No title

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

9Ethics of Mathematics. Reuben Hersh's article "Mathematics and Ethics" (Mathematical Intelligencer, Summer 1990) mentions only trivial ethical problems in pure mathematics. He misses the big one. Mathematical research occupies the lives of a large proportion of the world's best thinkers, w h o could have been doing some very useful things otherwise. Either the theorems they discover have an intrinsic value comparable to medical discoveries, say, or inventions in telecommunications, or they don't. If not, it must be ethically unjustified for mathematicians to spend their lives finding them, and positively criminal to corrupt the youth by attracting them into the discipline. If theorems do have value in themselves, it w o u l d be good to say so and stop selling the subject to the public on such predominantly utilitarian grounds. James Franklin School of Mathematics Umverslty of New South Wales Kensington 2033 Austraha

Fractal Theory I would like to suggest an Hegelian solution to the controversy between Steven Krantz and Benoit Mandelbrot (Mathematical Intelligencer, Fall 1989) about the significance of fractal theory. Mathematics always comes with tension between the pure and applied, the abstract and concrete; every mathematician must choose some position on this board. Fractal theory has its roots in the mathematics that was fashionable in the first quarter of our century. It involves a mixture of several fields developed in that period: dynamical systems, geometric function theory, Brownian motion, Lebesgue integral, etc. Looming in the background are the nineteenth century ideas of Felix Klein and Sophus Lie that the "symmetries" of a situation determine the fundamentals. 4

Mandelbrot was himself trained in the tradition of very abstract, pure French mathematics. He realized that what he dubbed fractal theory was an important applied complement to this tradition. Perhaps this "rebellion" against the Bourbaki dogma is his own Hegelian antithesis? Unfortunately, the mathematicians at the major mathematical centers who dominate the agenda are, by and large, not sympathetic to research like his that tries to combine ideas from different parts of the board; he had to be theatrical to overcome deeply ingrained prejudices. Krantz is also right in pointing out that most of the major ideas were in existence and the theorems were proved by the 1920s. In his earlier writings, Mandelbrot mentioned his intellectual debt to the ideas of Paul L6vy, the mathematical poet of Brownian motion theory (where m a n y of the beautiful intuitive ideas in the fractal work originated!), but that has tended to be forgotten in the recent popularizations. What is most important about this controversy is its lessons for the future interaction between mathematics and other disciplines, and the related educational questions. I cannot go into it here in detail; I am already infamous in certain circles for writing and preaching ad nauseam about this topic. However, I cannot resist pointing out that the most important moral of the fractal story is the opposite of the one that is implicitly assumed in the general scientific p r e s s - and by government agencies looking for something to fund that will lead to important applications or catch the public's e y e - - n a m e l y , that the concrete and pragmatic parts of mathematics are of most importance for practical results. Fundamental physics illustrates the principle in its purest form: relativity, quantum mechanics, and elementary particle physics are based on differential geometry, Hilbert space, and Lie group theory, not exactly subjects a student is likely to learn playing glorified video games on his or her PC. Even computer science--for all its triumphs of building better machines and more efficient software--is still tied to seemingly useless ideas generated by a few

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 19 1991 Spnnger-Verlag New York

mathematical logicians sitting around the Common Room at Princeton in the 1930s. I suggest that our role as mathematicians is to provide the mathematical fuel for new science! Robert Hermann 53 Jordan Road Brookhne, MA 02146 USA

.Poetry Neither in the articles of Jonathan Holden and W. M. Priesfley (Mathematical Intelligencer, Winter 1990), nor in the ensuing correspondence has anyone yet mentioned the work of the contemporary German poet, Hans Magnus Enzensberger. His Mausoleum, subtitled "thirty-seven ballads from the history of progress," is a latter-day Spoon River Anthology. Unlike Masters's graveyard, however, Enzensberger's crypts are peopled by real historical personages, among w h o m are Leibniz, Babbage, and Turing. In a later collection of Enzensberger's poetry there is also his Hommage ?l G6del, in which the first incompleteness theorem is stated and playfully contrasted with one of the tales of Baron Miinchhausen. That poem, moreover, was subsequently set to music by Hans Werner Henze in his Second Violin Concerto! This raises a new question: Apart from Tom Lehrer's facetious "tribute" to Lobachevski, has any mathematician besides G6del been memorialized in music?

For those who do not read German, Mausoleum is available in English; the title and publisher (Suhrkamp) are the same. A performance of Henze's Violin Concerto is also available, on British Decca LP HEAD 5. The accompanying program notes include the full German text of Enzensberger's poem, together with English and French translations of it. John W. Dawson, Jr. Department of Mathematzcs Pennsylvania State Unwersity York, PA 17403 USA

STATEMENT OF OWNERSHIP, MANAGEMENT, AND CIRCULATION (Required by 39 U.S.C. 3685). (1) Title of pubhcatlon: The Mathematical Intelhgencer. A. Publication No.: 03436993. (2) Date of filing:. 10/1/90. (3) Frequency of issue: Quarterly. A. No. of issues published annually, 4. B. Annual subscription price, $32.00. (4) Location of known office of pubhcation: 175 Fifth Avenue, New York, NY 10010. (5) Location of the headquarters of general business offices of the publishers: 175 Fifth Avenue, New York, NY 10010. (6) Names and addresses of publisher, editor, and managing editor. Publisher:. Springer-Vedag New York Inc., 175 Fifth Avenue, New York, NY 10010. Editor:. Sheldon Axler, Department of Mathematics, Michigan State University, East Lansing, MI 48824. Managing Editor: Spnnger-Vedag New York Inc., 175 Fifth Avenue, New York, NY 10010. (7) Owner. Springer Export GmbH, Tiergartenstrasse 17, D-6900 Heidelberg, West Germany;, and Spnnger-Verlag Berhn, Heldelberger Platz 3, D-1000 Berlin 33, West Germany. (8) Known bondholders, mortgagees, and other security holders owning or holding I percent or more of total of bonds, mortgages or other secunties: none. (9) The purpose, function, and nonprofit status of this organmatlon and the exempt status for Federal income tax p u ~ : has not changed during preceding 12 months. (10) Extent and nature of circulation. A. Total no. copies pnnted (net press run): Average no. copies each Issue dunng the preceding 12 months, 6273; no. copies single issue nearest filing date, 6437. B. Paid circulation: 1. Sales through dealers and carders, street vendors, and counter sales: Average no. copies each issue during preceding 12 months, 180; no. copies single issue nearest to filing date, 180. 2. Mall subscriptions: average no. copies each issue dunng preceding 12 months, 4794; no. copies single issue nearest to filing date, 4983. C. Total paid circulation: average no. copies each issue during preceding 12 months, 4974; no. copies single issue nearest to filing date, 5163. D. Free distribution by mail, carder, or other means. Samples, complimentary, and other free copies: average no. copies each issue dunng preceding 12 months, 572; no. copies single issue nearest to filing date, 572. E. Total distribution: average no. copies each issue dunng the preceding 12 months, 5546; no. copies single Issue nearest to filing date, 5735. F. Copies not distributed: 1. Office use, left-over, unaccounted, spoiled after printing: average no. copies each issue dunng the preceding 12 months, 662; no. copies single issue nearest to filing date, 702. 2. Return from news agents: average number of copies single issue during preceding 12 months, 65; no. of copies single issue nearest to filing date, 0. G. Total: average no. copies each issue during preceding 12 months, 6273; no. copies single sssue nearest to filing date, 6437. I certify that the statements made by me above are correct and complete.

Ute Bujard Vice-President, Production

6 THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

Karen V. H. Parshall*

A Study in Group Theory: Leonard Eugene Dickson's Linear Groups Karen V. H. Parshall

In 1901, a twenty-seven-year-old American mathematician, Leonard Eugene Dickson, published his first book. Entitled Linear Groups with an Exposition of the j Galois Field Theory, it represented a revised and expanded version of his 1896 University of Chicago doctoral dissertation and came out under the auspices of no less a publishing house than the distinguished German firm of B. G. Teubner [1]. By thus securing the Teubner seal of approval, Dickson's work not only came before one of the most important segments of the international mathematical community at the turn of the century, but it also served to build its author's reputation at home and abroad. How different would the reception of Dickson's ideas have been had the book appeared not with Teubner but with an American firm? With no well-established scientific publishers in the United States around 1900, who would even have taken on the task of producing a technical, research-level mathematics text? And if an American press had assumed the responsibility, w h o m would Dickson's work have reached? The American mathematical community? Probably. But the strong and influential German mathematical public? Most probably not. Although just idle speculation given the historical facts of the matter, these sorts of questions would have shaped the present article had events not turned Dickson from his original publication plans. Instead, though, we trace a sequence of events, beginning at the University of Chicago, which led to Teubner's publication of what has been called "a milestone in the development of m o d e r n algebra," Dickson's Linear Groups [2].

* C o l u m n Editor's address: Departments of Mathematics and HIStory, Umverslty of Virginia, Charlottesville, VA 22903 USA.

