis. Recall that a set of real numbers is Borel if it can be generated from the collection of open intervals by the operations of taking countable unions and intersections. The projective sets are those generated from the Borel sets by closing the collection of Borel sets under the operations of taking continuous images and of taking complements. Thus, if A c I~ is a projective set, then so are I~\A and f(A), where f : I~ --, ~ is a continuous function. Every projective set can be uniquely specified by a Borel set and a finite number of continuous functions. The point of view here is that in the set-theoretical zoo, the projective sets are rather tame. A set P c ~ is a perfect set if it is a closed set with no isolated points. It is a standard fact that if P c II~ is a perfect set, then there is a bijection of P with I~. So counterexamples to the Continuum Hypothesis cannot be perfect sets. A slightly harder result is that a counterexample cannot be Borel. In fact, if A c I~ is an uncountable Borel set, then A contains a perfect subset. If A is an arbitrary set of reals and A contains a perfect set, then there is a bijection of A with I~. Here are two effective versions of the Continuum Hypothesis. 1. Suppose A c R is an uncountable projective set. There exists a bijection of A with I~. 2. Suppose A c ~ is an uncountable projective set. A contains a perfect set. The first version asserts there is no easy counterexample to the Continuum Hypothesis (regarding projective sets as easy to define). The second version is perhaps a little more subtle. It, in essence, asserts that there is no easy counterexample to the Continuum Hypothesis for a simple reason. Clearly the Continuum Hypothesis implies (1). It does not imply (2). As with the Continuum Hypothesis, these effective versions of the Continuum Hypothesis are also not solvable on the basis of the axioms of set theory. However, unlike the situation with the Continuum Hypothesis, these versions are provable if one assumes large cardinal axioms. So G6del was partly correct. In fact, the influence of large cardinals is perhaps more subtle and interesting than even G6del might have hoped. What are large cardinal axioms? A large cardinal axiom asserts the existence of a very large infinite set. There is no formal definition, but there is a long list of axioms generally regarded as large cardinal axioms. Here are two. A cardinal ~ is strongly inaccessible if (a) whenever X is a set and IXI < ~, then IYI < where Y is the set of all subsets of X, (b) no set of cardinality ~ is the union of fewer than many sets of cardinality less than ~. 32 THEMATHEMATICALINTELLIGENCERVOL.16,NO.3,1994

Zero is strongly inaccessible (the axioms are vacuously true for 0). The least infinite ordinal, w, is also strongly inaccessible. The axiom that asserts that there is an uncountable strongly inaccessible cardinal is our first example. Here is another example. A cardinal ~ is measurable if there exists a nonprincipal ultrafilter U on ~ such that U is ~-complete; i.e., the intersection of fewer than ~ many sets in U is also in U. Again a; is a measurable cardinal. The axiom which asserts the existence of an uncountable measurable cardinal is our second example of a large cardinal axiom. It is perhaps not immediately obvious that a measurable cardinal is, in fact, large in any sense. However, if ~ is an (uncountable) measurable cardinal, then ~ is strongly inaccessible and, further, there are necessarily many strongly inaccessible cardinals smaller than ~. It follows that the existence of a measurable cardinal implies the consistency of the existence of a strongly inaccessible cardinal. Therefore (and this is a weaker claim), if the theory Z F C + There is a measurable cardinal is consistent, then so is the theory Z F C + There is a strongly inaccessible cardinal. This suggests the following partial ordering on large cardinal axioms. Suppose r and r are large cardinal axioms: ~1 -~ ~2

if the consistency of Z F C + r implies that Z F C + r is consistent. Now it seems that if r ~ ~2 and ~)2 ~ ~)1, then for any cardinal ~ satisfying r there are cardinals 6 < which satisfy r This is certainly the case for strongly inaccessible cardinals and measurable cardinals. The striking empirical fact is that the partial ordering defined above is a total ordering of the large cardinal axioms and further that it is a wellordering. This has not been proved for all of the large cardinal axioms which have been proposed to date; however there is compelling evidence that this will eventually be accomplished. As a consequence it is very likely that all the current large cardinals form a wellordered hierarchy. Implicit in this discussion has been the assumption that the axioms for set theory are consistent. The same applies to large cardinal axioms. By the results of Cohen and G6del, the consistency of the Continuum Hypothesis and the consistency of the negation of the Continuum Hypothesis are each provable from the consistency of the axioms of set theory. However, any large cardinal axiom implies that the axioms for set theory are consistent, and so by G6del's second incompleteness theorem we cannot hope to prove

that large cardinal axioms are consistent, even if we assume that the axioms for set theory are consistent. This is an important point and illustrates an essential difference between large cardinal axioms and axioms like the Continuum Hypothesis. The belief in large cardinal axioms is necessarily one based on faith. The structure that has emerged in the study of large cardinals is persuasive evidence that some nontrivial initial segment of the axioms are consistent. Some large cardinal axioms have been proposed that turned out to be inconsistent. However, the inconsistencies have always turned out to be rather easy. Of course there is no a priori reason w h y an inconsistency could not be a deep and difficult result, obtained only after a rich structure theory has been developed, like the "contradiction" in a M6bius strip. Related to the large cardinal hierarchy is a hierarchy of independence. One route to defining this hierarchy is to extend the partial ordering of large cardinal axioms to arbitrary propositions about sets. Thus, if r and r are propositions, we define r

_ r

if the consistency of Z F C + r implies the consistency of Z F C + r This is the hierarchy of consistency strength. It is believed, and this has been a very successful point of view, that the large cardinal axioms are cofinal in this partial ordering. But much more is empirically true: Given a proposition r either the theories Z F C + r and Z F C are equivalent or there is a large cardinal axiom r such that r162

and 0-- 1, n2i-1 = 7(nl,..., n2i-2). The strategy 7- is a winning strategy for Player I if Player 1 wins every run of the game in which Player 1 plays according to T. Similarly a strategy r is a winning strategy for Player 2 if Player 2 wins every run of the game in which Player 2 plays according to r. Clearly, only one of the players can have a winning strategy. Using the Axiom of Choice one can construct a set A for which neither player has a winning strategy in the game GA. Note though that for many specific sets there is a winning strategy for one of the players. For example, if A = [0, 1/2], then the game GA is determined. Player I simply chooses 0 for the first move. This guarantees a win for Player 1 no matter how either player plays from that point on. So there is a trivial winning strategy for Player I in this game. The game GA is determined if there is a winning strategy for one of the players. Projective Determinacy is the assertion that the game GA is determined for all projective sets A. Under the hypothesis of Projective Determinacy a comprehensive theory of the projective sets has been developed [2]. For example, assuming Projective Determinacy, it follows that every uncountable projective set contains a perfect subset and so the sentence r is true. Projective Determinacy also implies that the projective sets are Lebesgue measurable. determining axioms. There is little a priori evidence that Projective DeterI briefly discuss these axioms which were originally minacy is a plausible axiom or even that it is a consisproposed by Mycielski and Steinhaus [3] over 30 years tent axiom. However, the theory that follows from the ago. Suppose A c IL We associate to the set A a game GA assumption of Projective Determinacy is so rich that, a as follows. There are two players. The players alternate posteriori, the axiom is both consistent and true. The leschoosing either 0 or I at each move, with Player I play- son here is an important one. Axioms need not be a priori ing first. After infinitely many moves, an infinite binary true. sequence At first glance, the axiom of Projective Determinacy seems to have little in common with large cardinal axioms, but over the last 10 years a unified theory of large ( n l ~ n 2 ~ . 9 9 , n k ~ 9 9 "1

34

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

cardinals and determinacy has been developed. In particular, there are large cardinal axioms which imply Projective Determinacy. Thus, assuming that large enough cardinals exist, every uncountable projective set contains a perfect subset. This is a remarkable effect of the existence of large cardinals on rather small sets. I mention that all known proofs of this are somewhat involved (there are approximately three different proofs at this point). Here is another lesson. Basic consequences of the axioms need not be straightforward. A remark. One consequence of the unification of large cardinals and determinacy is the following experimental fact: Any large cardinal axiom that implies that every uncountable projective set contains a perfect set in fact implies Projective Determinacy [1]. A second remark. For a wide class of propositions ~b, if the consistency of Z F C + ~b implies the consistency of Z F C + Projective Determinacy, then ~b actually implies Projective Determinacy. This has been verified in a number of nontrivial instances. There are obvious limitations; nevertheless, the phenomenon is a genuine one. There still remains the question of the Continuum Hypothesis. The experience with large cardinals clearly demonstrates that it is worthwhile to look for new axioms. It seems equally clear that it is reasonable to hope our understanding of sets will evolve to the point where it is possible to resolve the Continuum Hypothesis. There have been numerous results obtained over the last several years which support this point of view. It is known, for example, that in the context of large cardinals the Continuum Hypothesis "settles" all questions that are as complex as the Continuum Hypothesis. Here complexity is a formal notion derived from an analysis of quantifiers. Another example is a recent result which for the first time identifies a canonical model in which the Continuum Hypothesis is false. This model derives from considerations of large cardinals and determinacy.

References 1. D. Martin and J. Steel, Iteration trees. J. Amer. Math. Soc. 7 (1994), 1-75. 2. Y. N. Moschovakis. Descriptive Set Theory: Studies in Logic and the Foundations of Math. Vol. 100: Amsterdam: NorthHolland (1980). 3. J. Mycielski and H. Steinhaus. A mathematical axiom contradicting the axiom of choice, Bull. Pol. Acad. 10 (1962), 1-3. Department of Mathematics, U-C Berkeley

Theorems Everywhere (by the column editor) Shakespeare writes of finding "tongues in trees, books in running brooks, sermons in stones, and good in every-

thing." If he had been mathematically inclined he might also have found theorems somewhere too, perhaps in clouds. Indeed, I expect people who are "mathematically aware" often bump into theorems in unexpected places. As an example, I recently recalled a puzzle-game which used to appear, perhaps still does, in children's magazines. You are challenged to get, say, from SHIP to DOCK in the fewest possible "steps," as illustrated by the following possible solution. SHIP, SHOP, CHOP, COOP, COOK, COCK, DOCK. (The fact that this game involves four-letter words has suggested the possibility of more ribald versions, for people who have nothing better to do with their time.) Here then is the S H I P - D O C K THEOREM. In any solution of the problem, there must be a word at least two of whose letters are vowels. I am, of course, not proposing this as a challenging problem for the readers of this magazine, but it may be useful in other ways. As already mentioned, it is an example of a theorem that comes up in "everyday life." Further, the result could conceivably be considered to be applied mathematics, being useful in solving the ship--dock problem: it suggests starting in the middle with the double-vowel word rather than at either end. Also it provides another illustration (along with the bridges of K6nigsberg, Pappus's Theorem, etc.) for convincing people that mathematics is not the study of numbers, as many of them seem to believe. Most interesting to me, however, are possible pedagogical uses of the result. Obviously, no mathematical background is needed to understand the problem. In limited experiments I have tried stating the result and asking people to explain why it must be so (I try to avoid using the word "proof" which seems to induce an instant panic reaction in some people). The results have been varied, from (a) near hostility, "Look, I stopped taking tests when I got out of school," to (b) embarrassment, "Let's change the subject," to (c) a glimmer of light, "It's obvious because the vowels in Ship and Dock are in different positions" (you're on the right track), to (d) "Well, I can see it all right but I can't explain it" (where have I heard that before?), to (e) "It's because every word must contain at least one vowel" (good, you're almost home!), to (f) the complete argument, and, better still, (g) the pleasure from seeing how the logical pieces fall together. Of course it's (g) that keeps us going and makes the practice of mathematics such a rewarding profession. As teachers, part of our job is to try to move our students away from case (a) toward case (g). Unfortunately, so far as I know, no one has yet figured out how to do this. Perhaps the best we can do is to go on scanning the clouds for more theorems. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

35

The Teachings of Hua Loo-Keng: A Challenge Today? C. Schweigman and S. Zhang

This article has been written to draw the attention of the international community of scientists once more to the work and life of the great Chinese mathematician Hua Loo-Keng. He led the mathematical sciences in China through a period of great political turmoil, reoriented matematical research, and above all put a great effort into the popularization of mathematics. In mass campaigns he taught hundreds of thousands of Chinese workers how to apply simple mathematical methods in their daily work. This article focuses on Hua's teaching of mathematics to the masses rather than on his theoretical work. We believe that he merits further attention for a variety of reasons. First, his life and work illustrate the interaction between science and political development. It is a two-way interaction. His work on applied mathematics was influenced very much by social and political developments in the People's Republic of China, and at the same time his work influenced the political processes. The interaction between socio-political development and science is still an important issue, although the debate on this issue seems to have faded into the background. What determines the development of science and the agenda for research? The tremendous problems faced by humanity, such as degradation of environment or development problems, may require much more conscious choices of direction in scientific activities. The life and work of Hua Loo-Keng reveal the far-reaching consequence of the choices he made, choices which were not evident and remain debatable. A second theme is this. There exists a persistent prejudice among mathematicians that only pure mathematics requires the highest level of intellectual creativity, and that application of mathematics in practice is of a lower standard, not a challenge for real scientists. Hua's work on the application of mathematics in the real world displays much creativity and intellectual power. Last but not least, we have tried to illustrate what type of applications Hua dealt with during the mass campaigns, and to discuss critically the merits of his approach. We start with a survey of Hua's life and work in the context of China's political developments. In "Some Nec36

essary Conditions" we try to explain why Hua Loo-Keng was so successful in his mass campaigns. The following section illustrates Hua's teaching during the mass campaigns. We end with an evaluation of the lessons that can be drawn from his life and work, in both industrialized and developing countries. Here use will be made of one of the authors' experiences in Africa.

Life and Work A good deal of attention has been given to the political developments in China up to 1972 and to Hua's role in them by Salaff [16]. Moreover, many bibliographic notes about Hua's life have been published (e.g., Halberstam

THE MATHEMATICAL INTELL[GENCER VOL. 16, NO. 3 (D 1994 Springer-Verlag New York

[5], Hu [6], Hua and Wang [12[, Li, et al. [13], Wang, et al. ent from the previous one at his home school. He was a secretary-librarian. There are many stories about his re[22], Wang [24], Zeng [26, 27]). Hua Loo-Keng came from a rather poor family in markable method of reading books from the library. He China. He was born in November 1910 in a small town would take a b o o k read the table of contents and introcalled Jintan in Jiangsu Province. 1 After 9 years of school- duction, and then turn off the light and imagine writing, his father sent him to a vocational school for accoun- ing the book himself. After a while he checked with the tancy in Shanghai. After 189years he had to return home book. In this way he went through all the essential points to assist his father in running a small grocery shop. In his quickly and enjoyed the reading more. Soon he began to 9 years at school, Hua Loo-Keng was a troublesome boy, follow advanced courses in the department. In 189 years, and his school record was not as good as one might imag- he finished all the courses and became a lecturer. This ine. His mathematical exercises were not neat but were was very unusual in a society with rigid hierarchies, as full of alterations--he was always looking for different Hua Loo-Keng had no academic diploma. It was there that Hua developed his interest in addiand better methods of solution [6]. His talents in mathematics were first recognized by his tive number theory. He once sent out three papers at school mathematics teacher, Wang Wei-Ke. When Hua the same time for publication in foreign journals and all returned home after his short stay in Shanghai, his fa- were accepted. His name became known abroad; Hardy ther was not happy about his work in the shop for he invited him in 1936 to visit Cambridge, where he stayed seemed to be obsessed with reading irrelevant books 2 years. Hardy advised him to finish a doctoral thesis, on mathematics. Wang Wei-Ke arranged a job for Hua but Hua decided not to, for fear this would restrict his in the school. His job was simply to clean up the class- study. In fact, Hua had no academic degree until 1979, rooms and to assist other teachers if necessary, but this when he received an honorary doctorate from Nancy in gave him the opportunity to read the books on algebra, France. His approach to study at Cambridge was characgeometry, calculus, etc., that Wang Wei-Ke obtained for teristic, always following the path which he believed to him. He was then about 18 years old. One year later, he contracted typhoid fever. The doctor believed that there He would read the table of contents and introwas no chance he would survive, but he did, although duction, and then turn off the light and imagine the disease left him lame in his left leg for the rest of his writing the book himself. life. This period was crucial for his career. First, he became convinced that doing mathematics would be the be most useful, however unorthodox it might be. While only suitable profession for him due to his poor physi- at Cambridge, he published more than 10 papers on varcal and material conditions. Second, he had to develop ious problems in number theory. Some of these are still a w a y to learn everything by himself. Later he always significant [10]. emphasized in his teaching that independent thinking In 1938 Japan invaded China. Hua felt that his counis of great importance [23]. Although he was extremely try needed him, so he canceled his planned visit to the talented, he always gave the credit for his achievements Soviet Union and returned to China. From 1938 to 1945 to his hard work and method of study. he was a professor in the South-West United University In 1930 he read a paper published in a journal in Shang- in Kunming, Yunnan Province, in the southern part of hai written by a professor called Su Jia-Ju. The paper was China. This university was formed by the merging of Beiabout a solution method for finding roots of polynomials jing University, Qinhua University, and Nankai Univerof order 5. Hua found errors in the paper, and submit- sity, which were dispossessed by the Japanese. During ted a note entitled "Reasons why Su Jia-Ju's method for the 8 years of the Sino-Japanese War, it was one of the solving an algebraic equation of order 5 cannot hold" very few Chinese academic institutions. Many famous to Science Journal (Shanghai), where it was published. Chinese scientists were trained there. Kunming was in This paper was written in an unconventional manner but the unoccupied area but suffered a lot from bombardwas extremely clear and concise. This struck Professor ments. The working conditions were very poor. Some Xiong Qin-Lai, the director of the Mathematics Depart- of his important work was completed during that time, ment of Qinhua University, one of the best universities in e.g., his well-known book Additive Prime Number Theory, China. Professor Xiong was even more surprised when although this was published in China only many years he learned that Hua Loo-Keng had no formal mathemat- later. ical education, and he invited Hua to come to Qinhua. From 1946 to 1949 he worked in the United States, first At the beginning, his job in Qinhua was not too differ- at Princeton and then at the University of Illinois, having been sent with several other young scientists to the United States by the Nationalist Government of Chiang 1 All the Chinese names in this article will be spelled according to the Kai-Shek. Some claim (e.g., [16], p. 213) that by sending "Pin Yin" system which is now standard in the People's Republic of China, except internationally accepted names like Chiang Kai-Shek, these scientists to the United States the Nationalist Government hoped to get access to advanced technological Hua Loo-Keng, and Mao Tse-Tung. THE MATHEMATICALINTELLIGENCERVOL.16,NO. 3,1994 37

Hua Loo-Keng

knowledge, in particular nuclear science, but there is no reason to believe that such access was possible. During that period, Hua wrote outstanding theoretical papers on number theory and many other topics, such as group theory and functions of several complex variables. When Mao Tse-Tung removed Chiang Kai-Shek and the People's Republic of China was created in 1949, Hua made the choice to return to China. In an open letter to all Chinese students staying in the United States, he tried to convince them to go back as well. He argued that "a foreign garden can never be your own garden," and that the new China needed their skills. Hua Loo-Keng returned to China in 1950. Unlike some of his returning colleagues, Hua was well received. He was appointed Director of the Mathematics Department of Qinhua University, and 2 years later became Director of the Institute of Mathematics of the reorganized Academia Sinica. Like all university staff members, Hua became very involved in reform campaigns. Students and teachers were organized in groups to study Marxism-Leninism and the Thoughts of Mao Tse-Tung and to analyze their own behavior and attitudes. In these years, Hua Loo-Keng wrote an article (see [6]) that contained much self-criticism. In this article, entitled "We should have only one tradition, the tradition of serving the people," Hua criticized the elitist traditional system of education in China, and the orientation of Chinese research toward Western countries. He urged students to break with selfish traditions and to serve the people. He apologized for his mistakes and recognized the leadership of the Communist Party in the process of personal 38

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

transformation. He avowed his determination to teach basic principles of science and technology to the masses. In the early fifties, Hua continued his theoretical work. Three books were published, Additive Prime Number Theory, Introduction to Number Theory, and Harmonic Analysis of Functions of Several Complex Variables. In those years, theoretical publications were still appreciated in China, and the third book won the Natural Science Prize of Academia Sinica in 1957. Meanwhile, he also wrote articles for newspapers and magazines especially about education. Learning is a stepwise process, he wrote, not a process of repetition, which has been the bane of education in China. Students must think independently, confront obstacles in the scientific research process, and have the courage to be creative. He pleaded for steady, industrious, and innovative work. His essays were later published as a book (see [23]). During that period, Hua represented China at various conferences abroad, such as the World Council of Peace in 1954 and the N e w Delhi Conference of Asian Nations. In 1956 he was elected a member of the Central Committee of the China Democratic League. This league, the core of which consisted of intellectuals and artists not members of the Communist Party, functioned as a "democratic party" under the leadership of the Communist Party, within the so-called United Confrontation, uniting noncommunists inside and outside of China. In the period of the First Five Year Plan (1953-1957) the People's Republic of China enjoyed one of its most flourishing periods. In 1956 and early 1957, the so-called Hundred Flowers Blooming Together campaign started. The campaign was initiated by the Communist Party to encourage intellectuals to speak out and criticize the Party. It was said that this would help the Party. Many intellectuals responded. The China Democratic League suggested to the State Council that students should be assigned jobs according to their expertise and that scientists should be free from overwhelming social and political activities. These ideas, along with other suggestions and criticisms, were later considered to be antisocialist and against the leadership of the Party. The campaign was soon followed by a massive rectification of intellectuals, and later by the antirightist movement, in which hundreds of thousands of intellectuals and senior university students were physically attacked. Being a self-made expert, Hua Loo-Keng managed to survive in this great political turmoil. He again made public statements of self-criticism, and was very depressed at that time. Hua continued to be a member of the Central Committee of the China Democratic League, and as such he participated in the second session of the Congress (1959). He joined the Communist Party of China only in 1979. In 1959, Hua was appointed Vice-President of the newly founded China University of Science and Technology, which was separated from Academia Sinica in 1958 with the aim of rapid promotion of advanced science and technology in China by teaching large numbers

of students, mainly from working-class families. In that ering from the crisis. Liu Shao-Qi, the Chairman of the period, Hua wrote some volumes of his textbook Intro- People's Republic of China at that time, also held the duction to Advanced Mathematics. These volumes are sim- portfolio of economic affairs. Liu stimulated large-scale ple, clear, and interesting, and yet show no lack of rigor. industralization, especially in the urban centers; he started to dissolve people's cooperatives and to allow They attracted a lot of young people in China. private profits. Mao Tse-Tung, as Chairman of the ComIn the late fifties, economic life in China underwent changes. Mao Tse-Tung believed that communist devel- munist Party, was supposed to be in charge of ideologiopment in China would consist of three stages. The ini- cal matters, and he emphasized rural development and tial stage was called the "new democratic" stage allow- small-scale industrialization in the rural areas. Both Mao ing private ownership. It was to be followed by the "so- and,Liu feared revisionism in the Party and the dancialist" stage emphasizing collective ownership; the final ger of class enemies taking over the leadership. Once Stage was the "communist" one where private and col- more, many cadres, intellectuals, and artists were critilective ownership were to be abolished. During years of cized as "bourgeois" trying to reinstall capitalism. This economic progress, especially after the good harvest in time, the criticism of the intellectuals heralded the disas1958, Mao believed that the "socialist" stage could be en- trous Great Proletarian Cultural Revolution. Hua's experiences and those of his students during tered and that this stage could be gone through quickly. The Party set out three ideological lines, which were the years 1958-1965 in the field of transportation and known as the "Three Red Flags," one of which was the agriculture encouraged him to extend his activities. In Great Leap Forward. Under these flags many changes 1965, together with a team of students, he launched were introduced. People's cooperatives were imposed a mass campaign to promulgate simple operations reon a large scale both in the rural areas and in the ur- search and statistical methods in the industrial sector of ban centers. A nationwide rural backyard steel-making China's economy and to implement these methods in campaign was started. Many initiatives were taken to practice. Not long afterward, in 1966, the Great Proletarindustrialize and modernize the country. A flood of ex- ian Cultural Revolution was started by Mao Tse-Tung, tremely exaggerated statistical data appeared, relating to against what he called the bourgeois headquarters of Liu Shao-Qi. A long period of political turmoil began, both industrial and agricultural sectors. This was followed by severe famine from 1959 to 1962. which officially ended in 1976. Hua was under enormMillions of people died of hunger in Anhui Province ous political pressure and had an extremely difficult time, especially at the beginning of the Cultural Revolualone. The crisis was blamed on bad weather during these 3 years, on loans by the Soviet Union that had to tion. A lot of his manuscripts, including the uncompleted be repaid immediately, and on the sudden withdrawal Introduction to Advanced Mathematics, The Input-Output of experts by the Soviet Union after relations between Method, and Mathematical Methods in Transportation Problems were confiscated and lost [23]. Hua Loo-Keng had China and that country were broken in 1958. some personal contacts with Mao Tse-Tung, and Mao Responding to the Great Leap Forward campaign, many scientists tried to contribute to economic devel- wrote to Hua on several occasions, encouraging him to opment. Hua Loo-Keng was very active among them. At continue working with and for workers and peasants. that time his keen interest in applications of mathematics to actual technological and economic problems in China H e w r o t e verse in praise o f linear p r o g r a m m i n g . began. Together with some of his students, he started in 1958 to visit transportation departments to promote Hua mentioned these contacts in one of his speeches [1] the use of operations research methods. He initiated the after the fall of the "Gang of Four." Mao's protection must use of input-output analysis in formulating national eco- have helped Hua through these troubles. Actually Mao started the cultural revolution in the ednomic plans. During that same period, intellectuals and students ucational sector. He encouraged the "rebellion" of young were also encouraged to do physical labor. Hua and his students to challenge all authorities, organizations, and students participated in agricultural activities. This expe- leaders. He believed that previous rectification moverience led him to start to apply linear programming and ments had not been successful in eliminating the danger graphic methods to agriculture [11]. He put a lot of effort of the restoration of capitalism by the existing enemy into it, giving great publicity to the results and starting a 9class inside the Communist Party. This required a new campaign to explain his approach in popular language. revolution from the masses up to the top of the Party. The style of his campaign was characteristically Chinese. The massive "rebellion" resulted in chaos throughout the Before groups of local people in Shandong Province, he country. No organization could function; no one could gave vivid illustrations of the use of graphical methods stay out of the "revolution." Many leaders of the Comto solve linear programming problems. He wrote rhymes munist Party were attacked physically by "red guards" and verse in praise of linear programming, organized on- (high school or university students), and later the persecution extended to academic authorities and others. the-spot workshops and radio broadcasts, etc. From 1963 to 1965, the Chinese economy was recov- Many were attacked, publicly humiliated, put in jail, or THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

39

killed. For almost 4 years no university enrolled a single student. Different political groups arose each claiming to be loyal to Mao, and fought against each other. Mao said later that the Cultural Revolution was one of the two important achievements in his life; the other was the creation of the People's Republic of China. In these years Mao introduced ("The Order of the 7th of May") a new educational system. Schools were to be set up as communities, where agricultural, industrial, and military training had to be combined. Many of these "May 7" cadres' schools and communist universities were formed. "The educational system needs a revolution," Mao said; "education must be combined with physical labor and with basic production activities." The word "practice" was exaggerated absurdly, being taken to mean instant and visible usefulness in everyday life. Nearly all academic activities were ignored and stopped, all "theoretical authorities" were persecuted, and all traditional and academic systems were criticized and abandoned. The less formal education one had, the better. During the Cultural Revolution, Hua Loo-Keng struggled to continue his campaign of applying and popularizing mathematical methods. That he concentrated on the industrial sector is not surprising, for industrialization was considered to be the backbone of development, not only heavy industry in the urban centers but also rural industry providing equipment for farm mechanization and development. The policy to make industry the leading factor of agricultural development resulted in a great number of new factories and enterprises. Hua Loo-Keng and his team traveled to hundreds of cities throughout China; they visited thousands of factories, giving lectures. Wang Yuan, who collaborated with Hua for many years, mentions that millions of workers and professionals attended his lectures ([23], p. 11; [12], p. xiv). Sometimes his audience for one lecture could be more than a hundred thousand people ([12], p. 2). The usual routine was for the problem-solving team to visit a factory for a week. The visit started with working sessions in the factory to discuss production problems. Later in the week Hua would give a lecture in the main hall on Optimum Seeking Methods and Overall Planning Methods, after which the team visited individual workshops to explain how to apply those methods. The team claimed to have solved 10,000 problems in one province alone [12]. These sessions provided the team of troubleshooting mathematicians with a permanent flow of new practical problems. During that time Hua Loo-Keng wrote two readers, Popular Lectures on Optimum Seeking Methods (published 1971) and Popular Lectures on Overall Planning Methods (published 1965). The Optimum Seeking Methods and the Overall Planning Methods were not only applied to industry, they were also used in shops. The campaign was so popular that the readers were distributed to almost every technical department and institution throughout the country. An educational film called "Optimum Seeking Methods" was also made and widely 40 THEMATHEMATICAL INTELL1GENCERVOL.16,NO.3,1994

shown [1]. Never had mathematicians reached so many people. The campaign went on when the "Gang of Four" was crushed (soon after Mao Tse-Tung died in 1976) and the Cultural Revolution ended. After Deng Xiao-Ping took office in 1978, the open-door policy was announced. People started to concentrate on the development of the country's economy rather than on political struggle. In this transformation period, the work of Hua LooKeng was very much appreciated. In 1983, while he was still alive, a 6-hour-long television series was made and broadcast in China about his early life-- unprecedented in China, for a scientist. Hua Loo-Keng died of a heart attack in 1985 while giving a lecture in Tokyo. In total he wrote more than 10 books and 200 scientific papers. He became known to almost everybody in China for his work in popularizing mathematics and optimization techniques, and is regarded as a national hero.