Born in I n d e p e n d e n c e , Iowa in 1874, Dickson moved to Texas early on, where his father prospered as a merchant and real estate investor [3]. After getting his primary and secondary education in the public schools of his h o m e t o w n of Cleburne, the y o u n g Dickson proceeded to the University of Texas for his undergraduate education and came under the tutelage there of the dynamic George Bruce Halsted. Thus lured into mathematics, Dickson pursued an unusually strong curriculum that placed special emphasis, not surprisingly, on Halsted's primary areas of interest, Euclidean and non-Euclidean geometry. In 1893, he earned his Bachelor's degree as the valedictorian of his class and stayed on at his alma mater for one more year as a teaching fellow while earning his master's degree, again under Halsted's supervision. With this new credential in hand, Dickson applied for doctoral fellowships at both the venerable Harvard a n d t h e u p s t a r t U n i v e r s i t y of Chicago for the 1894-1895 a c a d e m i c year. W h e n H a r v a r d c a m e through with an offer, Dickson accepted, only to change his mind later w h e n the Chicago award also materialized [4]. Thus, rather than opting for a Harvard just coming into its own mathematically (thanks to the then recent appointments of Gottingen Ph.D.'s William Fogg Osgood and Maxime B6cher), Dickson chose a Chicago boasting the strongest mathematical faculty in the country [5]. Under the direction of Eliakim Hastings Moore and his colleagues, Oskar Bolza and Heinrich Maschke, the Chicago department easily covered all of the major areas of late nineteenthcentury mathematical research and provided its students with a well-rounded, state-of-the-art education. Furthermore, consonant with their university's overarching philosophy, these three men strove to train ind e p e n d e n t researchers capable of making original

THE MATHEMATICALINTELLIGENCERVOL 13, NO 19 1991 Spnnger-VerlagNew York

7

contributions to their chosen fields. The transplanted Texan thrived in this atmosphere and, after just two years in the program, earned one of Chicago's first two mathematics Ph.D.'s. Working under E. H. Moore's direction, Dickson picked up on and greatly expanded upon a topic that had actively engaged his adviser's research interests shortly before Dickson's arrival at Chicago. In his 1893 paper on "A Doubly-Infinite S y s t e m of Simple Groups" delivered to the Mathematical Congress of the World's Exposition held in Chicago in August of that year, Moore had presented his findings on a new class of finite simple groups [6]. As early as 1878, the groups of order (1/2)q(q 2 - 1) for q prime (which we now call PSL2(q) = SL2(q)/( +-1)), had arisen in work of Felix Klein [7]. By the fall of 1892, Moore had discovered a simple group of order 360 that did not fit into this or any of the other five known classes of finite simple groups; and by the spring of 1893, Frank Nelson Cole had uncovered yet another non-conforming simple group, this one of order 504 [8]. In his Chicago Congress paper, Moore succeeded in uniting these two new simple groups into a class of doubly-infinite groups of order (1/2)qn(q2~ - 1), for q prime and for (q,n) # (3, 1). He showed, in fact, that all of the groups in this expanded class (PSL2(q"), in modern notation) were simple [9]. As a prelude to this result, however, Moore opened his paper with a key theorem on the underlying finite field with qn elements. Specifically, he proved that "[e]very existent field F[s] is the abstract form of a Galois-field GF[q"]; s = q"" [10]. Following in the tradition stemming from the ideas of Evariste Galois, Moore defined a Galois field as follows: given an indeterminate X, take an irreducible monic polynomial fiX) zq[X] of degree n over Zq, the finite field with q elements. The Galois field GF[q"] is the collection of qn equivalence classes of Zq[X]/ff(X)). His isomorphism theorem then enabled Moore to bring the theory of Galois fields, as developed particularly by researchers like Joseph Serret and Camille Jordan, to bear successfully in his group-theoretic setting. Following in his adviser's footsteps, Dickson also sought to extend the theory of finite fields and to explore its connections with group theory. He began his dissertation by examining the following set-up: take a polynomial 6(X) ( F[X] of degree k < p~ in an indeterminate X, where F = GF[p"], p is prime, and n E Z +, and define a substitution (or mapping) ~b : F ~ F by 6(6). Dickson termed this mapping a "substitution quantic of degree k on pn letters," or SQ[k;pn], if, again in modern terminology, it was bijective. In the first part of his thesis, he completely determined the representations of degree k < 7 and obtained partial results in degrees 7 and 11 [11]. His dissertation's second part focused on the application of the representation-theoretic results of part 8 THEMATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

one to the theory of the general linear group GLn(F), where F = GF[pn]. As Dickson freely acknowledged, his work generalized that of Camille Jordan as presented in the Trait~ des Substitutions of 1870, for there Jordan limited himself to GL,(F), where F = GF[p] [12]. Dickson thus aimed to rework the theory at what he viewed as a more appropriate level of generality. In so doing, he managed to prove theorems like this one for SLm(F), where F = GF[p"]: if Z denotes the center of SLm(F), then SLm(F)/Z is simple except when (m,n,p) = (2,1,2) or (2,1,3) [13]. Clearly, this theorem pushed Moore's 1893 result on doubly-infinite systems of simple groups through to triply-infinite systems, and Dickson credited his adviser for the idea of pursuing such a line of research [14]. With his doctoral research completed, Dickson, like so m a n y aspiring American mathematicians of his era, decided to undertake post-doctoral studies abroad [15]. In light of his thesis research and his general group-theoretic bent, it is little wonder that he sought out Sophus Lie in Leipzig and then Camille Jordan in Paris during the academic year 1896-1897. Upon his return to the United States in 1897, Dickson took up an instructorship in the mathematics department at the University of California in Berkeley, a group s p e a r h e a d e d by the Sylvester- and Klein-trained Irving Stringham. Despite the relatively high level of mathematical activity in the San Francisco area at this time (as witnessed by the 1902 establishment of a San Francisco Section of the AMS), Dickson opted to leave California in 1899 for an associate professorship back at Texas. In the three years after his graduation from Chicago, Dickson had energetically followed through on the various lines of inquiry suggested by his thesis research and had published almost thirty articles on related topics. In addition, he had resolved to transform his dissertation into a research-level exposition for the mathematical public on linear groups and their underlying Galois fields. Still, this latter project proved difficult both on the West Coast and in Texas due to the scarcity of resources like books and journals. Writing to E. H. Moore from Texas in December of 1899, Dickson expressed his frustrations on this score. He wrote that "I am trying to get Wiman's paper. In case you have access to the Stockholm Journal, could you spare the preprint temporarily? At any rate, I should like to borrow it when you are through with the comparison. My task is growing all the time! But I shall not give u p - - e v e n if it takes a couple of years m o r e - - & mean to have a book worth publishing" [16]. Evidently, Moore and his colleagues at Chicago had no doubts that the manuscript, whenever it reached completion, would merit publication. In fact, so sure were they of their former student's abilities that, early in 1900, they invited him back to Chicago, but this time as an assistant professor on the permanent faculty.

In reading Dickson's prospectus, Klein learned that The book here announced proposes to treat of linear congruence groups, or more generally, of linear groups in a Galois field, a subject enriched by the labors of Galois, Betti, Mathleu, Jordan and many recent writers. It seemed desirable to give an elementary but exhaustive exposition of the theory of Galois fields (endliche Korper [sic]). In treating linear groups, I have not confined myself to the correllations [sic] of the results of published papers, but have succeeded in introducing marked simplifications. This is especially true in the study of linear groups on m radices defined by a quadratic invariant. Having determined the structure of the senary linear groups, that of the groups for m > 6 is derived in a very few pages and without the difficult calculations given in the published papers [18].

W h e t h e r inspired b y this offer (which he immediately a c c e p t e d ) or w h e t h e r f u r t h e r a l o n g w i t h his project than he h a d led Moore to believe at Christmas time, by the spring of 1900 Dickson p l a n n e d to submit his m a n u s c r i p t to a n A m e r i c a n p r e s s l a t e r t h a t summer. In discussing his options with his adviser, he was counseled by Moore to set his sights higher than an American publishing house and to write to Felix Klein in the h o p e s of securing a G e r m a n publisher. Taking Moore's advice to heart, Dickson p e n n e d this letter to Klein on April 7, 1900: Dear Sir: I am takang the liberty to send you the enclosed prospectus of a book on "The theory of linear groups, with an exposition of the Galois field theory." I should feel very much honored to have your opinion of the desirability of such a book. It is probable that I shall publish it this summer in America; but, at the suggestion of Professor Moore, it has occurred to me that, if your sympathies could be enlisted, it might be accepted by Herr Teubner for his Sammlung. If that prove feasable [sic], I would make no offer to an American house, but take the MS with me on my trip to Europe this summer and submit it to your detailed cnticism. Trusting that I may hear from you and that I may be pardoned for my intrusion, I am With high esteem, yours most f a i t h f u l l y . . . [17]

Given that Dickson's book did come out with T e u b n e r and not with an American firm, it seems safe to conclude that Klein's sympathies were i n d e e d enlisted as a result of his positive reaction to Dickson's prospectus. With the book's appearance in 1901, h o w ever, D i c k s o n ' s a p p r o a c h to linear g r o u p s , to the t h e o r y of Galois fields, and to the determination of the finite simple groups entered the wider public d o m a i n for scrutiny and evaluation, and the j u d g m e n t p r o v e d favorable. O n e of the most telling e n d o r s e m e n t s of Dickson's insistence on the p o w e r a n d p o t e n c y of linear g r o u p s as an algebraic tool came from his main English-lang u a g e c o m p e t i t o r , William B u r n s i d e . As D i c k s o n n o t e d in the preface to his work, "[s]ince the appearance in 1870 of the great w o r k of Camille Jordan o n substitutions and their applications, there have b e e n m a n y i m p o r t a n t a d d i t i o n s to t h e t h e o r y of finite groups. The books of Netto, Weber and Burnside have b r o u g h t u p to date the t h e o r y of abstract and substitution g r o u p s " [19]. Burnside's book, entitled Theory of Groups of Finite Order, a p p e a r e d in its first edition in 1897, j u s t f o u r y e a r s p r i o r to t h e p u b l i c a t i o n of Dickson's treatise. There, Burnside rather bluntly dismissed the entire subject that Dickson w o u l d choose to treat. H e explained: It may be asked why, in a book which professes to leave all applications on one side, a considerable space is devoted to substitution groups; while other particular modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with properties of substitution groups, it would be difficult to find a result that could be most directly obtained by the consideration of groups of linear transformations [20]. Judging from the 1911 second edition of his book, h o w e v e r , D i c k s o n ' s Linear Groups--and c e r t a i n l y Georg Frobenius' work on the t h e o r y of g r o u p representations a n d g r o u p characters p r o v i d e d Burnside with a n y n u m b e r of examples of results most effectively treated using linear g r o u p s [21]. Without menTHE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991 9

tioning any specific authors by name, Burnside had to admit that "[v]ery considerable advances in the theory of groups of finite order have been made since the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds" [22]. Leaving aside the admittedly important contributions of Frobenius that especially justified this last statement, just what sorts of results did Burnside and other interested readers find within the covers of Dickson's book [23]?