S o m e Necessary C o n d i t i o n s What can possibly explain H u a Loo-Keng's extraordinary success in bringing mathematics to the masses? First, his personality was clearly crucial. Hua himself was self-made and very hard-working. Lacking formal academic training himself, he did not put much stress on it but promoted self-learning again and again. This was not in the Chinese educational tradition, which was authoritarian, being based on the Confucian Philosophy of preservation of nature and society. Students were expected to be docile, copying lessons that were dictated to them, the reverse of self-learning. On the other hand, the idea of explaining what is complicated (including mathematics) in simple terms is deeply rooted in Chinese culture. There is a Chinese saying "Difficult In, Easy Out," which is taken as a principle for scholars and teachers: One should study difficult theory, master it, and make it easy to understand for others, and Hua Loo-Keng's teaching did just this. Hua's popularization of mathematics corresponded very well with some of Mao Tse-Tung's thoughts on education. In his essay "On Practice" [15], first published in 1935, Mao Tse-Tung was very critical of the elitist nature of the traditional Chinese educational system. He recommended teachers to "speak in the popular language" and "to make what you say interesting" [21]. Mao's views were reflected during the First Plenary Session of the Chinese People's Political Consultative Conference on 21 September 1949: "Efforts should be made to develop national sciences, to place them at the service of industrial, agricultural and national defense construction; ... scientific knowledge should be popularized" (Art. 43). Of course, Hua's examples were all taken from the workplace, from daily practice. A further contribution to the popularity of Hua's cam-

paign came from the historical context. China had been humiliated by the Western powers, physically and psychologically. For thousands of years, China had considered itself as the center of civilization, the "Central Kingdom" of the world. Then in the 18th and 19th centuries the Western powers showed up with much more advanced technology; China was completely defeated and forced to accept various humiliating treaties, and the dream seemed to be over. Mixed feelings toward Western ideology and technology developed: a feeling of inferiority and at the same time, and for different reasons, of superiority. The Chinese learned the importance of military capacity heavily dependent on modern science and technology. There was a strong desire to catch up with the Western powers as quickly as possible, Especially after the creation of the People's Republic of China, hopes were vested in the Communist Party. The Chinese people supported the Party's policies with tremendous enthusiasm. In these circumstances, it is not surprising that Hua's mass campaign, aiming to increase production, was warmly received. The many mass movements of the time provided the environment for Hua's campaign, and his prestige and the success of his campaign contributed to the general popularity of such movements. J Although Hua Loo-Keng rarely expressed his own political views explicitly in his writings, he certainly supported some of Mao Tse-Tung thoughts. For instance, Chapter 3 of Popular Lectures on Optimum Seeking Methods [8] deals with the leadership of the Party and association with the masses. Of course, at that time in China every writer was supposed to show his/her dedication to Mao Tse-Tung's thoughts and to the leadership of the Party. Nevertheless, it seems very likely that Hua Loo-Keng was himself dedicated to socialist development and challenged by Mao's call to base theory on practice. Mao classified experts as either, "red experts" or "white experts," representing "proletarian" and "bourgeois" elements, respectively. Hua Loo-Keng was apparently considered a "red" expert, although criticized by some of Mao's followers for putting too much emphasis on mathematical expertise in promoting young researchers. Even his works on pure mathematics were criticized as typical examples of theory lacking in practical value [23]. It is said that Hua Loo-Keng did not enter a library for 10 years during the Cultural Revolution and could only do theoretical research late at night in his home [6].

O p t i m u m S e e k i n g M e t h o d s a n d Overall Planning Methods In this section we will illustrate the type of mathematics discussed by Hua in the factories: Optimum Seeking Methods and Overall Planning Methods. In public lectures, Hua concentrated on methods to minimize or maximize functions of one variable, a type

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with respect to 1618, by 1382 g. Two experiments should be carried out corresponding to C and D, with carbon contents of 1618 and 1382 g. If the experiment with 1382 g showed the better result, then the part of the paper strip to the right of 1618 could be torn off. He folded the new paper strip again in the middle and marked the point E symmetric to 1382, by 1236 g. A n e w experiment with carbon contents of 1236 g should be carried out, and the results of the experiments with 1236 and 1382 g compared. Supposing that the 1382 experiment gave a better result, H u a then explained that the left side of the paper strip from 1236 could be torn off as well. This procedure went on until the differences in experimental results were sufficiently small. He showed that every time the length of the paper strip had been reduced by a constant fraction: 0.618. He stated without proof that this is actually the best possible approach in a certain sense. In his reader, H u a discussed cases where more factors (variables) need to be considered. In fact, he dealt with the two-factor case, which he could illustrate using a square piece of paper with the two sides representing the factors. In both one- and two-factor cases, generalized convexity of the objective function is required. H u a managed to explain that without getting into technicalities. Moreover, he introduced in his early popular reader [8] m a n y other methods, such as steepest descent and some other simple direct methods (without using derivatives). He also explained h o w to treat cases where more than one experiment is allowed at one time (parallel experiments). In the later edition of his book Optimum Seeking Methods [9], m a n y mathematical proofs were provided as well. H u a often referred to Mao Tse-Tung's teaching which says, "In every complicated process influenced by m a n y factors, there m u s t exist one factor which dominates the others. Once this key factor has been found, others will follow more easily." Based on this, H u a suggested that for a complicated practical problem all possible influencing factors should be analyzed and classified according to their importance. By ignoring less important factors, problems of only small sizes would be obtained. In m a n y cases this would already give satisfactory results. Actually, he was advocating quick and effective heuristics. In m a n y published examples this approach is, indeed, followed (see also below). In his reader, H u a Loo-Keng claimed that his approach of the O p t i m u m Seeking Methods provided the means to reach the target of the Communist Party "to construct socialism in a fruitful, quick, good, and economical way." Overall Planning Methods deal with problems in project management. In a project several tasks have to be fulfilled. Tasks have precedence restrictions and the execution of each task takes a certain time. In which order should the tasks be carried out in order to finish the whole project in the shortest time? Overall Planning Methods include the Critical Path Method (CPM), the Program Evaluation and Review Technique (PERT), etc. H u a LooKeng introduced such problems in his Popular Lectures 42 THEMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994

on Overall Planning Methods (see [7, 23]) with a simple example of making tea. Water has to be boiled, a tea pot and cups have to be washed. There is an obvious w a y to order the jobs to save time. He structured the situation by drawing a network representing jobs and relations, which was easily accessible to the audience. "This is a simple matter, but as an introduction it leads to a m e t h o d in production management," he explained. "If it gets a bit more complicated, really one might get confused by decisions to be made. In m o d e r n industry, often it is not as simple as making tea. There are more tasks, hundreds and thousands or even more with complex relations. Often it can be the case that everything is ready except for a few spare parts, so the completion time is delayed. Or you make an effort to speed u p the process, but do not concentrate on the key points. So by overworking d a y and night, finally you finish a task in a hurry. But then you find out that you actually have to wait for some other tasks to be finished." These lectures impressed a lot of people. They easily accepted these "clever methods" in their daily life and work. The optimization and planning methods are intuitively easy to understand and the mathematics involved is elementary. Hua did not explain difficult mathematical concepts like integration or differentiation but helped people to develop a sense of optimization and planning. The results of the mass campaigns have been reported to be enormous. Hua included in his reader [8] the following example, reported by the Shanghai Oil Refining Factory. We accepted at the end of 1969 a research task to find an additive to lower the condensation point of certain lubrication oil. To our knowledge, this additive should consist of five materials. The composition of the additive would affect the condensation point very much. Also, the amount of additive in the oil has an effect on the result. Using our own experience and foreign literature, we had done more than 100 tests in half a year within the range indicated by the foreign literature. The best results had the effect of decreasing the condensation point from - 16~C to -42 ~C. We thought it was the final result and were about to stop the experiments. Just then, Comrade Hua Loo-Keng came to popularize Optimum Seeking Methods. We spent merely 2 more weeks doing more than 10 tests and decreased the condensation point further to -46~ The procedure was as follows: 9 We found that the composition of the additive was the key factor in this matter. The more additive the better, it seemed, in bringing down the condensation point, but the effect did not increase noticeably for more than 0.5% additive. So we fixed the amount at 0.5%. We further found that among these five materials, two of them could be dropped. Since the three remaining materials A, B, and C add up to 100%, the problem reduced to a two-factor problem. 9 The amount of A was first fixed at 25%. According to experience, the range of amount of B was set from 0.100 at 0.600 mole. Applying the Optimum Seeking Method to determine the amount of B we found a composition in which the amount of B was 0.134 mole, with condensation point -46~ 9 We then fixed the amount of B to be 0.134 mole and tried to find the best amount of A. Finally, we found it was still around 25% (in weight).

In the monograph of Academia Sinica, 1977 [1], in total 451 applications were listed. Factories reported applications, all claiming successful results. We quote here two examples to illustrate the presentation. The first one from Long Quan Wu Limestone Ore Division of the Capital Steel Company is entitled "Using Optimum Seeking Methods in Finding the Composition of Explosives." In our mining area, we used to use the No. 2 rock explosive, which is costly and difficult to obtain. Later we used other explosives, but the result was not ideal due to the low blasting power. The workers in our mining area therefore applied the 0.618 method in testing compositions of ammonium-coal explosives. The right composition of diesel oil and wood powder was found with the following formula: ammonium nitrate/diesel oil/wood powder = 100/3/6.5. The resulting explosive has power similar to that of the No. 2 rock explosive, but with three times lower costs. Every explosion can thus save 12,000 yuan for our country, and that amounts to 300,000 yuan per year. Another short report was from a match factory in the remote region Guanxi. Every year, about 1,000 m 2of wood for producing matches in our factory would get too dry and would be spoiled. It was then used as firewood, and so caused a big waste. We applied Optimum Seeking Methods to adjust the temperature to make the wood moist and also to adjust the corresponding processing procedures. As a result, the quality of matches made of such wood rose to meet the national first-class standard. After optimum seeking, from the end of 1974 to the end of 1975 we had used 1,000 m 2 of previously abandoned wood for producing high quality matches. This saves up to 38,000 yuan. As a matter of fact, the popularity of the campaign can also be seen from the great number of books on popularizing Optimum Seeking Methods published during and after the Cultural Revolution. Applications were shown in the reader by Hua and his team [8] and in the collection of applications compiled by Academia Sinica in 1977 [1]. Many provincial towns compiled books on Optimum Seeking Methods with examples. Most provinces and big cities had their own "Office of Popularizing Optimum Seeking Methods." Some of the books, many compiled by these offices, are Refs. [2, 4, 20, and 25].

Discussion The political changes in China largely determined the direction of Hua's scientific work in the later part of his life. On the other hand, Hua had a great impact on the implementation of political ideas, both by his work and by his scientific prestige. His popularity was due to his original and pathbreaking activities, but making him a hero also gave him power in the political process. Hua being a strong-willed independent thinker had to establish a balance between his role as independent scientist and the political use that was being made of his work by party and government. As a prestigious intellectual holding high office he was responsible not only for his success stories, but also for the political use that

could be expected to be made of them. Many mathematicians all over the world have been challenged by Hua's "mathematics for the masses." Although there is good reason to admire his original thoughts and initiatives in popularizing mathematics, the full assessment of Hua's merits depends on a profound evaluation of his political role as well. We do not embark here on such an evaluation but observe that he contributed support to a political regime that has been responsible for many disasters, failures, and mistakes. Some aspects of Hua's work were important in the specific context of his time, whereas others are of permanent significance. The importance of Hua's practice depended on the relatively low educational level of Chinese workers and the technological backwardness of industry. The simple repetition of those mass campaigns would be inappropriate today, for one of their major purposes is now served to a large extent by the Chinese national system of education. The papers of Hua and his team on the popularization of applied mathematics mainly describe mathematical methods. These papers are (see also [22]) fascinating and show how very difficult it is to explain mathematical methods in simple language. Indeed, it is a task that perhaps could only be performed by a great mathematician. Hua called his approach popularization of mathematics. Although he developed in his papers many valuable ideas on transfer of knowledge, nowhere did he explicitly analyze the educational principles involved. Some key words (reinvention, discovery, intuitive methods, etc.) show up both in Hua's papers and in standard works on mathematics education [3], but the intentions are very different. Hua was not addressing students, but people who had almost no background in mathematics. Although mathematics education is increasing in many countries, there remain many who are not familiar with mathematics. They would gain very much from the popularization methods of Hua and his colleagues. Hua and his team did not recount their experiences with mass campaigns in the thousands of factories. We do not know how problems were identified, could the problems really be framed as optimization and planning problems, were the results implemented or was implementation hampered by organizational constraints, etc. Hua Loo-Keng himself mentioned that he regretted very much not having time to describe his field experiences in detail. It is quite likely that such reports would have shown that the results were not always satisfactory, because most problems in practice are much more complex than the presentations in Hua and Wang's book [12]. Production planning, whether in small workshops or in large industrial enterprises, requires a much wider variety of skills than operations research techniques alone. Usually there are many important factors that cannot without great difficulty be incorporated in mathematical models: hierarchical decision relations, uncertain demand and supply, etc. Mathematical models can play an THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

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important, but only partial role. Real solutions can only be the result of an extensive process of interaction between planners and technicians at the workplace and the mathematical modelers. It is a pity that Hua and his team, who certainly faced hundreds of unexpected problems, did not write about them. Still we should give full credit to some impressive results of Hua's activities. People became aware that they could, indeed, increase production, and workers and technicians started experiments to do so on a large scale. During the last two decades the mathematization of society has been rapid in industrialized countries. Personal computers have played an important role. In industry, agriculture, finance, and public services, operations research and statistical techniques are frequently used. Nevertheless, the implementation of results is spotty. The role of operations research has been severely criticized: it is too sophisticated, too theoretical, researchers do not understand the real problems, uncertainty makes the models of limited value, etc. Communications between experts in different fields and between experts and 44 n~EMATHEMATICAL INTELLIGENCERVOL.16,NO.3,1994

nonexperts have become extremely important. We need to find, or at least to try to find, common languages to understand each other. Hua Loo-Keng's practice showed us that it is possible to build this sort of bridge. This is perhaps Hua's most important lesson for future generations. He was able to communicate with laymen on the workfloor and with technicians, not only by being an excellent teacher but also because of his profound interest in the practical problems at stake. Hua and his team were strongly challenged by them and used all their intellectual power and mathematical skills on them. A merger between practice and theory became reality. Although problem-solving mathematical techniques may be primarily developed for large-scale industries, at an advanced level of technology, the experiences in China have shown that they can offer a lot in low-tech situations as well. This point may be of special importance to developing countries. The problems are tremendous: poor infrastructure, risky food production, poor market structure, lack of credit systems, etc. Within the research community few understand the problems with

which the poor have to struggle. Village organizations, farmers' cooperatives, regional branches of government departments, and other local bodies do not always have access to the offices where the financing of research is being decided. In addition, they often lack the experience needed to come up with specific research proposals, in particular research involving mathematics. Yet there are many problems arising in everyday surroundings where such research might well be considered, and where mathematical modeling and operations research could make a valuable contribution. Hua Loo-Keng and his colleagues would call on mathematicians, statisticians, and other scientists who are usually wrapped up in university lecture-rooms to step down and enter field work. This need not be done by Hua-like mass campaigns. Let us illustrate by experiences at the University of Dar es Salaam in Tanzania [17,18]. In order to make courses in operations research as relevant as possible, the students, who nearly all came from the rural areas, returned during vacations to make analysis of the agricultural problems in their villages with the help of relatives, neighbors, and village functionaries. They examined whether it might be useful to contract a loan at a bank, whether they ought to acquire an ox-drawn plow, whether it would be best to plant early or late, with an eye to the spread of labor, what acreage should be designated for communal farming, and many other things. An example arose when the Tanzanian Food and Nutrition Centre, occupied with a large-scale program of food and agriculture instruction, had to advise villagers on the size of acreage for maize, which was the main food crop in the villages. We consider a certain village and estimate, on the basis of size and composition of the population, the yearly total maize requirement: r kg. Suppose the average maize yield can also be estimated: # kg/ha. If every year the total acreage under maize in the village were equal to r/# ha, then in many years a maize shortage would occur. To decrease size and frequency of shortages, more land would have to be cultivated with maize, but the question is: how much more? Is it 10%, 30% or even 50% more than r/# ha? The answer depends on two factors. What level of risk of maize shortage is acceptable? What is known about the variability of yield, in particular the probabilities of low yields? Let Y denote the maize yield per ha, modeled as a random variable with unknown cumulative distribution function F(y). Let the acreage under maize, which is a decision variable, be denoted by x; then the food shortage, denoted by Z(x), can be written as Z(x) = max(0; r - Yx). Suppose one wants to accept a maxim u m probability c~of occurrence of a shortage. It is easy to see that then x should be at least r/y~, with y~ such that F(y~) = c~. This seems to be a reasonable approach, but will often lead to a high value of x. Moreover, what is a proper choice of c~?Another drawback of this approach is that the size of shortages is not taken into account.

An alternative is based on a balancing of the cost of labor against the consequences of shortages. Suppose the labor costs of cultivating 1 ha of maize are w, and let w = c#, with c the average labor costs of producing I kg of maize; further assume that every kilogram of shortage is purchased at price p. The labor required to cultivate an area x can then be evaluated by the "labor costs" cpx; the costs of purchasing the shortage are pZ(x). The expected value of the annual "costs" of cultivating an area x is = E(cux +

pZ(x)).

For the calculation of 9(x) it would be necessary to know the probability distribution F(y) of the maize yield. Estimation of F(y) should preferably be based on observed yields. If no, or only very few, yield-data are available, it can be useful to gather information from farmers on the yield distribution, e.g., minimum, maximum and most likely yield. This may result in a rough estimation of F(y). The value of x that minimizes 9(x) depends on p and c only through p/c. If, in case of shortage, food has to be flown in from abroad, p will be large. This type of consideration may lead to a reasonable estimate of p/c. For the village concerned and estimated F(y), a choice of p/c of 3-6 would result in 10-20% more land for maize. To evaluate the consequences, one could also calculate the probability of shortage. If x is taken to be 10% (20%) larger than r/#, a shortage will occur in about 4 (2) of 10 years, which is quite often, and the average shortage will be around 5% (3%) of the yearly requirement. This type of result has also been found in other examples: High probabilities of shortage are acceptable as long as the expected shortage remains relatively low; as a decision criterion the expected shortage seems to be more useful than the probability of a shortage. The results were worked on at the University and discussed by students and staff. These studies, which made fruitful use of mathematical modeling and simple operations research methods, brought up many crucial points for discussion, for instance, that what was best for a village need not be best for the government; that the selling prices of produce were not always in reasonable proportion to the amount of agricultural labor; what could be done about the lack of a market within reach where produce might be sold; were the farmers to bear the risks of fluctuating world market prices themselves, or would it be better to accept an income that was stabilized by the government but on average much lower; etc. Students began to dare to look at problems in the village in an integrated manner. The mathematical approach offered a means of giving an analysis of the problems, and the interaction between the computation of alternatives and the discussion of political decisions or of the necessity of governmental measures caused the real problem to become clearer. This is not a cure-all. Such studies can usually have only an indirect effect: N e w suggestions may be made; existing plans or ideas may be supported or rejected. AcTHE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

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tually solving farmers' problems is usually a very complicated process; it m a y take years and great effort before, for instance, traditional practices are changed in order to accept a new agricultural method. In developing countries, farmers' initiatives to improve living conditions, like cooperative farming, installation of cooperative cereal banks, etc., deserve a lot of scientific support. Recently, an interdisciplinary research project on food security on the Central Plateau in Burkina Faso, West Africa was set up. One of the objectives was to determine areas of priority for future field research. It resulted [14, 19] in field research projects centered around farmers' cooperative initiatives to overcome the agricultural crisis on the Central Plateau. In these projects mathematicians work together with farmers, interviewers, extension officers, economists, etc. There is a strong need for mathematicians and quantitative scientists who are prepared to work in interdisciplinary teams and to develop a profound interest in farmers' problems. They have to be very good at mathematics and apply all their intellectual energy to explore h o w mathematical modeling can be used to s t u d y the practical problems. Moreover, they have to be able to explain in c o m m o n language w h a t they are doing and what their results are. In short, disciples of Hua Loo-Keng are required. Both as pure and as applied mathematician, H u a LooKeng showed great originality. Some people m a y argue that his move from pure mathematics to down-to-earth applications is a loss for science. This point of view is not ours. The permanent flow of practical problems provides mathematicians with a rich resource of theoretical problems as well. Daily-life problems can be a challenging inspiration even for pure mathematicians.

Acknowledgments The authors thank Dr. David Cappitt, Roehampton Institute, London, Professor Wang Yuan, Academia Sinica, Beijing, and Dr. Chandler Davis for valuable comments.

References 1. Academia Sinica, Selected Results of Popularizing Optimum Seeking Methods on National Wide Scale, Office of Studying and Popularizing Applied Mathematics & Chinese Institute of Scientific and Technological Literatures, Beijing: Scientific and Technological Literatures Press (1977) (in Chinese). 2. Beijing Group, Optimum Seeking Methods and Its Applica.tions, Beijing Group of Popularizing and Applying Optimum Seeking Methods, Beijing: Beijing People's Press (1972) (in Chinese). 3. H. Freudenthal, Mathematics as an Educational Task, Dordrecht: D. Reidel (1973). 4. Guandong and Guanzhou Office, Optimum Seeking Methods and Examples, Guandong Province and Guanzhou City Office of Popularizing Optimum Seeking Methods, Guandong: Guandong People's Press (1972) (in Chinese). 5. H. Halberstam, An obituary of Loo-Keng Hua, Math. Intell. 8 (1986), 63-65. 46

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6. X. Hu, Paths to Success of Scientists, Jiangsu: People's Press of Jiangsu Province (1982) (in Chinese). 7. L. Hua, Popular Lectures on Overall Planning Methods (with supplements), Beijing: Chinese Industry Press (1965) (in Chinese). 8. L. Hua, Popular Lectures on Optimum Seeking Methods (with supplements), Beijing: National Defence Industry Press (1971) (in Chinese). 9. L. Hua, Optimum Seeking Methods, Beijing: Academic Press (1981) (in Chinese). 10. L. Hua, Selected Papers, Berlin: Springer-Verlag (1983). 11. L. Hua, et al. Application of mathematical methods to wheat harvesting, Chinese Math. 2 (1962), 77-91 (in Chinese). 12. L. Hua and Y. Wang, (revised and edited by J.G.C. Heijmans), Popularizing Mathematical Methods in the People's Republic of China, Boston: Birkhafiser (1989). 13. S. Li, et al. (eds.) Dictionary of Names in Modern China, Beijing: Chinese International Broadcast Press (1989) (in Chinese). 14. A. Maatman, C. Schweigman, T. Thiombiano, and J. Van Andel, Food security and sustainable agriculture on the Central Plateau in Burkina Faso: Observations on a research agenda. Tijdschrifl voor Sociaal-Wetenschappelijk Onderzoek van de Landbouw, The Netherlands. 15. T. Mao, Selected Works ofMao, Beijing: Foreign Languages Press (1975). 16. S. Salaff, A biography of Hua Loo-Keng. Science and Technology in East Asia (Nathan Sivin, ed.), New York: Neale Watson Academic Publications Inc. (1977). 17. C. Schweigman, Doing Mathematics in a Developing Country: Linear Programming with Applications in Tanzania, Dares Salaam: Tanzania Publishing House (1979). 18. C. Schweigman, Operations Research Problems in Agriculture in Developing Countries, Khartoum: Khartoum University Press and Dares Salaam: Tanzania Publishing House (1985). 19. C. Schweigman, T. Thiombiano, and J. Van Andel, Etude interdisciplinairede risquesdans l'approvisionnement alimentaire du plateau Mossi au Burkina Faso, Universit6 de Groningen, Pays-Bas and Universit6 de Ouagadougou, Burkina Faso (1989) (in French). 20. Sichuan Office, Elementary Applications of Optimum Seeking Methods, Sichuan Province Post Administration Office, Beijing: People's Post Press (1985) (in Chinese). 21. F. Swetz, Mathematics Education in China: Its Growth and Development, Cambridge: MIT Press (1974). 22. J. Wang and M. Gu (eds). Dictionary of Names, Shandong: Shandong Education Press (1986) (in Chinese). 23. Y. Wang, D. Chen, L. Ji et al. (eds.), Selected Work ofHua LooKeng on Popular Sciences, Shanghai: Shanghai Education Press (1984) (in Chinese). 24. Y.Wang, Hua Loo-Keng. Biographiesof World Famous Mathematicians (Part I), Beijing Academic Press (1990) (in Chinese). 25. T. Xie and G. Tan, Principles and Applications of Optimum Seeking Methods, Zhejiang: Zhejiang People's Press (1979) (in Chinese). 26. S. Zeng (ed.), Brief Introduction to World Famous Scientists, Beijing: Science and Technology Literature Press (1983) (in Chinese). 27. S. Zeng (ed.), Dictionary of Names in Scienceand Technology, Beijing: Chinese Youth Press (1988) (in Chinese). Department of Econometrics University of Groningen 9700 AV Groningen The Netherlands

The Status of the Kepler Conjecture Thomas C. Hales

In 1990, Wu-Yi H s i a n g a n n o u n c e d that he h a d p r o v e d the Kepler conjecture, the conjecture that no arrangem e n t of spheres of equal radius in 3-space has density greater than that of the face-centered cubic packing. Since that time, a w i d e r a n g e of books, journals, a n d n e w s p a per articles has described his r e m a r k a b l e accomplishment. The 1992 y e a r b o o k of the Encyclopzedia Britannica announced, "Without d o u b t the m a t h e m a t i c a l event of 1991 w a s the likely solution of K e p l e r ' s s p h e r e - p a c k i n g p r o b l e m b y Wu-Yi H s i a n g " [10]. Barry C i p r a reported in Science, on March 1, 1991, "Last spring ... the first thing H s i a n g did w a s to pick the oldest, hardest, unsolved p r o b l e m in the subject [of classical g e o m e t r y ] . . . . The second thing he did w a s solve it. "1 H e w a s honored for his w o r k in J a n u a r y 1993, at the joint meetings of the AMS-MAA, b y being invited to deliver a plenary address entitled "The proof of K e p l e r ' s conjecture on the sphere-packing p r o b l e m . " As a result of such a n n o u n c e m e n t s , m a n y are p r o n e to accept H s i a n g ' s solution to the s p h e r e - p a c k i n g problem. Even if H s i a n g w i t h d r a w s his claims, s o m e m i g h t continue to believe, for years to come, that the p r o b l e m has been successfully solved. It has b e c o m e necessary, therefore, to write this article on the status of the Kepler conjecture, to correct the public record. The Kepler conjecture is still a conjecture, a n d not a theorem. The best k n o w n b o u n d on density (0.773055)

is contained in a 1993 p a p e r b y D. J. Muder. (The facecentered cubic packing has density 7 r / v / ~ , or approxim a t e l y 0.74048.) True, H s i a n g has issued a preprint on the s u b j e c t - - a n d a revision, a n d a revision of a revision. His recently published article describes a p r o m i s i n g general p r o g r a m to p r o v e the conjecture, but this p r o g r a m is yet to be carried out [7]. His early preprints contained serious g a p s a n d flaws, a n d m a n y of the experts in the field h a v e c o m e to the conclusion that his w o r k does not merit serious consideration. 2 2The earliest preprint repeatedly makes the unstated assumption that all triangles contain their circumcenter. Thus, his argument only applied to an extremely special family of packings. Hsiang incorrectly claimed in an early preprint that the truncated local polyhedron was always a subset of the local cell. He said this (false) claim was "easy to see" and gave no explanation. Without a carefully established connection between the truncated local polyhedron and local cells, many of his results have limited significance. Ian Stewart's article contains a description of some of the other errors of his early work (see Ref. 11).