Although various facts regarding abstract fields and Galois fields had lain scattered about prior to Dickson's undertaking, his book represented the first systematic treatment of finite fields in the mathematical literature. First of all, Dickson presented a unified, complete, and general theory of the classical linear g r o u p s - - n o t merely over the prime field GF[p] as Jordan had done - - b u t over the general finite field GF[pn], and he did this against the backdrop of a well-developed theory of these underlying fields. Although various facts regarding abstract fields and Galois fields had lain scattered about prior to Dickson's undertaking, his book represented the first systematic treatment of finite fields in the mathematical literature. With this as his point of departure, Dickson then painstakingly explored the structure of the classical linear groups, determining, a m o n g other things, their normal subgroups and their orders [24]. He next worked out the isomorphisms existing between the various groups [25] and codified all of the findings in tables exhibiting the then-known simple groups of order less than a billion [26]. Of particular significance in these tables, he highlighted the two non-isomorphic groups of order (1/2)8! = 20160, namely, the alternating group on eight letters A 8 and the projective special linear group PSL3(4 ) [27]. Furthermore, he noted the isomorphism of As and PSL4(2). Thus, although two other pairs of projective special linear groups s h a r e d the same orders, that is, IPSL2(5)I = IPSL2(4)I = 60 and IPSL2(7)I = IPSL3(2)I = 168, the two groups in each pair are also isomorphic. From these facts two curious phenomena emerged: 1) the existence of three pairs PSLm(q), PSLm,(q') (where q and q' n o w d e n o t e p o w e r s of p r i m e s ) h a v i n g t h e p r o p e r t y t h a t IPSLm(q) I = IPSLm,(q')I while (m,q) ~ (m',q'); and 2) the existence of a projective special linear group with the same order as an alternating group [28]. 10

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

In two short papers published in the Communications in Pure and Applied Mathematics in 1955, Emil Artin masterfully settled not only the obvious open questions concerning the extent of these particular phenomena but also the issue of such concurrences of order among all of the finite simple groups known in 1955. Relative to the two types of coincidence apparent from Dickson's tables, Artin proved in his first paper that Dickson had indeed isolated the only three instances of the first sort as well as the only six cases of the second [29]. In his second paper, he established the more general result, namely, that the only coincidences among the orders of the finite simple groups (as of 1955) were precisely those Dickson had documented [30]. In addition to its immediate and obvious impact on the problem of classifying all of the finite simple groups, Dickson's book also drew out some of the intimate connections between finite group theory and geometry. Of special interest, Dickson situated the problem of determining the twenty-seven straight lines on a cubic s u r f a c e - - a problem dealt with by Jordan in his TraitS--within his own generalized theoretical framework. He thereby succeeded in giving a more conceptual, group-theoretic justification of the fact that this problem was the same as that of trisecting the periods of a hypereUiptic function of four periods [31]. As these examples should make clear to the modern reader, a young and energetic but virtually unknown American, Leonard E u g e n e Dickson, had i n d e e d written "a book worth publishing" by the summer of 1900. Its appearance--most likely with Felix Klein's e n d o r s e m e n t - - i n the Teubner series ninety years ago this year signaled ever more clearly the arrival of the Americans on the mathematical scene. What's more, it marked their official entrance into an area in which they would excel throughout the twentieth century, namely, the theory of finite simple groups.

References 1. Leonard Eugene Dickson, Linear Groups wzth an Expositzon of the Galois Field Theory (Leipzig: B. G. Teubner, 1901; reprint ed., New York: Dover Publications, Inc., 1958). Dickson's thesis appeared as "The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group," Annals of Mathematics 11 (1897), 65-120, 161-183, or The Collected Mathematical Papers of Leonard Eugene D~ckson, (A. Adrian Albert ed.) 5 vols., New York: Chelsea Publishing Co. (1975), 2: 651-729. 2. Wilhelm Magnus, Introduction to the Dover edition, Linear Groups, p. v. 3. Several biographical sources on Dickson exist, but see, for example, A. Adrian Albert, "Leonard Eugene Dickson 1874-1954," Bulletin of the American Mathematical Society 61 (1955), 331-345; Raymond Clare Archibald, ed., A Semzcentenmal Hzstory of the American Mathematical Society 1888-1938, 2 vols., New York: American Mathe-

matical Society (1938); reprint ed., New York: Arno Press (1980), 1, 183-194; Charles C. Gillispie, ed., The Dictionary of Scientific Biography, 16 vols., 1 supp., New York: Charles S. Scribner's Sons (1970-1980), s.v. "Dickson, Leonard Eugene," by Ronald S. Calinger. 4. D. Reginald Traylor with William Bane and Madeline Jones, Creatwe Teaching: Heritage of R. L. Moore, Houston: University of Houston (1972), 28. 5. On the atmosphere at Chicago and in its Mathematics Department at this time, see Karen Hunger Parshall, "Eliakim Hastings Moore and the foundmg of a mathematical community in America, 1892-1902," Annals of Science 41 (1984), 313-333; reprinted in Peter Duren, et al., ed., A Century of Mathematics in America--Part II, Providence: American Mathematical Society (1988), 155-175. The developments at Chicago are viewed from the broader perspective of late nineteenth-century mathematical developments in America in Karen Hunger Parshall and David E. Rowe, The Emergence of an Amer-

6.

7. 8.

9.

10. 11. 12. 13. 14. 15.

16.

17.

18.

ican Mathematical Research Community: J. J. Sylvester, Felix Klein, and E. H. Moore, forthcoming in the joint American Mathematical Society/London Mathematical Society series in the History of Mathematics. Eliakim Hastings Moore, "A doubly-infinite system of simple groups," Mathematical Papers Read at the International Mathematical Congress Held in Connectzon with the World's Columbian Exposztzon: Chzcago 1893 (E. H. Moore, Oskar Bolza, Heinrich Maschke, and Henry S. White, ed.), New York: Macmillan & Co. (1896), 208-242. (Hereinafter cited as Congress Papers.) Felix Klein, "Uber die Transformation der elliptischen Funktionen und die Auflosung der Gleichungen ftinften Grades," Mathematische Annalen 14 (1879), 111-172. E. H. Moore, "Concerning a congruence group of order 360 contained in the group of linear fractional substitutions," Proceedings of the Amerzcan Assoczat~on for the Advancement of Sczence 41 (1892), 62; and Frank N. Cole, "On a certain simple group," 40-43 in Congress Papers. Moore, "A doubly-infinite system of simple groups," 238-242. Ibid., 211. Dickson, "The analytic representation of substitutions," 68-120 or 652-706. Our notation for the prime changes here in order to conform to Dickson's usage. Camille Jordan, Traitd des Substztutzons et des Equatzons alg~briques, Paris: Gauthier-Villars (1870). Dickson, "The analytic representation of substitutions," 135 or 721. Ibid., 67 or 653. In a sample space of 320 "active" members of the American mathematical community in the years from 1891 to 1906, 112 or 35.0% reported spending some time studying abroad. See Della Dumbaugh and Karen Hunger Parshall, "A profile of the American mathematical research community: 1891-1906," to appear. Leonard E. Dickson to E. H. Moore, December 19, 1899, University of Chicago Archives, E. H. Moore Papers, Box 1, Folder 19. As always, I thank the University of Chicago for permission to quote from its archives. Leonard E. Dickson to Felix Klein, April 7, 1900, Klein Nachlat] VIII, Archive 528/1, Nieders~ichische Staatsund Universit/itsbibhothek (NSUB), Gottingen. I thank the library for permission to quote from its archives and Dr. Helmut Rohlfing, the director of the library's Handschriftenabteilung, for his help and hospitahty during my recent research trip to Gottlngen. Leonard E. Dickson to Felix Klein, April 7, 1900, Klein Nachlai~ VIII, Archive 528/2, NSUB, Gottingen. Dickson

gave his simplified treatment of this point in Linear Groups, 208-216. 19. Dickson, Linear Groups, ix. 20. William Burnside, Theory of Groups of Finite Order, Cambridge: University Press (1911); reprint ed., New York: Dover Publications, Inc. (1955), viii. The preface to the first edition was reprinted in the second. 21. Among the pertinent works by Frobemus, see Georg Frobenius, "Ueber Gruppencharaktere," Sitzungsberzchte

22. 23.

24.

25.

26. 27.

28.