1Cipra and lan Stewart have also given more recent reports, documenting emerging gaps and errors of earlier versions. Cipra wrote, "Three years ago, Hsiang announced a solution to a centuries-old problem in solid geometry: proving that the densest possible packing of spheres is the 'face-centered' cubic arrangement often seen in a stack of oranges. But Hsiang's proof has still not won acceptance by other experts in the field" [1]. Stewart reported, "Experts in the field of sphere-packing have.., become increasingly dissatisfied with the proof because of errors in his papers and because of a large amount of detail which he omitted" [11]. For other popular accounts of Kepler's conjecture see Refs. 4, 9, and 12. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3 (~ 1994 Springer-Verlag New York

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H s i a n g h a s c o n s i s t e n t l y m a i n t a i n e d t h a t his p r o o f s a r e c o m p l e t e a n d t h a t t h e p r o b l e m is s o l v e d . E x p e r t s in t h e field d i s a g r e e . T h e s e c o n d e d i t i o n o f C o n w a y a n d Sloane's book on sphere packings states that there are " g a p s i n t h e a r g u m e n t a n d t h e m a t t e r c a n n o t y e t b e reg a r d e d as s e t t l e d . " M u d e r w r i t e s a b o u t H s i a n g t h a t " t h e s t a t u s of t h e s e c l a i m s is u n r e s o l v e d . " By m i s r e p r e s e n t i n g t h e n a t u r e o f t h e s e o b j e c t i o n s 3 a n d b y f a i l i n g to c o n f r o n t t h e difficulties, 4 H s i a n g i n v i t e s a r e s p o n s e . T h i s a r t i c l e w i l l p o i n t o u t s o m e of t h e g a p s in t h e a r g u m e n t a n d w i l l c o m m e n t o n w h a t m i g h t b e d o n e to s e t t l e m a t t e r s .

Critical C a s e A n a l y s i s H s i a n g ' s article r e l i e s e x t e n s i v e l y o n w h a t h e calls critical case a n a l y s i s . To p r o v e K e p l e r ' s c o n j e c t u r e , h e h a s m a n y i n e q u a l i t i e s to e s t a b l i s h , s a y of t h e f o r m f ( x ) > 0, for all x i n a c o n f i g u r a t i o n set S. By critical case, h e m e a n s a n e x p e r i m e n t a l test case. 5 H e selects a n e l e m e n t xo E S, v e r i f i e s f ( x o ) >_ 0, a n d c o n c l u d e s o n t h e b a s i s of t h e exp e r i m e n t t h a t t h e i n e q u a l i t y h o l d s for all x E S. O f t e n h i s critical c a s e s a r e e x t r e m e p o i n t s of t h e s e t S; in s o m e instances they are mean values. Sometimes he considers whole families of examples. We should not underestim a t e t h e v a l u e of e x a m p l e s , e s p e c i a l l y w e l l - c h o s e n o n e s , in p r o v i d i n g h e u r i s t i c e v i d e n c e for d i f f i c u l t c o n j e c t u r e s . But in t h e e n d , h o w e v e r h e p i c k s his test cases, e m p i r i c a l

3 Cipra reports that Hsiang "regards the controversy over his proof as a dispute about subjective standards of how much detail a properly written mathematical proof has to contain." Hsiang says that it "has nothing to do with me, and it has nothing to do with sphere packing" [1]. It is true that a few of the objections may be considered a dispute about how much detail to give. For instance, on page 819 of [7], he tells us that a "straightforward computation will show that the volumes of the local cells ... is bounded below by 4v~ - 0.065." He repeats this type of exposition throughout the article (he tells us the bound on volume; we must guess the method that produces the bound). This is a shortcoming about how much detail to provide. Many of these bounds come as the solutions to constrained nonlinear optimization problems in 30 or more variables. Hsiang casually asserts each unproved bound and moves ahead. He never properly formulates the constraints of the optimization problems, and he never gives general methods that can be used on such problems (except for critical case analysis and numerical checking). For the most part, my criticism does not pertain to how much detail is provided in the exposition. My criticism pertains to the substance of his argument and its defects. 4 When I pressed Hsiang for proofs of particular propositions in 1990, 1991, and 1992, he was unwilling to divulge them, promising that they would be presented in detail in a revised version. The requested proofs were absent from the revision, and from the published account. His unwillingness to share his arguments with others has been puzzling. When I presented Hsiang with a detailed account of gaps and errors in his 1991 preprint, he simply responded, "Your letter made me realize that I cannot assume that the average mathematician knows too much about elementary spherical geometry. I will make sure that the revised version reflects this fact" [8]. 5 Hsiang may not agree with this characterization of critical case analysis. Nevertheless, experimentation by another name is still just experimentation. 48 THEMATHEMATICALINTELLIGENCERVOL.16,NO.3, 1994

Figure 1. Thirteen-Sphere Problem. Take 13 vectors v l , . . . , v13 in 3-space, representing the centers of spheres, and determine the minimum of 1

13

h(v, . . . . ,v,3) = ~ y ~ (llv,[l-2), i=l

subject to the constraints Ilvill > 2 and Ilvi - vjH > 2, for all i ~ j. This particular arrangement, discovered by Schiitte and van der Waerden, has an average buckling height of roughly 0.04557. There are three rings of four equally spaced spheres and one additional sphere at the north pole.

v e r i f i c a t i o n b a s e d o n l i m i t e d e x p e r i m e n t a t i o n is i n a d missible as proof. A g l a n c e at his a r t i c l e w i l l r e v e a l m a n y e s s e n t i a l i n e q u a l i t i e s t h a t a r e b a s e d o n critical case a n a l y s i s , rather than rigorous argument. 6 The reader must decide whether the results, verified empirically by checking a f e w t e s t cases, a r e reliable. I f i n d t h a t m a n y of his inequalities are believable, although yet unproved. Some of t h e k e y i n e q u a l i t i e s a r e false.

6 There is his trade-off scheme on page 764, his argument showing the disadvantage of adjacent large long forks on page 768, and his argument for the impossibility of 7-forks on page 774. His buckling height bounds rely on circumstantial evidence stemming from the special case of uniform buckling heights. One should also mention his argument about the optimal extensions of stars on page 806, Lemma 11, his basic volume estimates on pages 810-814, and his core packings of 5n-type on page 818 (see Ref. 7). Many of the methods of Hsiang's earlier preprints were left obscure. He announced difficult results as "easy to see" without any indication of methods. Many arguments in his article are still unclear, but he is unabashedly explicit about the use of critical case analysis (see pages 764, 769, 774, 776, 782, 791,802, 805, 812, 815, and 816 in Ref. 7). This constitutes a priori grounds for the rejection of his article.

A First Example o f Critical Case A n a l y s i s

Consider the following example of critical case analysis. It is known that at most 12 nonoverlapping spheres can be tangent to a given central sphere of the same radius. An old problem in the theory of packings, and an essential step in Hsiang's approach, is the estimate of how close a 13th sphere can come to being tangent. We formulate this problem precisely in Figure 1. The vectors represent the centers of unit spheres surrounding a central sphere at the origin. Hsiang's formulation has the additional constraint that IIvi II ~ 2.18, for all i. The function h ( v l , . . . , v13) is what Hsiang calls the average buckling height. An arrangement is buckled if at least one of the spheres is not tangent to the given central sphere. The individual terms (11viii - 2)/2 are called the buckling heights. To solve this problem (under the additional constraint IIv~11~ 2.18), Hsiang examines the case that all the vectors have a uniform length: IIv~II = llvj II, for all i and j. Making a few rough estimates, he concludes without rigorous justification that h(vl,...,v~3) ~ 0.0316 in the case of uniform lengths. The estimate he needs, however, is that h ( v l , . . . , v13) ~ 0.025, for all choices of v l , . . . , V13, and not just for the case he considered of uniform lengths. Yet he abruptly terminates his argument after this one special case, with this one-sentence argument: "Since 0.0316 is more than 25% larger than 0.025, the above area-estimate also implies the lower bound of 0.025 for the averaged buckling heights for type II core packings in the nonuniform case" [7], p. 784. (Type II means 13 vectors, as formulated in Figure 1.) The general principle he advocates in this sentence is that one may prove a general inequality by showing that it holds by a considerable margin in a special case. If the foothills, which we see, are no more than 1000 feet, then the mountain peak, hidden by the clouds, cannot be more than 1250 feet. Such handwaving would not merit serious consideration, if this work had not been proclaimed "without doubt the mathematical event of 1991." Students who resort to such tactics jeopardize their grade-point averages. For a professional, it is hardly imaginable.

only raises the question of 13 spheres; he does not attempt a proof. Giinther discusses the problem and shows that there can be at most 13 tangent spheres but leaves the possibility of 13 spheres open. Only Hoppe makes a serious attempt to prove the impossibility of 13 spheres. Hoppe's argument is undermined by the use of a flawed triangulation of the sphere. He triangulates the sphere as follows. Fix n points on the sphere and join every pair of points by an arc of a great circle. Whenever two arcs intersect, eliminate the longest arc until there remains a net of spherical triangles whose vertices are the given n points. Hoppe claims that this procedure always leads to a triangulation with the property that the diagonal joining opposite vertices of adjacent triangles is longer than the common edge. (In Figure 2, the dashed line represents the diagonal.) Simple examples show that Hoppe's claim is incorrect. In the arrangement of five points shown in Figure 3, the edges of the pentagram have lengths x I > x 2 ~ X3 x4 > xs. We regard the vertices of the pentagon as five of the n points on the sphere, and we have joined every pair of points, as Hoppe requires. Comparing the lengths of

Figure 2. A Characteristic Property of Hoppe's Triangulation.

Historical G r o u n d s for S k e p t i c i s m

The 13-sphere problem, mentioned earlier, is a classical problem dating back to a famous discussion between Isaac Newton and David Gregory. Their question was whether 13 spheres could be arranged tangent to a 14th sphere. Newton suspected that this was impossible; only a proof was lacking. The explicit version of the GregoryNewton problem, or 13-sphere problem, asks just how small h(vl, 9 9 V13) can be'. The 13-sphere problem has a long history of highly dubious proofs. Three 19th-century mathematicians, Giinther, Bender, and Hoppe, are often cited as the first to have solved the 13-sphere problem. However, Bender

Figure 3. Hoppe's Phantom Triangulation. The triangulation leads to results that Hoppe did not anticipate. The vertices of the outer pentagon represent five points on the surface of a unit sphere. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3, 1994

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Figure 4. Flawed Triangulations. edges at intersections E, B, and then C, w e get triangulation (1) of Figure 4. Comparing lengths at intersections in the order A, B, and then C, we get triangulation (2). Comparing the lengths at intersections E, A, and then B, w e get no triangulation at all (3). Thus, H o p p e ' s proced u r e does not always lead to a triangulation, and even if it does, it does not have the necessary p r o p e r t y that the diagonal is longer than the c o m m o n edge (see triangulation (2)). Another h a n d w a v i n g solution to the problem of 13 spheres was given b y Boerdijk in 1952. The 13-sphere problem is equivalent to the question of w h e t h e r it is possible to place 13 disks or circles of radius 7r/6 radians on the unit sphere w i t h o u t overlap. To estimate the empty area on the unit sphere not occupied b y disks, Boerdijk identifies a system of plug-shaped regions. H e assumes that these plugs are disjoint, although they clearly are not (Figure 5). If this oversight is corrected, his argument breaks d o w n entirely. 7 The first p r o p e r solution to the problem of 13 spheres was given b y K. Schiitte and B. L. van der Waerden, in 1953. A few years later, a brief, reliable solution to the problem was given b y Leech. If we try to use Leech's argument to get an explicit lower b o u n d on the average buckling heights h ( v l , . . . , V13) for 13 spheres, we get disappointing results. On a related topic, in 1942-1943, Fejes T6th published a proof of the b o u n d of 0.7547 on the density of sphere packings. The statement of the theorem contains a footnote saying, "In the proof, we have relied to some extent