29. 30. 31.

der PreuJ3ischen Akademze der Wzssenschaften zu Berhn (1896), 985-1021; "Ueber die Primfactoren der Gruppendeterminante," op. cit., 1343-1382; "Ueber die Darstellung der endlichen Gruppen durch lineare Substitutionen," op. cit. (1897), 994-1015; "Ueber die Darstellung der endlichen Gruppen durch lineare Substitutionen II," op. cir. (1899), 482-500; and "Ueber die Composition der Charaktere einer Gruppe," op. cit., 330-339. These works may also be found in Georg Frobenius, Gesammelte Abhandlungen, (Jean-Pierre Serre ed.) 3 vols., Berlin: Springer-Verlag (1968). Burnside, v. For a masterful historical treatment of Frobemus and his work, see Thomas Hawkins, "The origins of the theory of group characters," Archwe for Hzstory of Exact Sciences 7 (1971), 142-171; and "New light on Frobenius' creation of the theory of group characters," op. czt. 12 (1974), 217-243. For each of the classical linear groups (that is, the general and special linear, the umtary, the symplectic, and the orthogonal groups), Dickson presented a formula for its order. Remarkably "modern," his order formulas were displayed in a form suggestive of those for the finite Chevalley groups, based on the Bruhat decomposition and involving the exponents of the Weyl groups. See, for instance, Roger W. Carter, Szmple Groups of Lze Type, New York: John Wiley & Sons (1972). I thank my resident expert in algebraic groups, Brian J Parshall, for pointing this out to me. Fifty years later, Jean Dieudonn~ showed that Dickson's list of isomorphisms was, in fact, complete. See Jean Dieudonn~, On the Automorphisms of the Classical Groups with a Supplement by Loo-Keng Hua, vol. 2, Memoirs of the American Mathematzcal Society (1951). Dickson, Linear Groups, 308-310. Ibid., 309. As Dickson pointed out, this non-isomorphism had first been proven using brute force by the American female mathematician, Ida May Schottenfels in "Two non-isomorphic simple groups of the same order 20160," Annals of Mathematics (2)1 (1900), 147-152. Schottenfels was one of the two women who emerged among the sixty-two "most active" participants in the American mathematical research community between 1891 and 1906. In all, seventy-one women surfaced in a total sample space of 1061. See note 15 above, and Della Dumbaugh and Karen Hunger Parshall, "Women m the American Mathematical Research Community: 1891-1906," to appear. Here, I have used not Dickson's now antiquated notation for these groups but rather that used in J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Fimte Szmple Groups, Oxford: Clarendon Press (1985). Since these groups are all simple, they are also denoted simply by L,,(q). Emil Artin, "The orders of the linear groups," Communications in Pure and Apphed Mathematics 8 (1955), 355-365. Emil Artin, "The orders of the classical simple groups," op. czt., 455-472. Dickson, Linear Groups, 303-307. THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

11

The Opimon column offers mathematicians the opportumty to wrzte about any issue of interest to the mternatwnal mathematical commumty. Dzsagreement and controversy are welcome. An Opmwn should be submitted to the editor-in-chief, Sheldon Axler.

The Happy Formalist J. M. Henle Over the past ten years there has been a renaissance in the philosophy of mathematics and with it the birth of a genuinely new approach. My first exposure came with Reuben Hersh's provocative paper, Some proposals for reviving the philosophy of mathematics, but it has roots extending back at least to Lakatos and Putnam. 1 This new approach, which I will call "'quasi-empiricism," was most engagingly presented in Davis and Hersh's The Mathematical Experience and was recently crystallized in Thomas Tymoczko's fine anthology, New Directions in the Philosophy of Mathematics (which includes the paper by Hersh mentioned above). I have always considered myself a formalist. I found the renewal of debate both exhilarating and troubling. On the one hand, it was exciting to watch the emergence of a new philosophy. I enjoyed The Mathematical Experience tremendously; it is a beautiful book and the questions it raises are stimulating. On the other hand, many of the writers were hard on formalism, in some cases, almost abusive. The emotional content of their attacks took me by surprise. I found that often I did not recognize my o w n beliefs in their characterizations of formalism. The purpose of this paper is to present my understanding of formalism and answer the attacks made upon it. In the course of this I will also describe the major ideas of quasi-empiricism and make some criticisms of my own. I apologize in advance for the cursory summaries of the principal philosophies of mathematics. I have attempted to extract the most salient features, omitting ideas more characteristic of individual philosophers.

Good introductions to most of these are better found in Putnam or Wilder. For a more complete picture of quasi-empiricism, the reader should see Hersh or Tymoczko. Finally, I would like to thank Tom Tymoczko for many helpful comments and suggestions. Those familiar with his work will understand, of course, that the views expressed here are not his.

Formalism The basic premise of formalism is, in the w o r d s of Haskell B. Curry, that "mathematics is the science of formal systems. ''2

1 See T y m o c z k o . 2 C u r r y , p. 56.

1 2 THE MATHEMATICALINTELLIGENCERVOL 13, NO 19 1991 Spnnger-Verlag New York

Rather like the concept of "computable function," formal systems can be defined in many ways. Let us say here that a formal system consists of: a. a formal language (a collection of symbols together with unambiguous rules for forming these into statements of the language), b. a collection of statements (the axioms of the system), and c. a system of inference (a collection of unambigu o u s rules for determining w h e n one statement follows directly from other statements). In a given formal system, the theorems are those that follow, in a finite number of steps, from the axioms. The study of a formal system lies in the study of these propositions. The formalist thesis, which is rather like Church's thesis, 3 is simply that all of pure mathematics can be imbedded in formal systems. There are really no other codicils or claims. Note that we refer only to "pure," or abstract, mathematics. What is called applied mathematics is not the same and is not within the compass of formalism.

Platonism, Intuitionism, and Logicism The Platonist is concerned with mathematical truth. Platonists believe that truth can be recognized outside of formal systems. They consider, for example, that the continuum hypothesis 4 has an absolute truth value, although it has been shown to be neither provable nor disprovable in standard formal systems. The intuitionist is concerned with mathematical morality. The intuitionist believes that certain mathematical practices are proper and some are not. They consider, for example, that the principle of the excluded middle is not justified, s The formalist I describe in this paper is neutral on truth. Formalist truth is a relative truth, depending as it does on the axioms and rules of the system. This does not, however, deny a formalist the opportunity of affirming something more universal. A formalist may in fact seek truth or even justice. He or she may prefer one formal system to another on moral, religious, or political grounds. Such criteria,

3 Church's thesis ,s that every intmtlvely computable functmn is formally computable, that is, that every function or algorithm that seems computable to us is in fact a member of the set rigorously defined by Church (and equivalently by Kleene and Tunng) Since the thesis was formulated m 1936, no counterexample has been found 4 The continuum hypothesis asserts that every subset of the real numbers can be put into 1-1 correspondence with either the natural numbers or with the real numbers s The principle of the excluded middle asserts that every mathematical statement is either true or false.

though, have nothing to do with formalism. The hallmark of the formalist is tolerance. A Platonist w h o denies the Axiom of Choice, say, may be scornful of mathematicians who use it. In contrast, the formalist, whatever her or his persuasion, will recognize the validity of work in any formal system. Logicism appears very similar to formalism, asserting that all of mathematics can be reduced to logic. Curry points out that this leaves logic undefined. 6 There are few a d h e r e n t s of logicism today. The modern manifestation might be called "categorism," the view that all of mathematics can (and should) be reduced to category theory. The formalist is opposed equally to all injunctions and prohibitions. I regard formalism the way I do the Bill of Rights of the Constitution of the United States. Formalism is o p p o s e d to all forms of mathematical bigotry. Its thesis guarantees the right of anyone to practice deduction and induction in any formal system. I repeat, this does not prevent the formalist from forming opinions. Formalists may be liberal or conservative, religious or atheist, deconstructionist, vegetarian or Rotarian. What it does prevent is the imposition of those views on others.

Quasi-empiricism The philosophies described above all prescribe a foundation of some sort for mathematics. Quasi-empiricism rejects the need for a foundation. The answer to the question "What is mathematics?" is that mathematics is what mathematicians do. What they do may not d e p e n d on an abstract philosophical creed, but rather on the tastes and temper of the place and time. The quasi-empiricist sees mathematics as a cultural enterprise. Mathematicians behave at times as scientists, at other times as artists or grocers. To understand mathematicians, we must observe them, note their similarities to and their differences from people in other walks of life. To the quasi-empiricist, the proper question is not "What is mathematics?," but "What does it mean to do mathematics?"

Scope A first criticism of formalism is that virtually everything can be presented in a formal system. Music, for example, law, puff pastry, or football. I draw a distinction between imbedded and modelled. We can construct a mathematical model of the solar system, but that model is only a reflection of reality. We can model international trade, the transmission of disease, and the ten commandments, but no econo6 Curry, p. 65. THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991 13

mist, biologist, or ethicist would argue that any one model captures all important features. When I say that a particular area of mathematics can be imbedded in a formal system, I mean that there is a system such that the theorems of that system are exactly the theorems of the given area. This suggests a second criticism, that not all of mathematics can be so imbedded. What about intuitionist mathematics, for example, or model theory, or metamathematics? In fact all of these can be imbedded. Model theorists, for example, may argue that they are really studying models, not formal systems. In fact, all the m o d e l s t h e y s t u d y can be f o u n d in set theory. Whether they choose to look at it that w a y or not, they are working in ZF.7 Intuitionists may be unhappy in a Kfipke model, 8 but the discomfort is only philosophical. They can argue that the formalization is inappropriate or distasteful. They cannot argue that it is impossible.

Belief in a large number is no more daring, I should think, than belief in T o l s t o y ' s War and Peace. Metamathematicians may claim that they perceive truths that cannot be expressed in the system. G6del's sentence of arithmetic, for example, states in code that it cannot be proved. Certainly the truth of that sentence cannot be acknowledged in arithmetic, but it can in a larger system, ZF, or some simple extension of arithmetic. We only claim that the metamathematician's work can be embedded in a formal system, we do not specify in advance the formal system, guarantee its completeness, or even its consistency. I do not mean to belabor the point, which is technical and philosophically trivial. The logicists are fight that mathematics can be reduced to logic, but the choice of logic is arbitrary. It can also be reduced to set theory, arithmetic, geometry, or knot theory. Of course, Galois theory loses significance, intelligence, and beauty w h e n it is arithmetized, but the structure and inferences remain.

Ambiguity I can imagine some worry being expressed about the loose description I have given of a formal system. One can make this much more precise (see, for example,

7 Zermelo-Frankel set theory 8 See Fitting. By "Kripke m o d e l , " I mean the formal system as set up by Fitting, not s o m e larger system with models.

14

THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

Curry). The simplest solution is technological. The word "unambiguous" can be replaced by the stricture that a suitable computer can be programmed to arbitrate any dispute concerning the grammar and rules.