Figure 5. Boerdijk's Overlapping Plugs. When estimating the empty areas near disk 1, Boerdijk identifies a plug (A) extending radially from the first disk. When estimating the empty area near disk 2, he identifies a plug (B) extending radially from the second disk. Boerdijk's argument is invalidated by the overlap of the two plugs, which he assumed were disjoint.

solely on intuitive observation [Anschauung]." The point of his proof at which he fails to give a rigorous argument is the explicit b o u n d on the 13-sphere problem. H e tries to justify this by saying, "This treatment seems to us to be warranted, since essentially w h a t matters to us is only a rough estimate, but an exact estimate is p u r s u e d only with difficulty." As time passed, Fejes T6th became increasingly explicit about the assumption he made. In 1952, Boerdijk gave a counterexample to the strongest formulation of Fejes T6th's estimate. A weaker estimate, which still implies the b o u n d on density, remains an open problem. To Fejes T6th's credit, he d o c u m e n t e d the assumption in his original paper. His books isolate the u n p r o v e d assumption and s h o w carefully h o w the b o u n d of 0.7547 w o u l d follow from a lower b o u n d on average buckling heights

h(vl,..., v13). The subject is thus littered with faulty arguments and a b a n d o n e d methods. A f e w - - including Schfitte, van der Waerden, Leech, and Fejes T 6 t h - - h a v e taken the care to give reliable arguments. If Fejes T6th, w h o d e v o t e d decades of productive research to the theory of packings, was forced to renounce his sphere-packing b o u n d because of an incomplete treatment of the 13-sphere problem, then w e should be all the more skeptical of Hsiang's work, which dismisses the general case of the same problem with one faulty sentence. On a problem of considerable historical significance, it is not sufficient for Hsiang to argue that because the case he tried works b y a comfortable 25% margin, things must also w o r k in g e n e r a l s

7To give rigorous content to Boerdijk'sarguments, we take the Delaunay triangulation of the unit sphere associated with n marked points vl,..., vn (the centers of the disks). Assume that the points v~ are separated by at least ~r/3radians. We take the set of all Delaunay triangles with a given v~ as vertex, and call this a star. The region within the star lying outside of the disks of radius 7r/6 radians, placed at each point v~, is called the empty area. Suppose that for every star we knew that the area 2r(1 - cos(Tr/6))of a disk plus one-third the empty area was more than 41r/13. Then, as each bit of empty area is contained in exactly 3 stars (for each Delaunay triangle is contained in exactly 3 stars), 8 The problem runs deeper. Hsiang gears earlier lemmas to the spe13 disks combined with the total complementary empty area would cial case he considers, making it impossible for the reader to correct have total area exceeding 13(41r/13),the area of a sphere, a contradic- Hsiang's argument without substantial rewriting. For example, Subtion. This is more or less Boerdijk's argument. However, for the star lemma 7 makes the hypothesis that points are separated by 0.98 racontaining four equilateral triangles of side ~r/3 (which forces a fifth dians. This is harmless in the case of uniform lengths, considered by isosceles triangle into the star), the disk area plus one-third the empty Hsiang, because points are separated by at least 1.012 radians. But area is only about 0.948, less than the requisite 4~r/13 ~ 0.967. Thus, in the general case, points might be separated by as little as 2 arcsin(1/2.18) ~, 0.953, and so Sublemma 7 cannot be applied. his argument is not easily redeemed. 50

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

Fejes T 6 t h ' s P r o g r a m

1/12, if Si and Sj particular system of weights9: wij are close but not equal. Fejes T6th conjectured that the inequality of Lemma 1 holds for every packing if his system of weights is used. Because of our restrictions on #, the inequality of the lemma (for fixed j) involves only the local cell Cj and some of the local cells that share a face with Cj. (See Figure 6.) Inequalities such as this express the hope that it is sufficient to consider two layers of local cells (a cell Cj and some of its adjacent ceils). Fejes T6th gives some geometrical motivation for his conjecture that the fundamental inequality should hold, but he does not attempt a proof. In conclusion he writes, "With the proof of this inequality the whole problem [the Kepler conjecture] would be settled .... Thus it seems that the problem can be reduced to the determination of the minimum of a function of a finite number of variables, providing a programme realizable in principle. In view of the intricacy of this function we are far from attempting to determine the exact minimum" [2], pp. 299-300. Hsiang attempts to carry out Fejes T6th's program (using the parameter # = 2.18 and the unique system of weights wij for which wij is independent of j, for all j with Sj close to Si). To do so, he must make careful estimates of the volumes of local cells Ci and then compare these volumes against the constant 4v~. In the next section, we will take a closer look at some of these estimates. =

In 1953, Fejes T6th proposed a program to prove the Kepler conjecture [3], pp. 174-181. To describe his program, we begin with a system of weights. Fix a constant p in the range 2 _< p < 2vf2. We say that two spheres are close if the distance between the centers is less than p. Let {Si} be the set of spheres in a packing. For every pair Si, Sj of spheres in the packing, we select a weight wij. We assume that the weights are in the range 0 G wij < 1, that w~j -- 0 if Si and Sj are not close, and that Y~ wij = 1, for all i. A localcell (also called the Voronoi ce~lor Dirichlet region) in a packing of spheres is associated with each sphere So. By definition it is the convex polyhedron formed of all points in space closer to the center of So than to the center of any other sphere in the packing. Assume that the spheres of the packing have radius 1. By adding spheres to the packing if necessary, we may assume that no point of Euclidean space has distance greater than 2 from the center of some sphere. The local ceils are then of bounded size. The following lemma was proved by Fejes T6th for a particular system of weights described below. LEMMA 1. Let wij be a system of weights for a packing. Let

Ci be the local cell around the sphere Si. If the fundamental inequality wij(vol(Ci) 4x/2) _> 0 -

i

holds for all j, then the packing has density at most

/ v3 .

Proof. Let U be a large region in Euclidean space, for instance, a ball of large radius centered at the origin. Let Su be the union of all the spheres of the packing that are contained in U. Then the ratio vol(Su)/vol(U) is a close approximation to the density if U is sufficiently large. Since ~-~j wij = 1, we see that Z

s~r

wij vol(Ci)

i

is approximately the volume of U, whereas ~ s l cu ~ i wij4x/2 is approximately 4x/-2 times the number of spheres in U, that is, (4v~)vol(Su)/(4~r/3). Thus, the inequality of the lemma, summed over all spheres Sj in U, implies that vol(Su) < 4~r + cu - 71" + eu, vol(U) - 3(4v~) v/~ _

_

Figure 6. Local Cells in the Face-Centered Cubic Packing. Kepler's conjecture would follow from a bound on the volumes of the local cells adjacent to a given cell.

where ev is a small contribution from the boundary of U. The contribution from the boundary tends to zero as the radius of U tends to infinity. In the limit, we obtain the upper bound of 7r/v/~ on the density, as desired. Thus, if a system of weights satisfying the inequality of the lemma can be found for each packing, then Kepler's conjecture is true. Fejes T6th only considered one

9 It follows that w i i = 1 - Y ~ j ~ i w i j . For w i j to be non-negative, Fejes T6th m u s t choose # small e n o u g h that at m o s t 12 centers c o m e within distance # of the center of Si. This calls for a n explicit b o u n d on the 13-sphere problem. Fejes T6th p r o p o s e s # -- 2.0534 [2], p. 299. THE MATHEMATICALINTELLIGENCERVOL.16, NO. 3, 1994 5 1

A Second Example of Critical Case Analysis

Consider Hsiang's reliance on critical cases in the section (w of his paper that estimates the volume of local cells. Section 11 has the important function of using "volume estimation techniques ... to establish lower bound estimates for the volumes of the resulting local cells of various kinds of local extensions. Such estimates will provide a set of convenient tools for the proof of Theorem 2." Theorem 2 is his main theorem, the solution of the Kepler conjecture. One expects the estimates of this section to be carefully established, for he tells us they provide the tools for the proof of Kepler's conjecture. The estimates are developed in a series of three examples and a lemma (Lemma 11). The examples of Hsiang's article are not incidental: Key propositions rely on estimates put forth through examples. No proofs are given of the estimates in the examples, although the first example tells the reader that the estimate follows from the "same computation" as that of the first example of the previous section. Turning back to the earlier example, we find that the computation consists merely of two experimental test cases. After giving these two test cases, Hsiang claims, without further comment, that the general multidimensional estimate will follow by "straightforward numerical checking." Thus, we are led to conclude that the general estimation techniques are based on a few experimental cases and "numerical checking." The estimate of the first example admits an easy counterexample. The problem he poses is this. Select 12 points

on the surface of a unit sphere subject to 3 constraints1~ (1) The first point is located at the north pole (spherical coordinates (0, q~) = (0, 0)). (2) The next five points are located on the circle ~b = 7r/3. (3) Every pair of points has separation at least ~r/3 radians. (At times, we use the angle to measure distances between points on the unit sphere.) Now, arrange the 12 points so that the polyhedron formed by placing a plane tangent to the unit sphere at the given 12 points has the smallest possible volume. Hsiang estimates that the volume will always be at least 4 v ~ - 0.07. Here is the counterexample (see Figure 7). Arrange the five points evenly around the circle of colatitude q~= 7r/3. Arrange the next layer of five points evenly around the circle ~b = 2.04, but shifted by an angle of ~r/5 from the previous layer of five points. Place the 12th point at the south pole. Then the volume is about 4 v ~ - 0.096, smaller than Hsiang's lower bound. This looks like a small amount, but it is significant in terms of the scale relevant to this problem. By w a y of comparison, the regular dodecahedron has approximate volume 4v'2 - 0.1066. Under normal circumstances, when a counterexample is uncovered, we analyze the proof to find the hidden assumption or error in reasoning. In this case, there is no possibility of doing so, as no proof is given. Yet it is hardly surprising that counterexamples can be found, for Hsiang's method is, by his own admission, merely numerical checking. What about the other examples and lemma? The main result of the section (Lemma 11) is false. The proof of the lemma slights what I consider the most interesting family of configurations. These are the configurations (described in the next paragraph) with exactly 11 close neighbors and 1 other neighbor. The proof is little more than a cursory analysis of a special critical case. No proof is given of the estimate in the second example. I suspect the estimate of the third example may be correct, but again no proof is given. After finding problems with other major estimates of this crucial section, I let this last example pass unscrutinized. Methodological

Figure 7. Counterexample to Hsiang's Lower Bound. Hsiang claims that volumes of certain local cells will always be at least 4 v ~ - 0.07. This local cell has a smaller volume. Ten of the spheres are equally spaced around two circles of colatitude ~b= lr/3 and ~b=2.04. The other two spheres are at the poles. 52 THE MATHEMATICALINTELLIGENCERVOL.16, NO. 3, 1994

Problems

Analyzing Hsiang's lemma (Lemma 11) in further detail, we can show that either he did not perform the calculation he said he did or he made a major error in performing it. He claims without proof that the configurations just mentioned--those with exactly 11 close neighbors and 1 other neighbor--are easily dismissed. (By definition, one sphere is close to another if their centers are at most 2.18 apart. Two spheres are neighbors if they have adjoining local cells.) He states that in this case "it is quite straightforward to combine the volume formulas of the truncated tangent subpolyhedrons and the volume esti10 H s i a n g gives an additional constraint related to buckling heights. The c o u n t e r e x a m p l e satisfies this extra constraint.

mation lemmas of w to s h o w that the v o l u m e of the core part (of the resulting local cell) already exceeds 4 v ~ . " The truncated tangent subpolyhedron, defined b y Hsiang, is a E convex p o l y h e d r o n closely related to the local cell, w h o s e v o l u m e is easier to estimate than that of the local cell. Because he omits the proof of this "straightforward" claim, we repeat the a r g u m e n t ourselves. If we use the m e t h o d s he says he used, w e get an answer that is n o w h e r e near the 4 v ~ b o u n d he gives. We try his m e t h o d on a parti4ular configuration with 11 close neighbors a n d I other neighbor. We place the center of a central sphere at the origin. The 11 close neighbors are spheres centered at points pl,. 9 pll, at distance 2 from the origin. The other neighbor has its center at a 12th point p12 at distance 2.18 from the origin (or at 2.18 + e, if we wish to make the distance strictly greater than 2.18). The 12 points are located approximately at the centers of the faces of a dodecahedron. We determine their precise position b y Figure 8, s h o w i n g the separation of pairs of points. The point p12 is not shown. The v o l u m e of the local cell is s o m e w h a t larger than Figure 8. A Methodological Error.This configuration depicts 4v~. Let us a p p l y the m e t h o d of v o l u m e estimation that a local cell whose volume cannot be accurately estimated by he used to give a lower b o u n d of 4v~. We are instructed the techniques developed by Hsiang. to use the " v o l u m e formulas of the truncated tangent s u b p o l y h e d r o n s " and his v o l u m e estimation lemmas (of ods to establish a b o u n d is incorrect. It means that he his w According to Hsiang's definition of truncated tangent subpolyhedron, p12 is replaced b y a point P~2" really offers no proof of the lemma, and that his "set of a point representing the center of a unit sphere resting convenient tools for the proof of Theorem 2 [the Kepler conjecture]" remains unsubstantiated. 12 on three of the other unit spheres. The truncated tangent s u b p o l y h e d r o n is a cell obtained after replacing the nonclose neighbor/912 with P~2, whose separation from other points is s h o w n in square brackets in Figure 8. Note that R e d u c t i o n s to Critical Cases three of the distances are 2, corresponding to the three At times, Hsiang proposes a reduction a r g u m e n t to jusspheres on w h i c h the n e w sphere rests. The n e w cell, w h i c h contains the truncated tangent tify his treatment of only a few special cases. Consider his polyhedron, has a v o l u m e slightly less than 5.6131, which S u b l e m m a 7. He quickly excludes all but a few special on the scale relevant to this problem is considerably less cases with the claim "By means of size decreasing dethan 4 v ~ ~ 5.6568. The other estimation techniques that formations, one m a y reduce the above proof to the more he cites are not relevant for the particular example we critical case" [7], p. 774. Is it really possible to reduce to his critical case b y means of size-decreasing deformations? have constructed. So in this case, Hsiang's m e t h o d s yield 5.6131, not 4x/2 as claimed. The importance of this example is that it escapes the 12The unsubstantiated results of Section 11 are used throughout the m e t h o d s and techniques that Hsiang has introduced to following two sections--the two sections giving the "proof" of the solve the problem. ~1 His claim to have used these meth- Kepler conjecture. In these two sections he does not explicitly carry out computations of volumes of local cells, relying extensively instead on the estimates of Section 11. For example, note the flow of his argument 11Another example of methodological difficulty is his claim to have in Case/4: "The same proof as that of Lemma 11 will show that the found a particular volume estimate of certain slabs. He writes, "The volume.., is at least equal to 4v'2 - 0.015 .... It is quite easy to show that the volume of the central local cell is also at least 4V'2 - 0.015. above estimate implies that the volume is at least 1.06 times hj if hj < 0.052 and it is at least equal to hd if 0.052 < hj ~ 0.09" [7], p. 757. If Therefore, the total volume of thirteen local cells is at least equal to ... 52V'2+ 0.55." we turn to his "above estimate," we find that it is For another example, consider the reliance on Section 11 in Case IV [7], p. 825. "It follows from the results of w that the volumes of the (~tan5) (hj-~h2)" modified local cells of the former type are at least equal to 4v~ - 0.015 This is less than hd --not at least hj as he claims--for hj > 0.076. while that of the remaining ones are at least equal to 4v'~ - 0.065. (An earlier preprint contains the incorrect estimate of 1.1hj .) The esti- Hence, the possible volume deficit is at most equal to 0.345 while the mate that the slab is at least hj is used throughout the article. Perhaps previous contribution is at least 0.4086. This proves Theorem 2 [the the slabs do have volumes at least hj, but this does not follow from Kepler conjecture] for the case of core packings with thirteen close neighbors." Hsiang's stated method of estimation. THEMATHEMATICALINTELLIGENCERVOL.16,NO.3,1994 53