Existence Another objection has been raised: We at least regard formal systems as real, even if we doubt the reality of sets, numbers, and functions. Aren't we then Platonists, too? It's hard to know where to begin to answer this. Most of us, after all, are mathematicians and not philosophers. We are not used to doubting those things that we write with, that we sit on, or that we use to brush our teeth. Formal systems have nearly that class of reality, in that they can be completely described in print and can be programmed on computers. One may argue that such programs are at least as abstract as the integers that can be used to code them. Aren't we at least Platonist about the integers? Some of us, perhaps, but after all, the program that is needed to arbitrate, say, arithmetic, is a very finite thing. It can be coded by a single integer. I don't have to believe in all of the integers to work in arithmetic. Belief in a large number is no more daring, I should think, than belief in Tolstoy's War and Peace. Finally, what exists and what doesn't is not the concern of formalism. We don't actually deny the reality of mathematics, we just don't assert it. In fact, there is no real conflict between formalism and Platonism. One might just as well accuse me of Platonism since I appear to believe in the existence of scholarly journals, Reuben Hersh, and the philosophy of mathematics (whatever that is).

Mathematical Practice A more important criticism of formalism is that it tells us nothing about mathematical practice, or worse, that it gives us the wrong impression of it. Few mathematicians, after all, pursue their craft solely by formal proofs. An argument can also be made that truly formal proofs are extremely rare in mathematics. In answer, I must agree that formalism does indeed say nothing about mathematical practice. It would be nice to know h o w it is that we do/discover mathematics. We don't scorn such knowledge, but we make no claims to it. What formalists do claim is that we k n o w w h a t mathematics is, and this is no small matter. One can argue that philosophy has seen few such achievements. I believe, for example, that it is still unsettled as to what art is, what history is, or for that matter, what philosophy is. I have heard a philosophical colleague argue that the only real accomplish-

ments of philosophy are either mathematical or trivial (or both). The formalist definition of mathematics, I think, is neither. Formalism is limited. But those w h o extrapolate to suggest that formalists view the art of mathematics as consisting only of formal deduction are exposing their prejudice and not ours. It is often pointed out that truly rigorous mathematics would be deadly. It would be impossible to write or to read, but formalists do not do it or insist that the formalization be carried out. The validity of a piece of mathematics rests on the reasonable belief that it can be done.

Mathematical Epistemology In Goodman's view, formalists are "surfacist." They are led to their views by a superficial observation of mathematicians at work. Noting that they are engaged almost exclusively in formal manipulations, they conclude that that is what constitutes mathematics and nothing else. In particular, he says that formalists deny the reality of intuitive constructions. As we noted before, this is entirely false. The basis of formalism is philosophical, not empirical, and we do not deny anyone's reality. We are agnostics, not atheists. Goodman and others are excited by the similarities between science and mathematics. But these similarities, no matter h o w strong, are not as significant as those things that set mathematics apart. The important point is not that mathematics shares heuristics with physics and geology, but that it has its o w n unique ways of understanding.

most mathematicians make sound decisions on what they themselves should study. I am also a pessimist. I fear that these same mathematicians can make terrible decisions about what others should study. Time after time we mathematicians plead the cause of basic research. When bureaucrats cry that our particular field has no practical significance, we point to all the fields w h o s e applications arose only after hundreds of years. We defend research for its o w n sake. We argue that ultimately the benefits to society will come, and that it is impossible to determine n o w the value of a theorem in the future. It is ironic that even as we suffer from the narrow-mindedness of n o n m a t h e m a t i c i a n s , we are n a r r o w - m i n d e d ourselves.

I have heard a philosophical colleague argue that the only real accomplishments of philosophy are either mathematical or trivial (or both). Mathematical Logic The argument is sometimes made that foundational philosophies exclude the nonlogician from the practice of philosophy. This may be a valid criticism of logicism and intuifionism, but hardly of formalism. All one needs for formalism is to understand the axiomatic m e t h o d , an u n d e r s t a n d i n g p o s s e s s e d by virtually every educated mathematician.

Games Mathematical Aesthetics The accusations here are similar to those above. Formalism says nothing about what mathematics is good and what is bad. Quite true, and we are almost proud of our silence. Too often have mathematicians scolded their peers for work not to their taste. Too often have the prejudices of the world entered into what should be an oasis of intellectual freedom. Frederick the Great is reputed to have said: "I have no quarrel with mathematicians. They at least do not form sects." He was wrong. They do, and they can be pretty mean to nonbelievers. But couldn't there be here a serious flaw in formalism? Might not infinite toleration leave the discipline with no taste? Wouldn't even formalists agree that there are formal systems of no value whatever? Certainly we would. I do not think, however, that formalist toleration would lead to a plague of bad mathematics. Shabby papers are more reasonably the fault of our system of rewards and punishment for academicians. I am, first of all, an optimist. I believe that

Hersh describes formalism as "the philosophical position that much or all of pure mathematics is a meaningless game. ''9 Formalists do not assert that mathematics is meaningless. Its meaningfulness is obvious to all. Its meaning, however, may lie outside mathematics. The word "tree," after all, has no intrinsic meaning, not even an intrinsic pronunciation. We humans have given it meaning. The word "game" has an unfortunate history. Formalists have used it in the past, mostly for its shock value, I think, but also as a defensive move, to deflect criticism from colleagues who insist that they work on something "real." Even today mathematicians will sometimes take evasive action in the face of charging Platonists. In the hands of others, the word may be used perjoratively, either to deny that formalists can think seriously about the world, or else to suggest that no one

9 Hersh (m Tymoczko),p 12 THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

15

with our narrow views could actually enjoy mathematics. Both of these are slanders not worthy of reply.

The Incompleteness Theorem Every description of formalism includes tales of Hilbert insisting that axiom systems be consistent, and how hopes that we could ever prove them consistent were dashed by G6del's incompleteness theorem. The (usually unstated) conclusion is that formalism is thus a failure. All this happened before I was born. I became a formalist knowing full well the problems with consistency. The formalism I affirm coexists (as we all do) with undecidability and uncertainty. Rather than dismay, I face the situation with delight. The fact that I may be operating from inconsistent premises is a little exciting. Researchers in set theory (my field) live particularly close to the edge, working in systems far more powerful and dangerous than arithmetic or analysis. My own doctoral thesis was nearly wiped out two months before my defense. The experience left me with a healthy sense of mortality and a greater respect for mathematics. I like to think that we have learned that paradoxes lead to wisdom, not grief. Rather than d o u b t the validity of formalism, this uncertainty has led me to doubt truth. The existence of different models of arithmetic 1~ tells me not just that we may never know the truth, but that there may be no truth to know. These, however, are my own views and not formalism.

Neoformalism Some readers may be wondering if the formalism expressed here isn't different from the classical variety. It doesn't seem controversial, does it? H o w can you argue with a message of intellectual freedom? I have w o n d e r e d a b o u t this too, b u t a careful reading of Curry's Outlines of a Formalist Philosophy of Mathematics, first drafted in 1939, reveals little beyond what I have written. There are no inflammatory statements, no righteous denials of reality, morality, or truth. Curry does differ from earlier versions associated with Hilbert and others. There is no fixation on consistency, for example. Bourbaki, often regarded as a paragon of formalist practice, is not as enlightened as Curry in many ways. 11 There is, for example, no ac-

10 N o n s t a n d a r d m o d e l s , c o n s t r u c t e d in a larger s y s t e m s u c h as set theory 11 Mathias (Ignorance). 16 THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

knowledgment in Bourbaki of Godel's work until the 1950s. 12 If it seems appropriate, the term "neoformalism" could be used to describe this package of ideas now fifty years old. I have no objection.

Mathematical Pedagogy A serious charge against formalists is that they are partly responsible for the " n e w math," and all our current educational problems. 13 The argument blames university professors who decided that school mathematics should be taught axiomatically and then influenced teachers, writers, and publishers to adopt their views. I will not argue the merits of the new math here. It may have been a mistake (though I liked it a great deal when it was practiced on me, and it seems to have produced the largest generation of mathematicians in history). There are, however, many theories to explain w h y our students today are so poor and so poorly motivated. Some progress has been made in recent years, but on the whole, the problem of teaching mathematics is extremely difficult and insights are elusive. I do object to those who fault formalism. It is irrelevant whether individual formalists had anything to do with the new math. Formalism is silent on the subject of pedagogy, as it is on many subjects. To blame formalism is rather like blaming John Philip Sousa for the Vietnam war. To put it another way, my students never need convincing that an answer exists to any problem. Are they Platonists? Most of my students have trouble with induction. Are they intuitionists? And what about those premeds in the back who, impatient with my attempts at motivation, cry "Just tell us h o w to do it!" Are they quasi-empiricists?