Suppose that we attempt to salvage his argument. If several objects do not fit into a particular region, then they cannot fit into a subregion. So a rigorous argument could be made by taking size-decreasing deformations to mean contractive deformations. Unfortunately for Hsiang, specific examples show that contractive deformations do not exist in the given context.

\j\J

\J

Nonexistence of Contractive Deformations

Continuing our analysis of Sublemma 7, let us consider one particular configuration not on his list of critical cases and show that it does not admit a nontrivial contractive deformation. We give the eight points p0,..., p7 on the The problem he has posed is this. Is it possible to ar- surface of a unit sphere, arranged so that their convex range 13 points on a unit sphere subject to the 2 con- hull forms a polyhedron with seven faces around a given straints that (1) the angular separation of each pair of vertex p0. By showing that it does not admit a contracpoints is at least 0.98 radians and (2) the convex hull of tive deformation, we show that, even with sympathetic the 13 points is a polyhedron with the property that 7 misreading, Hsiang's argument remains flawed. of its faces meet at a single vertex? Hsiang says it is not Figure 10 illustrates the eight vertices forming seven possible. His conclusion may be correct. But this much is faces. The nine edges marked with a dot each have length clear: It is impossible to reduce to his special case as he 0.98 radians. These are the edges ~vi, pj] for claims. (i, j) = (5, 4), (4, 3), (3, 2), (2, 1), (7, 6), (6, 5), By relying on size-decreasing deformations, he repeats an error for which he was duly criticized. Ian Stewart, (0, 6), (0, 3), and (0, 1). (1) reporting some of the early errors exposed by Conway, stated, "Hsiang's earliest preprint definitely contained We also assume that the pentagon (pop3p4p5p6) is reguerrors .... An equally fallacious claim made by Hsiang lar, that the quadrilateral (poplp2p3) is a square (meaning was that if several objects do not fit into a particular re- equiangular), and that pl lies on the circumscribing cirgion, then they cannot fit into a region of smaller area" cle of the triangle (POP6P7).These conditions determine (see Figure 9) [11]. the arrangement of points. The remaining edge [Pl, pT] Figure 9. A Fallacious Claim. The two disks cannot fit inside a 3 x 3 square but can fit inside a 2 x 4 rectangle of smaller area.

Figure 10. A Rigid Star. Arrangement (A) of eight points on the sphere does not admit a contractive deformation. The back side (B) of the same arrangement shows that the complement of the union of disks of radius 0.98 centered at the eight points is a pentagonal region with extreme points at ri~. 54

THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 3,1994

can then be shown to have approximate Euclidean length 1.268, or 1.374 radians. We choose the point r q to be the point on the unit sphere farthest from p0, located at 0.98 radians from the points p~ and pj. We take an arrangement that is already v e r y close to Hsiang's set of critical cases. Nevertheless, it does not admit a nontrivial contractive deformation that w o u l d bring it to one of the critical cases. By contracting the edge ~Vl~ P7] to 0.98 radians (without increasing the lengths of the edges already of length 0.98), w e w o u l d bring the arrangement to a critical case. However, w e cannot simply pivot p7 a r o u n d the vertex P6 toward Pl because such pivoting is not contractive. (It cuts off a corner of the c o m p l e m e n t a r y region at 1"17.) To be precise, given an arrangement A of eight points p 0 , . . . , P 7 arranged to form seven faces a r o u n d P0, we let C(A) be the u n i o n of the disks of radius 0.98 radians placed at each of the points Pi. In the cases w e study, C(A) will be simply connected. We say that a deformation A' is contractive if C(A') c C(A). We take our deformation to be a collection of continuously differentiable curves {pi (t)} on the unit sphere, with pi (0) = pi. We assume that the curves define an arrangement At satisfying the constraints of the problem for all sufficiently small t. Ifi particular, the given edges must each have length at least 0.98 radians, and the points must form seven faces a r o u n d the vertex p0. We say that the deformation is nontrivial if p~(0) • 0, for some i. L E M M A 2. The arrangement given above does not admit a nontrivial contractive deformation.

Proof (Sketch). We write d o w n the system of inequalities that a contractive deformation gives. We obtain nine inequalities from the condition that the edge Lvi(t), pj (t)] is at least 0.98 radians for (i, j ) m a r k e d as above. Taking (i, j ) = (0, 1), which is the ninth pair in Eq. (1), w e have [[P0 ( t ) -- P l (t)I[ 2 :

For example, we have I]p5(t) - r54112 = 4 sin2(0.98/2) + 510(t) for some non-negative function 210(t). The derivative gives the linear relation -2p~(0) 9?'54 ~- Zl0, a relation a m o n g the variables x5, ys, and zlo. We run through the indices (i, j ) in the order given b y Eq. (2) and introduce non-negative variables z m , . . . , z19. There are four additional relations that come from the condition that the convex hull of the points is a p o l y h e d r o n with seven faces meeting a r o u n d the vertex po(t). For example, the convexity condition for the faces (po(t)pl (t)p2(t)) and (po(t)p2(t)p3(t)) is that the origin and p3(t) lie on the same side of the plane passing through po (t), pl (t),and P2 (t). Otherwise stated, det(po(t) - p l ( t ) , po(t) - p 2 ( t ) , po(t) - p 3 ( t ) ) = 220(t) for some non-negative function 220(t). Our initial conditions on pi imply that z20 = 2~0(0) _> 0. We linearize this and the remaining three constraints coming from the faces (po (t)pi (t)pj (t)) and (po (t)Pd (t)pk (t)), for (i, j, k) = (6, 7, 1), (3, 4, 5), and (4, 5, 6). We obtain 23 h o m o g e n e o u s linear relations a m o n g the 16 + 23 variables xi, yj, zk, for i = 0 , . . . , 7, j -- 0 , . . . , 7, and k -- 1 , . . . , 23. They d e p e n d on the choice of vectors ei and e~, which are not given without first specifying coordinates for Pi. Using these relations we m a y eliminate the variables xi and yi b y expressing them in terms of the variables zi. These relations also give seven linear relations a m o n g the variables zi. These are linear relations we will label e q l , . . . , eq7. We also show an equation eq8, which m a y be described as the unique linear combination of eql, 999 eq7 in which the coefficient of z5 is 1, and which does not involve any of the variables zi, for i -- 9, 10, 13, 15, 18, 19.

eql:

0 = - z l + 1.101z10 + 0.763zu - 1.018z12 - 1.194z19,

eq2:

0 = -1.0z4 - 0.935z13 + 0.309z14 + 2.168z15

eq3:

- 2.037z16, 0 = -1.3099z2 - 1.0z3 - 1.0z5 - 1.177z6 - 1.3099z7 - 1.177z8 - 1.0z9 - 1.442z10

4 sin 2 (0.98/2) + 59 (t)

for some continuously differentiable function 29(t) satisfying zg(t) ~ 0, for t sufficiently small, and 29(0) = 0. Passing to an infinitesimal deformation, w e can linearize this equation. Set zi = 2~(0). We have z9 _> 0. Set p~(0) = xiei + yie~, where ei and e~ are linearly i n d e p e n d e n t vectors in the plane tangent to the unit sphere at pi. The derivative of our equation is -2p~(0) "p1-2p~ (0) "p0 = zg. This is a linear relation a m o n g the variables x0, Xl, Y0, Yl, and zg. Continuing in this manner, w e obtain nine homogeneous linear equations, b y running through the indices (i, j), in the order given in Eq. (1), and b y introducing additional non-negative variables z l , . . . , z9. The next 10 constraints come from the conditions that pi(t) and rij are separated b y at least 0.98 radians [i.e., rij is not in the interior of C(At)] for

- 1.427zn + 0.535z12 - 0.077z13 - 1.001z14 - 0.847z15 - 0.169z16 - 0.965z17 - 0.391z18 + 0.563z19,

ecl4:

0 = -0.660z2 - 0.207z3 - 0.297z8 - 0.297z9 - 0.7197zn + 0.270z12 + 0.284z13

eqs:

- 0.002z14 q- 0.141z15 - 0.705z16 - 1.0z20, 0 = +0.361z2 + 0.275z3 - 0.615z6 - 0.162z7 + 0.324z8 - 0.247z9 - 0.753z10 + 0.393zn - 0.147z12 + 0.021z13 + 0.276z14 - 1.04z15

(i, j ) = (5, 4), (4, 5), (4, 2), (2, 4), (2, 1), (1, 2), (1, 7), (7, 1), (7, 5), (5, 7).

+ 0.653z16 + 0.088z17 + 0.384z18 + 0.294z19 (2)

-

1.0z21,

THEMATHEMATICAL INTELLIGENCER VOL.16,NO.3,1994 5 5

eq6:

0 = --0.150z2 -- 0.743z3 -- 0.052z8 + 0.388z9 -- 0.063z10 + 0.105zn + 0.563z12 -- 0.0575z13 -- 0.744z14 q- 0.329z15 + 0.0655z16 -- 0.364z19 -- 1.0z22,

eq7:

0 = +0.766z2 + 0.585z3 + 0.823z6 + 0.823z7 + 0.688z8 + 1.017z9 + 1.111z10 + 0.668zn -- 0.623z12 + 0.045z13 + 0.585z14 + 0.862z15 + 0.171z16 + 0.194z19 -- 1.0z23,

eq8:

0 = +0.757zl + 0.633z2 + 0.579z3 + 0.233z4 + 1.0z5 + 0.757z6 + 0.429z7 + 0.262z8 + 0.171zn + 0.622z12 + 0.329z14 + 0.340z16 + 0.875z17 + 0.874z20 + 1.02221 q- 0.565z22 -t- 1.269z23.

in extending an arrangement of spheres, to fit the holes or pockets of the arrangement with additional spheres. His argument relied on this hypothesis repeatedly, even though no proof was offered. For example, in Figure 11, suppose that we start with the initial arrangement of seven spheres (circles) marked 1 , . . . , 7. Hsiang's hole-fitting hypothesis then states that the tightest way to arrange spheres in the second layer of the packing is to fit each hole with a sphere. A hole in this two-dimensional example is filled by placing a new sphere tangent to two spheres of the original arrangement. (In three dimensions, holes are filled by placing a sphere tangent to three other spheres.) Thus, we place spheres 8, 9, and 10 in the holes. We then place additional spheres in the remaining holes of the original arrangement. Once the holes are filled, additional spheres 11 and

Sixteen-digit precision was used in these calculations, although less is shown. We have the non-negativity constraints zi ~ 0, for all i. As all of the nonzero coefficients of equation eqs are positive, and as the variables are also non-negafive, we conclude that the variables z~ are all zero, except possibly for z9, Zl0, z13, z15, Z18, and Z19. The equations e q l , . . . , eq6 now become a homogeneous system of six equations in six unknowns. The determinant of this system is nonzero (approximately -0.498), so the remaining variables must also be zero. This proves that the deformation must be trivial, as claimed. This completes the proof. What is the significance of this negative result? Hsiang's early preprint omitted the argument for sevenfaced polyhedra; it merely remarked that "it is easy to see that no vertices.., have more than six forks." (The number of forks is the number of edges or faces surrounding the vertex.) The fact that this much analysis was required to study a single arrangement shows that those who challenged his "easy to see" claim had more than ample justification for doing so. He claims to use deformation arguments, and deformation arguments (properly developed), even if linearized, require the solution to large systems of equations. His packing bounds are dependent on this result. In later arguments he uses case-by-case arguments that list all relevant polyhedra with only four, five, or six faces around a given edge. Hence, we must put all his later conclusions on indefinite hold. One is left to conclude that his hasty reduction has no real substance to it and that his critical case remains an isolated test case. The Hole-Fitting Hypothesis

An obvious flaw in Hsiang's early preprints on Kepler's conjecture was his "hole-fitfing" hypothesis. A hole is a place in a partial packing where an additional sphere may be placed tangent to three spheres of the partial packing. He claimed that it was best, whenever possible, 56 THEMATHEMATICAL 1NTELLIGENCERVOL.16,NO.3,1994

Figure 11. A Counterexample to the Hole-Fitting Hypothesis. Hsiang argues that the best way to extend a packing is to place sphere 9 so that it touches spheres 3 and 4. The second diagram shows that a better arrangement is possible if sphere 9 is moved away from sphere 4.