Formalism in the Large Davis and Hersh's second book, Descartes" Dream, was less successful. The case having already been made, the authors turned polemical. In a final, rambling essay, they discuss formalism in general, "the condition wherein action has become separated from integrative meaning and takes place mindlessly along some preset direction. ''14 They associate this with the mathematical understanding of the word and follow it d o w n a path of horror. They are careful not to blame the h o l o c a u s t on formalism, b u t they come very close. 15

12 See, for example, Bourbala, 1948 13 T h o m , 1971. 14 Davis a n d H e r s h , 1986, p 282 15 I b i d , pp. 291-2

It was that essay, in fact, which propelled me to write this paper. The suggestion they made is odious and flawed in many ways. It was made seriously, h o w e v e r , and since I have a r g u e d for formalism in terms of its liberalism and benignity, I should answer it. The accusation is first of all confusing. Two contradictory approaches are made. At first, they appear to suggest that the Holocaust was in part a reaction against the perfection of mathematical formalism. They then abandon this and argue that formalism itself (the general variety) shares direct blame for this evil and so many others. The argument has a reasonable sound: Abstraction leads to a loss of meaning. Replacing humans by figures enables those in power to commit atrocities. Formalism insulates the actor from her or his crimes. Further, technological analysis leads to the desire to try technological solutions (such as nuclear weapons) for their own sake. We are now very far from mathematical formalism, but let me venture a reply. The chief argument is that one is able to commit a greater evil when one can put distance between oneself and the crime (for example, torture). I would argue that this may be true in the short run but not at length. In fact, it has long been a theory of mine that one is capable of the greatest evil only when one believes one is acting for higher purposes. There is a limit to avoidance. We can escape consideration of our acts only so long. There is no such limit on zealotry. The common thread running through the most terrible crimes is self-righteousness. The distinction here is not academic. If you decide to accuse formalism, you have, ironically, a formal strategy for avoiding the commission of great evil, namely abjure formalism. If instead you accept my argument, there is no easy path. We ourselves may be the enemy, and we must continually reflect on our actions. Not even our altruistic impulses can be trusted. It comforts us to place the onus entirely on a person, a people, or an ideology, but I can't be so comforted. We will never be free from villainy until we recognize its seeds in all of us. Returning to Davis and Hersh's essay, their first suggestion is pretty speculative. Even if true, attaching blame to formalism would be like accusing the victims of rape. Their second argument is not supported by facts. There is actually quite a lot known about mathematical tastes among Nazi mathematicians, and they were far from formalist. Indeed, they opposed axiomatization and associated it with Jewish mathematics. 16 It makes little sense then to conclude that formalist tendencies powered the Holocaust. Germany was a rightist totalitarian state. On the 16 Segal, 1986.

other side, in the Stalinist Soviet Union, the story is similar. Throughout the 1930s and 1940s, logic and idealistic mathematics were seen as capitalist tools. Their study was for many years a very dangerous activity. 17

Frederick the Great is reputed to have said: "'I have no quarrel with mathematicians. They at least do not form sects.'" He was wrong. What in Fact is Mathematics? This is the central question of any philosophy of mathematics. It is the question that formalism answers and quasi-empiricism doesn't. To say that it is what mathematicians do is to say, in effect, that there is no universal meaning in the word "mathematics." A philosophy that begs this question can hardly fault others for failing to address more collateral issues. Perhaps the essence of the dispute is that quasi-empiricists do doubt that there can be such a universal meaning. After all, what is commonly accepted as mathematics has changed dramatically through the years. This is true, but the same can be said about "clothing." In any case, what is philosophy good for if not for finding things universal? Formalism does not object to the research into mathematical practice that the quasi-empiricists have stimulated. Indeed, as a mathematician, I am more than a little flattered to be the object of so much study. On the other hand, mathematical practice is not mathematics (no more than cutting up pigs is biology, or literary criticism is poetry). Undoubtedly social scientists could be found who, given a grant, would chart h o w frequently mathematicians sharpen their pencils. They would collect and contrast data from algebraists, geometers, and non-Riemannian hypersquarers. They may even arrive at significant conclusions, but the significance w o u l d pale beside the question that remained unanswered: "What is mathematics?" Such research may be good sociology; it may be good psychology, intellectual history, or even cognitive science. As philosophy, it is pretty thin.

Tolerance As I have said, formalism is compatible with Platonism. Actually, in the eyes of a formalist, all the various philosophies are compatible. Formalism wants

17 See M a t h i a s (Terror). Matl'uas a t e s m a n y sources. The m o s t accessible m c l u d e reviews a n d reports from t h e Journal of Symbohc Logzc, vol. 14, p. 183 a n d p. 243 (1949), a n d vol 17, p p 124ff (1952). THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991 17

to say w h a t mathematics is. Platonism wants to tell us w h e n it is true. Intuitionism wants to tell us w h e n it is good. Logicism wants to tell us where it comes from, and quasi-empiricism wants to tell us w h a t it means to do it. The trouble begins w h e n each of these (except formalism) claims exclusivity. The formalism I have described m a y seem pretty tame. Beyond a brief definition of mathematics, it says nothing. In fact, it is tame, which makes me w o n d e r w h y it has been attacked so strenuously. It is all the more ironic that formalism should suffer such abuse, w h e n its o w n message is one of tolerance.

References 1. Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematzcs" Selected Readings, Englewood Cliffs, NJ: Prentice-Hall (1964). Second Edition, Cambridge University Press (1983). 2. N. Bourbaki, Foundahons of mathematics for the working mathematician, Journal of Symbohc Logic 14 (1948), 1-14. 3. Haskell B. Curry, Outhnes of a Formahst Philosophy of Mathematzcs, Amsterdam: North-Holland Publishing Co. (1951). 4. P. J. Davis, and R. Hersh, The Mathematical Experience, Boston: Birkhauser (1981).

Though you m a y not recognize the n a m e (yetS), you'll know our lineage well. Harper & Row a n d Scott, Foresman/Little, Brown were t w o of the most respected names in a c a d e m i c publishing; HarperCollins - the p r o d u c t of their union - not only enjoys that respect but is n o w o n e of the largest publishers of English-language books in the world. We've brought those resources a n d experience to bear in developing the most comprehensive a n d up-to-date line of mathematics texts on the market. One of these '91 titles is perfect for your course:

An Introduction to Differential Equations With Boundary Value Problems Andrews

Modern Algebra and Discrete Structures Lax

18

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

5. P. J. Davis, and R. Hersh, Descartes" Dream, Boston: Harcourt Brace and Jovanovich (1986). 6. M. C. Fitting, Intuitiomstzc Model Theory and Forcing, Amsterdam: North-Holland Pubhshing Co. (1969). 7. N. Goodman, Mathematics as an ob]echve science, Am. Math. Monthly 86 (1979), reprinted in New Dzrectmns in the Philosophy of Mathematics (T. Tymoczko, ed.), Boston: Birkhauser (1986). 8. R. Hersh, Some proposals for reviving the philosophy of mathematics, Advances m Mathematics 31 (1979), reprinted in New D~rectzons m the Phzlosophy of Mathematics, (T. Tymoczko, ed.), Boston: Birkh/mser (1986). 9. A. R. D. Mathias, The Ignorance of Bourbaki, manuscript. 10. A. R. D. Mathlas, Logic and Terror, manuscnpt. 11. Sanford L. Segal, Mathematics and German politics: the National Socialist experiment, Hzstorza Mathematica 13(2), (1986). 12. Ren6 Thom, Modem mathematics: An educational and philosophic error?, American Scientist 59 (1971), reprinted in New Dzrections m the Phdosophy of Mathematzcs (T. Tymoczko, ed.), Boston: Birkh/iuser (1986). 13. Thomas Tymoczko, ed., New Drrectzons in the Philosophy of Mathematzcs, Boston: Birkh/mser, (1986). 14. Raymond L. Wilder, Introduction to the Foundatzons of Mathematzcs, New York: John Wiley and Sons (1952). Department of Mathematics Smith College Northampton, MA 01063 USA

Mathematics With Applications in the Management, Natural, and Social Sciences Fifth Edition Lial/Miller/Hungerford If y o u ' d like to examine any of these titles, or learn more a b o u t HarperCollins' other fine mathematics texts, c o n t a c t your local HarperCollins representative or write: Mathematics Marketing M a n a g e r College Division HarperCollins Publishers 1900 East Lake Ave. Glenview, Illinois 60025

--" HarperCollinsPublishers

Did Poincar6 Say "Set Theory Is a Disease"? Jeremy Gray

One of the most colourful quotes in all the history of mathematics is attributed to Poincar6: "Later generations will regard Mengenlehre as a disease from which one has recovered.'" The quotation is perhaps a little spoiled by the intrusion of the German word for set theory, but doesn't that give it just the aura of verisimilitude required for maximum conviction? How fortunate to have an illustration of those vigorous expressions of opinion we all know mathematicians make in private but seldom confide to the page; how fortunate too that Poincar6 chose to lambast a subject we and all our students know, the hapless core of modern mathematics. Had he chosen, shall we say, the theory of higher order Bessel functions, the quote would surely be less well known. I wish, however, to suggest that Poincar6 never made such a remark. Since the quote is u s e d to b u t t r e s s an a r g u m e n t that Poincar6 was strongly opposed to the study of the theory of sets, if the quote goes down, then Poincar6's position can be seen to be richer, more interesting, and more profound, if less colourful. (Any reader who can find reliable testimony that Poincar6 did make this remark is cordially invited to let me know so that the source can be published.) I became suspicious of the p u r p o r t e d quotation when I realised that it was far cruder in its attitude toward the study of set theory than any other remark I could find by Poincar6. My suspicion deepened on seeing that none of the most scholarly recent writing on Poincar6 carries the remark. Gregory Moore [8], and Jean Cassinet and Michel Guillemot [1] between them devote many. pages to how Poincar6's views evolved during the 1900s, but they never allude to this remark. J. W. Dauben's equally reliable and thorough study of Cantor [2] also neglects to carry it. This suggests that none of these authors found it in an authentic text of Poincar6.

It is, however, quoted in a number of places. Morris Kline's [7] has it on page 1003. Amongst popular sources I cite Martin Gardner's [5], page 27. We can start with Kline's account, which is in a well-respected scholarly source. It runs: Poincar~37 remarked critically, "But it has happened that we have encountered certain paradoxes, certain apparent contradictions which would have pleased Zeno of Elea and the school of Megara . . . . I think for my part, and I am not the only one, that the important point is never to introduce objects that one cannot define completely in a finite number of words." He refers to set theory as an interesting "pathological case." He also predicted (in the same article) that "Later generations will regard [Cantor's] Mengenlehre as a disease from which one has recovered." If you consult any of the sources that Kline gives in footnote 37, the Proceedings of the Fourth International

THE MATHEMATICAL INTELLIGENCERVOL 13, NO 19 1991 Spnnger-Veflag New York

19

Henri Poincar4

Congress of Mathematicians, Rome, 1908, 167-182; the Bulletin des sciences math~matiques, (ser. 2), vol. 32, 1908, 168-190; extract in Oeuvres, 5, 19-23, you are referred to PoincarCs essay L'Avenir des Math~matiques. However, you will not find the remark. In the case of the Oeuvres none of the passage appears, because the editors chose to reproduce, as they say, only a part of PoincarCs essay. The other two sources, which are identical, only carry the passage about "certain paradoxes" pleasing to Zeno, the suggested remedy, and the suggestion that set theory is a pathological case. In particular, the remark we seek is not in the article from which the other remarks are taken. A faulty attribution of a source does not mean that Poincar~ did not pass his memorable judgement. Nonetheless, it is worth speculating on where Kline may have taken his information from. His account closely follows that of E. T. Bell, Men of Mathematics, first published in 1937. On page 558 of the Fireside reprint, 1965, we find essentially the same quotation down to the remark about using only a finite number of words. Bell then continued: Whatever be the cure adopted, we may promise ourselves the joy of a physician called in to treat a beautiful pathologic case. A few years later PoincarCs interest in pathology for its own sake had abated somewhat. At the International Mathematical Congress of 1908 in Rome, the satiated physician delivered himself of this prognosis: "Later generations will regard Mengenlehre as a disease from which one has recovered." 20

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

The English of these two versions of PoincarCs text differs up to and including "a finite n u m b e r of words." Then comes the remark about the pathological case. So far, both are fair translations of PoincarCs essay, which suggests that both Kline and Bell had independently consulted it. But the English of the quote we seek is given by Kline exactly as it is found in Bell, which suggests that this is (one of) Kline's sources. However, there are things wrong with the story as it is f o u n d in Bell. Although he quoted correctly from PoincarCs essay, L'Avenir des Math~natiques, he attributed it to 1905. However, as the French Bulletin and the Italian Rendiconti del Circolo Matematico di Palermo, 26, 1908, 152-168, where the essay was also printed, make clear, the essay was circulated at the 1908 International Congress, so it represents PoincarCs views of that date. Bell has conjured up a shift in PoincarCs thought. Perhaps Kline, on seeing that L'Avenir in fact dates from 1908, tidied up that error in Bell, only to make another, namely that the famous remark is to be found in the text of L'Avenir. Indeed, Bell does suggest that the remark was part of the speech. But he does not explicitly say so, which leaves open the possibility that it could have been a throwaway remark, either made at the time and then suppressed or else made in conversation. It is unlikely to have been made at the time, for as footnotes in the Bulletin and the Rendiconti make clear, Poincar~ did not in fact give the speech himself. He was temporarily ill, and Darboux read it for him. I must assume that Darboux stuck to the text, as it was circulated in advance in the form of a pamphlet published by the Circolo di Palermo. Poincar4 was associated with them at the time; at the Rome conference he adjudicated their prize competition, awarding a medal to Severi. Were it not too fanciful, one might imagine Darboux looking up and saying something like "'Of course, what Poincarl really thinks is . . .'" Perhaps, but none of the sources say that. Nor do they say that informally, in conversation, PoincarO made the remark. Nor did Bell attend the conference and catch the remark himself, for his name does not appear in the Elenco dei Congressisti given on page 12 of the Atti del congresso internazionale dei matematici, 1908, vol. 1. However, there is no need for such uncharitable speculation, for there is a source earlier than Bell from which he may well have taken the story, and which enables us to come close to proving a negative. This is an essay by James Pierpont called Mathematical rigor, past and present, given as an invited address at the American Mathematical Society's meeting in Nashville, 1928, and printed in the Bulletin of the AMS. Bell may even have heard the address, but we can be sure he read it, even though he does not cite it. He used some of Pierpont's phrases and also the same quotes from Brouwer that Pierpont gave, as well as following the main line of Pierpont's argument. Purists may

even feel that the essay has a little too much of Bell's own style of ruthless simplification. Bell could justifiably have been confident of Pierpont's mastery of his material, for Pierpont, then close to retirement as Erastus L. DeForest Professor of Mathematics at Yale, was a well-respected member of the American mathematical community who had shown a lifelong interest in the history of recent mathematics. The essay gives a good impression of the man G. D. Birkhoff said was for many years a source of inspiration at Yale, to quote from his obituary by Oystein Ore [9]. Ore points out that Pierpont was very widely read, and for many years borrowed more books from the Yale library than anyone else. This is what Pierpont wrote: PoincarO* at the Rome Congress (1908) went so far as to say "Later generations will regard the Mengenlehre as a disease from which one has recovered." His reference was to O. HOlder's Die mathematische Methode, 1924, 556. This is not an easy book to find, although the essay on HOlder in the Dictionary of Scientific Biography, vol. VI, 1972, 472-4 by Ernst H61der makes it sound as if it might be worth fuller examination. Pierpont's reference turns out to be footnote 4 on the last page of the book, where HOlder wrote POINCARI~ hat auf dem internationalen MathematikerKongre~ in Rom im Jahre 1908 gesagt, man werde sp~iter auf die Mengenlehre als auf eine uberwundene Krankheit zurtickblicken. I translate this as: Poincar6 at the International Mathematical Congress of 1908 in Rome said that one would later look back on set theory as a disease one has overcome. But HOlder gave no precise reference, and as we have seen there are no words to this effect in L'Avenir. Nor was HOlder at the Congress, for his name and that of Pierpont is also missing from the Elenco dei Congressisti. What has happened? I offer this reconstruction. PoincarO, whose own changing views on set theory are worth discussing in full, took the opportunity at Rome to comment on an issue agitating many mathematicians: the apparent paradoxes in the subject. He suggested that a remedy (a resolution of the paradoxes) would be found quite easily, for that is how I interpret his tone, and that it would be interesting to do so. He proposed one of his own, the suggestion that one restrict one's attention to just those objects definable in finitely many words. (This would severely curtail the extent of Cantor's paradise.) H61der, who was not at the Rome Congress but knew that PoincarO's address had caused a stir, offered at the end of his book a summary of what Poincarl had said. The beau cas pathologique became a disease, a not unfair translation possibly coloured by the gossip that would also have spread news of PoincarO's opinions. The optimism that Poincar6 conveyed be-

came the suggestion that one will look back on the disease and see that one has recovered from it. Pierpont unhappily took the summary for a remark that PoincarO actually made, perhaps giving HOlder as his reference and giving no reference to L'Avenir because he knew that there was none to be found. He added the phrase "later generations," which makes his version of the quote more memorable than my literal translation above, and left the term Mengenlehre in German because he knew Poincarr often used the German term. Moreover, by putting it in quotes and letting it stand alone Pierpont turned what had been offered as a summary into a single, highly quotable remark that could be repeated without, one might even say out of, context. Bell then took the story over from Pierpont, giving it his own twist. The remark does indeed read much more like a rejection of set theory than does the text of L'Avenir, as Bell saw, so he ascribed the remark and the text to different years. Kline recognised that L'Avenir belonged to the Rome Congress but interpreted Bell's account to mean that after all the quote m u s t be in there s o m e w h e r e . Whence the firm ascription of it to a text, backed up by no less than three references, where it cannot be found, and the impression, also given in the popular literature, that Poincarr was much more firmly opposed to the study of set theory than was indeed the case. As to PoincarO's actual views, these seem to have shifted over the years. Poincar6 first used the term Mengenlehre in the 1880s to describe early results and concepts that he attributed to Cantor in w h a t we would call point-set topology. He was very enthusiastic about them, and took the opportunity in 1885,

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

21

when secretary of the recently founded Soci~t~ Math~matique de France, to propose Cantor for membership (see their Bulletin, 13, 87). There is no reason to suppose that in later life he repented of his early enthusiasm for Mengenlehre in the sense of point-set topology. Mengenlehre could also mean either something like naive set theory or something like axiomatic set theory such as Zermelo was currently developing. In his Rome address in 1908, Poincar~ connected it to the famous paradoxes then under vigorous discussion. The paradoxes were variously regarded. W. Purkert and H. J. Ilgauds suggest, in their [11], that Cantor had known of the paradoxes of the largest ordinal and the largest cardinal for several years before Burali-Forti published the ordinal one in 1897. However, the way Burali-Forti and, for that matter Cantor himself, regarded their discoveries may not have been the way we do. A. R. Garciadiego [4], pages 39-41, suggests that the paradoxes only achieved significance in the course of being reworked by others. In Rome, Poincar~'s position on the paradoxes was to advocate using only those terms that can be defined precisely in a finite number of words. Yet the tenor of his remarks is upbeat: There are certain problems (nothing wrong with that in research), but one has the

joy of the doctor called to a beautiful pathological case. Set theory, by implication, has a disease, but Poincar~ did not say that set theory is itself a disease. Moreover, a doctor's attitude to a disease is not that of a friend of the patient: To a doctor a disease may be interesting, and Poincar~ spoke of one's joy at being called to the problem. In 1908 the issue of the paradoxes did not seem to be "make or break" for set theory. Over the next few years, Poincar~'s views hardened, as is described, for example, by P. Dugac, [3], 65-96, and W. Goldfarb, [6], 61-81. Poincar~ refined his idea of using only those definitions that involve only a finite number of words, to exclude what he called impredicative definitions. In this way he barred Russell's paradox of the least number not definable in fewer than one hundred words. He argued against Russell's attempt to base mathematics on logic and with equal vigour against Zermelo's axiomatisation of set theory. He took up what he regarded as a pragmatic, psychologistic position in defence of mathematical intuition, and did not allow the existence of any cardinal other than Alephzero (see his 'La logique de l'infini' (Derni~res Pens~es, Paris, 1913). He expressed his views with considerable vigour. But nowhere did he argue for a position nearly as sweeping as sometimes attributed to him, hung on the alleged remark that set theory, of whatever kind, was a disease. References

1. J. Cassinet, M. Guillemot, L'Axiome du choix dans les math~matiques de Cauchy (1821) ~t Godel (1940), Th~se pr~sent~e ~ l'universit~ de Toulouse (1983). 2. J. W. Dauben, Georg Cantor: his mathematics and philosophy of the infinite, Cambridge, MA: Harvard University Press (1979). 3. P. Dugac, Georg Cantor, and Henri Poincar~, Bolletino dz Stona delle Scienze Matematiche 4 (1984), 65-96. 4. A. R. Garciadlego, On rewriting the history of the foundations of mathematics at the turn of the century, Hzstoria Mathematica 13 (1986), 39-41. 5. M. Gardner, Mathematics Carmval, New York: Vintage Books (1965). 6. W. Goldfarb, Poincar~ against the logicistics, History and philosophy of modern mathematzcs (W. Aspray and P. Kitcher, ed.), Minneapolis: University of Minnesota Press (1988), 61-81. 7. M. Kline, A Hzstory of Mathematzcal Thought from Ancient to Modern Times, Oxford: Oxford University Press (1972). 8. G. H. Moore, Zermelo' s Axzom of Choice: its Origins, Development, and Influence, New York: Springer-Verlag (1982). 9. O. Ore, James Pierpont--in memoriam, Bulletin of the AMS 45 (1939), 481-486. 10. J. Pierpont, Mathematical rigor, past and present, Bulletin of the AMS 34 (1928), 23-53. 11. W. Purkert and H. J. Ilgauds, Georg Cantor, 1845-1918, Basel: Birkh/luser (1987).