12 should be set in place to produce the smallest possible local cell around spheres 3 and 4. The placement of sphere 11, equidistant from spheres 8 and 9, is dictated by the positivity of the second derivative of the tangent function on the interval 0r/3, 7r/2). In 1992, I objected strenuously to his repeated use of this hypothesis and presented him with the following counterexample. We begin with the same initial arrangement of spheres 1 , . . . , 7 and fill in holes 8 and 10 as before. But instead of following Hsiang's prescription with sphere 9, we shift it slightly to position 9'. The other two spheres, 11 and 12, must also be adjusted slightly to 11' and 12'. All other spheres are placed in a manner identical to the arrangement Hsiang would give. If we calculate the total area of the resulting local cells of spheres 1,..., 7, we find that the total area is smaller in this example than we have following Hsiang's hypothesis. 13 Thus, it is not true in general that the best w a y to extend an arrangement is by fitting the holes with additional spheres. In Hsiang's final revision, he has done his best to eliminate his hole-fitting hypothesis from his argument. 14 Yet the published version continues to invoke the hypothesis (see Case 13 [7], p. 818). But if the principle is not sound, heIs not justified in invoking it at all. 15

13 The areas of the cells around spheres 1 , . . . , 7 are easily compared. The cells around spheres 1, 2, 5, 6, and 7 are identical in the two cases. The cells around sphere 4 are mirror images of each other. Finally, the cells around sphere 3 are different because the spheres 4, 9, 11, and 8 are equally spaced around sphere 3 when the hole-fitting hypothesis is not followed. The positivity of the second derivative of the tangent function on the interval 0r/3, ~r/2) shows that the equally-spaced cell has smaller volume. 14 Hsiang, trying to recover from his error, has argued that although he uses this principle heuristically, his actual proofs never rely on it. (This argument was made by Hsiang in a lecture to the Mathematics Department at the University of Chicago, March, 1992.) Such a revisionist position is contradicted by numerous passages in the second part of his 1991 preprint (see [6], pages 6, 13, 57, 59, 60, 61, 62, 64, 67). He espoused the view that it is always best to fill in all the "big holes" whenever it is possible to do so without obstacle. The 1991 argument uses this principle (and not just heuristically!) in determining secondlayer arrangements of "completely separable configurations" and in the determination of second-layer arrangements of 5D-configurations. 15 In his 1993 publication, Hsiang continues to cling to this incorrect principle. In an attempt to justify the hypothesis, he argues, "Since the amount of non-touchingness of the above five added spheres will create at least that amount of additional volume to the peripheral parts while the possible total amount of decrements in the volumes of core parts of the local cells of that ten spheres is much smaller, the total effect of having non-touching extensions on the top of the above five big holes is strongly volume increasing" [7], pp. 818-819. He is handwaving here. This defense of the hole-fitting hypothesis, if accepted, would also prove the nonexistence of our counterexample. The flaw of this argument becomes apparent if we try to apply it to our counterexample of Figure 11. (We reproduce the relevant part of the second configuration of Figure 11 in Figure 12.) His argument emphasizes the large gap between spheres 4 and 9 in Figure 12. This is what he calls the "additional volume to the peripheral parts." His claim is that the improved spacing ("the possible total amount of decrements in the

Figure 12.

Detail

of Figure 11.

Hsiang's Contribution to the Subject If I have given the impression that Hsiang's work makes no worthwhile contribution to the subject of sphere packings, then I have given the wrong impression. Here is a brief summary of some of the contributions Hsiang has made to the subject. As mentioned in an earlier section (p. 51), Hsiang attempts to carry out Fejes T6th's program. Hsiang has improved the method by adjusting Fejes T6th's system of weights and the associated parameter #. The problem is then to prove the fundamental inequality of Lemma 1. He has offered a strategy, backed by empirical evidence, to prove the inequality. He says that we should make a systematic study of the volumes of the local cells Ci and that the local cells will have volumes greater than 4v~, except under very special circumstances. He has given an outline of an argument for the difficult case of cells with 13 faces. By studying the buckling heights and gully widths one can hope to quantify the interplay between the volume of one local cell and its neighboring ones. Hsiang's work gives evidence that a proof of the fundamental inequality will be difficult, but not impossible. There are other ideas, well known within the field, that Hsiang has extended. The strategies of truncation and the estimation of buckling heights, as well as their role in establishing packing bounds, are due to Fejes T6th. Fejes T6th's truncation is cruder than the one introduced by Hsiang. But strictly speaking, Hsiang's "truncated tangent polyhedron" is not a truncation at all, and this raises

volumes of core parts") of the spheres around sphere 3 is not enough to offset the increase in volume created by the gap between spheres 4 and 9. What Hsiang has failed to notice is that by moving spheres 4 and 9 apart, another surrounding sphere (12') is able to move closer to sphere 4, creating a net decrease in volume. By concentrating his attention on the changes in volume in a small neighborhood of the hole, he forgets that a small shift in one sphere allows all the spheres of the arrangement to shift and settle, and this can completely change the optimal placement of spheres far away from the hole under consideration. THE MATHEMATICAL1NTELLIGENCERVOL.16,NO. 3, 1994 57

subtle technical difficulties that await a proper resolution (see [7], p. 764). The strategy of studying the problem of 13 spheres by making explicit estimates of the minimal area of stars with n forks, for n ~ 6, goes back to Hoppe. But as we mentioned, Hoppe's argument is based on a phantom triangulation. Hsiang improves the method by using a reliable triangulation.

Conclusions In the end, I feel that Hsiang has missed the point of the subject of sphere packings. Many packing problems have geometrically intuitive solutions. One is offered a geometrically plausible arrangement that is claimed to be optimal. By performing some rough calculations, one rapidly sees that any other arrangement will almost certainly be worse. The difficulty of these problems stems from the fact that there is an unwieldy number of possible configurations and that uniform estimates are rarely found. One must devote an inordinate amount of energy to exclude families of configurations that seem most implausible to the imagination. Arguments in this field of mathematics would be far easier if the implausible configurations could be dismissed without proof. But rigor requires that proofs be given. One of the most unsettling aspects of his article is his deliberate and persistent use of methods that are known to be defective. The errors in his hole-fitting principle and his size-decreasing deformation were pointed out to him some time ago. His claims over the last 3 years that the next revision will answer all objections have grown tiresome. In conclusion, I offer a suggestion. First, Hsiang should withdraw his claim to have resolved the Kepler conjecture. Mathematicians can easily spot the difference between handwaving and proof. Then, Hsiang should isolate the statements in his article that he was unable to prove rigorously. He should show carefully how the Kepler conjecture would follow from these statements. In this way, his work would make an important contribution to the field. It would provide a concrete program that could eventually lead to a solution to the problem. Instead, by presenting experimental hypothesis as fact, he destroys the credibility of his own work.

Acknowledgments I would like to thank J. H. Conway, D.: J. Muder, and N. J. A. Sloane for many helpful comments.

3. L. Fejes T6th, Lagerungen in der Ebene auf der Kugel und im Raum, Berlin: Springer-Verlag, 1953. 4. David H. Freedman, Round things in square spaces, Discover, 13(1) (January 1992), 36. 5. S. Giinther, Ein stereometrisches Problem, Arch. Math. Phys. 57 (1875), 209-215. 6. W.-Y. Hsiang, On the density of sphere packings in E 3, II-- The proof of Kepler's conjecture, Center for Pure and Appl. Math. University of California, Berkeley, preprint PAM-535, September, 1991. 7. W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler's conjecture, Int. J. Math. 4(5) (1993), 739831. 8. W.-Y.Hsiang, personal communication, Letter to T. Hales, March 3, 1992. 9. I. Stewart, The kissing number, Scientific American 256(2) (Feb. 1992), 112-115. 10. I. Stewart, Mathematics, 1992 Yearbook to the Encyclopaedia Britannica, 1992. 11. I. Stewart, Has the sphere packing problem been solved?, New Scientist 134 (2 May 1992), 16. 12. I. Stewart, The Problemsof Mathematics, 2nd ed., New York: Oxford University Press, 1992.

Bibliography 1. C. Bender, Bestimmung der gr6ssten Anzahl gleich grosser Kugeln, welche sich auf eine Kugel von demselben Radius, wie die (ibrigen, auflegen lassen, Arch. Math. Phys. 56 (1874), 302-306. 2. A.H. Boerdijk, Some remarks concerning close-packing of equal spheres, Philips Res. Rep. 7 (1952), 303--313. 3. B. Cipra, Music of the spheres, Science 251 (1991), 1028. 4. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 2nd ed., New York: Springer-Verlag, 1993. 5. L. Fejes T6th, Uber die dichteste Kugellagerung, Math. Z. 48 (1942-43), 676-684. 6. R. Hoppe, Bemerkung der Redaction, Arch. Math. Phys. 56 (1874), 307-312. 7. W.-Y.Hsiang, On the density of sphere packings in E 3, I, preprint, 1990. 8. W.-Y.Hsiang, On the density of sphere packings in E 3-Kepler's conjecture and Hilbert's 18th problem, preprint, 1990. 9. W.-Y.Hsiang, Sphere packings and spherical geometry-Kepler's conjecture and beyond, Center for Pure and Appl. Math. University of California, Berkeley, preprint PAM528, July 1991. 10. W.-Y.Hsiang, On the density of sphere packings in/!;3, I, Center for Pure and Appl. Math. University of California, Berkeley, preprint PAM-530, August 1991. 11. J. Leech, The problem of the thirteen spheres, The Mathematical Gazette 40(331) (Feb. 1956), 22-23. 12. D.J. Muder, A new bound on the local density of sphere packings, Discrete Comp. Geom. 10 (1993), 351-375. 13. K. Schiitte and B. L. van der Waerden, Das Problem der dreizehn Kugeln, Math. Annalen 125 (1953), 325-334.

References 1. B. Cipra, Gaps in a sphere-packing proof?, Science 259 (1993), 895. 2. L. Fejes T6th, Regular Figures, New York: MacMillan, 1964. 58

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 3,1994

Department of Mathematics University of Michigan Ann Arbor, M148109-1003 USA

Jeremy Gray*

Otto Hi lder and Group Theory The German mathematician Otto H61der wrote his major papers on the structure of finite groups during the years 1888-1895, leaving his name on the Jordan-H61der theorem. But his achievements in that area were much greater, and they have recently been the object of a penetrating study by Dr. Julia Nicholson [1]. Some of her work has already been published and it is to be hoped that more willJcome out; it is a pleasure to report on it here. H61der was born on 22 December 1859 and studied engineering at the Stuttgart Polytechnikum in 1877 for a year before going to Berlin, as every aspiring young German mathematician did at that time. Carl Runge, a fellow student of H61der, recalled much later that Weierstrass's lectures left a deep and lasting impression, even though they were not polished and well constructed. Weierstrass would sometimes get in a muddle improvising a proof, only to put it right imperturbably next time. But Weierstrass was a sympathetic tutor, who listened attentively to his students and really responded to the question, unlike Kronecker, who could not be made to listen but always changed the subject straight away to talk about his own work. On the other hand, Kronecker was a far more approachable person, and many young people would be invited to his hospitable home [2]. Otto's son Ernst later wrote that his father's interest in algebra was primarily due to Kronecker, but there are reasons to doubt this. The year at Berlin seems to have influenced H61der considerably. When, as a young postdoctoral student working on his Habilitation, he found himself at Leipzig with Klein, the clash of styles prevented them from working together. H61der's interests were still in function theory (his discovery of the H61der inequality dates from this period). The two men met again, in 1886 in G6ttingen, where Klein lectured on the Galois theory of equations, and by then H61.der's interests seem to have been moving in an algebraic direction. The Faculty at

* C o l u m n Editor's address: Faculty of Mathematics, The O p e n University, Milton Keynes, M K 7 6AA, England.

G6ttingen soon wished to offer H61der an assistant professorship, but the Prussian Ministry of Culture, which oversaw all such appointments, vetoed it on the grounds of H61der's limited teaching experience. Instead, H61der took a post at T/ibingen, where he had taken his doctorate; or rather, he was offered such a post in May, but to the university's surprise the reply came not from Otto but his brother Eduard, a professor of law there. Otto had been admitted to a clinic in Erlangen suffering from some sort of mental collapse. The Faculty agonised over what to do, but courageously stuck to the view that since the post had been offered informally they had better go through with it and hope that H61der would make a speedy and satisfactory recovery. They made a formal offer of a position on 31 August 1889, which was accepted, and H61der's mental state improved steadily until he was able to give his inaugural address in June 1890. T/.ibingen was to be well repayed for its confidence in H61der; Ernst H61der called the years his father spent there the most fruitful of his father's life. Throughout those years H61der devoted himself to group theory. The starting point was the Galois theory of equations. The standard sources for this by then were Jordan's Traitd des substitutions (1870) and Eugen Netto's Substitutionentheorie (1882). In these presentations, what we arguably, following H61der, call the Galois group is defined as the group of permutations of the roots of the given equation. A central result was Abel's theorem, which states, in modern language, that there is a chain of radical extensions of the smallest field containing the coefficients (thought of as indeterminates) that culminates exactly in the splitting field. The implied contrast is with the case of equations with constant coefficients. There it was known that the radical extension might overshoot the splitting field, and that to obtain the splitting field it might be necessary to adjoin certain roots of unity, which Klein called auxiliary irrationals; as H61der was to show, adjoining them does not alter the Galois group of the extension.

THE MATHEMATICALINTELLIGENCERVOL.16, NO. 3 (~)1994Springer-VerlagNew York 59

Otto Hi~lder

H61der regarded the gap between radical extensions and splitting fields as a defect in the theory. To analyse it, he took up a suggestion that Klein had made to him and proposed a new, but equivalent, definition of what it is for an equation to be solvable by radicals. Jordan had offered a definition along these lines: An algebraic equation is solvable by radicals over a field K if its splitting field is contained in a radical extension of K. H61der proposed that an algebraic equation is solvable by radicals over a field K if its splitting field is contained in field Kn, where K = Ko < K 1 < ...