Otto H61der 22

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

Faculty of Mathematzcs The Open Universzty Milton Keynes, MK7 6AA, England

Jacques Hadamard Jean-Pierre Kahane

In introducing Jacques H a d a m a r d to the London Mathematical Society in 1944, G. H. Hardy called him the "living legend" in mathematics. Legendary work, flourishing over function theory, number theory, geometry, mechanics, ordinary and partial differential equations. Legendary figure, running the first mathematical seminar in the Coll6ge de France, travelling over all continents, involved in actions for human rights and world peace, celebrated among mathematicians and ordinary people, a little man with a great character, le petit p~re Hadamard, as we called him affectionately in the 1950s. Nevertheless, no mathematical library contains the whole mathematical work of Hadamard, because--unlike the work of much less important mathematicians--it was never collected and published in its entirety. No street in Paris wears his name. The l e g e n d n e e d s a revival, especially in France. The life of Jacques Hadamard extends over almost a century: 8 December 1865 to 17 October 1963; about the same period of time as David Hilbert (1862-1943). Henri Poincar6 (1854-1912) was more than ten years older, Emile Borel (1871-1956), Ren6 Baire (18741932) and Henri Lebesgue (1875-1941) younger. The turn of the century was extremely bright for mathematics in France. Without a doubt, Poincar6 was le prince des math~maticiens." In the younger generation Hadamard played a leading role, and I shall discuss it briefly in a moment apropos of set theory. Hadamard's life spans three wars on French soft, six political regimes in France, from Napoleon III to De Gaulle, and such events as the Paris Commune, the Dreyfus affair, the Russian Revolution, the rise of fascism in Europe, the Front Populaire, the shame of Vichy, the Resistance, Hiroshima, the cold war, the

first man in space. He was involved as a citizen in many aspects of our history. I shall divide this paper into two parts. In the first I shall try to evoke the long life of Hadamard and his involvement in events of the time without saying much of his mathematical work. By selecting a few topics or notions to which his name is attached, in the second part I intend to give an idea of the variety and the richness of his mathematical production.

THE MATHEMATICALINTELLIGENCERVOL 13, NO I9 1991 Spnnger-Verlag New York

23

children, and looked a most marvellous couple up to her death in 1960 after a marriage that lasted sixtyeight years), and obtained the Grand Prix des Sciences Math4matiques for his work on entire functions and in particular on the function ~ considered by Riemann: ~(t) = F

+ 1 (s - 1)'rr-s/2~(s)

1

(s = -2 + it)

where he proves that ~(t) is of the form ~(t) = C [ I

Jacques and Louise-Anna Hadamard on a visit to India.

Life and Time

Jacques Hadamard originated from a Jewish family of Lorraine. There are traces of Hadamards, printers in Metz, in the eighteenth century, and also of a very remarkable great grandmother who lived during the French Revolution. Before Jacques was born the family settled in the Paris area. His father taught humanities in high school, his mother was a good pianist. The war with Prussia, the defeat, the fall of Napoleon III occurred w h e n he was not yet five years old. What he remembered later from the siege of Paris (winter of 1870) was eating a piece of trunk of the elephant of the Jardin des Plantes. At school he was bright in every subject but mathematics. "To parents in despair because their children are unable to master the first problems in arithmetic I can dedicate m y example. For, in arithmetic, until the seventh grade, I was last or nearly last," he said in 1936. ("Je puis dddier mon exemple aux parents que ddsesp~re l'inaptitude de leurs enfants ~ triompher des premiers probl~mes d'arithmdtique; car en arithm~tique--jusque et y compris la cinqui~me--j'~tais le dernier ou d~peu pros".) He was particularly successful in Latin and Greek. Howe v e r - - d u e to an inspiring t e a c h e r - - h e discovered the beauty of mathematics, moved to science, prepared for the competitive entrance exams of Ecole Polytechnique and Ecole Normale Sup6rieure, and got first place in both, with the highest score ever seen at the Ecole Polytechnique. He chose Ecole Normale Sup6rieure (1884), where he studied under Jules Tannery ("the scientific guide") and Emile Picard ("the superb teacher"). After leaving Ecole Normale Sup6rieure he taught as a high school teacher (a pupil of his was Maurice Fr6chet) and prepared his thesis. In one year (1892), he got his doctorate (on functions defined by Taylor series), married his beloved mate Louise-Anna Trenel (a beautiful romance: they loved each other, he waited too long, she got engaged to another man, he jumped, pleaded, succeeded, they married, had five 24

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

1 -

,

The story deserves telling. The Acad6mie des Sciences had proposed a study of ~r(x), the number of prime numbers less than x, and everybody expected the Prize to be given to Stieltjes, who had just published a note saying that he had proved the Riemann hypothesis. However, the proof did not come, ~(t) has much to do with ~(x), and Hadamard got the Prize. Four years later, he actually proved the prime number theorem: "rr(x) - ei x. There are many anecdotes about Hadamard as a young man. According to his daughter Jacqueline, he was the model for le savant Cosinus, a kind of inteUectual comic of the early 1900s. Le savant Cosinus is incredibly forgetful in practical matters. Actually this was the case for Hadamard. Here is an example. He liked to pick herbs, and once took his little sister on an herb-picking expedition in the Alps; he settled her on the edge of a glacier, went on collecting leaves, went back home, and only then discovered that he had left the child behind in a dangerous situation. Another, even more dramatic example occurred much later, in 1940, w h e n he succeeded in leaving France for the United States, but left behind the briefcase containing the American visas. The years 1892-1912 were extremely fruitful from a mathematical point of view. After Bordeaux, Hadamard obtained a chair in Paris. In 1909, he was elected at the Coll~ge de France and in 1912 at the Acad4mie des Sciences. After 1909 the famous Hadamard seminar began at the Coll~ge de France, where every year he selected topics and speakers from the whole mathematical world. It was a happy time at home: bright children, music, good friends like Borel, Lebesgue, the physicists Paul Langevin and Jean Perrin and their German friend Albert Einstein, who played violin with Hadamard. However, it was not possible to ignore the external world. Captain Alfred Dreyfus, a relative of Hadamard, attached to the Deuxi~me Bureau of the French Etat-Major, had been accused of spying, was judged, condemned, and deported in rather strange circumstances, with no proof and a heavy climate of antiSemitism (1894). Hadamard did not feel involved at

the beginning. But the truth was revealed fact by fact: Dreyfus was innocent, another officer was guilty. France was divided into dreyfusards (Zola, Clemenceau, Hadamard) and antidreyfusards (most dignitaries in the Army and the Catholic Church). Only in 1906 was Captain Dreyfus reinstated in his rank and civil rights. Meanwhile, Hadamard took an active part in the Dreyfusan Ligue des droits de l'homme, in which he remained a member of the Central Committee until late in life and then was replaced by his daughter Jacqueline. L'Affaire Dreyfus played an important role in France and in Hadamard's life. As an example of the climate of the time, Charles Hermite once, crossing Hadamard, shouts: "'Hadamard, vous ~tes un traitre'" (you are a traitor). Then, before Hadamard can react, " V o u s avez trahi l' analyse pour la gdorn~trie" (you betrayed analysis for geometry). A typical bad joke of the time, when treason and treachery were key words of the social life in France. A tragedy of another magnitude began in 1914. Pierre and Etienne, the elder sons of Hadamard, were killed in 1916, and the obituaries written by Jacques Hadamard show that he was struck in a terrible way. Almost all the students of Ecole Polytechnique and Ecole Normale Sup6rieure were killed too. The First World War was a scientific disaster for France. Between 1918 and 1939 Hadamard was oriented to the left, mainly due to the rise of Nazism in Germany and its copies in France. He was a convinced antifascist. In 1938 he was one of the few in France--apart from c o m m u n i s t s - - t o be indignant towards the Munich a g r e e m e n t b e t w e e n Chamberlain, Daladier, Hitler, and Mussolini about Czechoslovakia. Here is a letter the H a d a m a r d s sent to their colleagues in Prague (copy due to Dr. Vladimir Korinek and Jacqueline Hadamard): Dear friends, We need not tell you how close we feel to you in these days of mourning. You at least do not wear any shame and can be proud of maintaining your honour untouched. The behaviour of your President, his constant dignity, are and will be admired all around the world, and hopefully will be retained by immanent justice. Above the Western governments who betrayed you and betrayed us we shake hands with you. Jacques and Louise Hadamard. (Mes chers amzs, En ces jours de deuil, est-zl besoin de vous dire comhen nous sommes proches de vous. Du moins n'avez-vous pas la honte et pouvez-vous avolr la fiert~ de dire que vous avez maintenu haut votre honneur. L'attitude de votre Pr~szdent, la &gnit~ dont il ne s'est pas ddpart~ un instant, ont fait l'admiraflon de tous et feront celle de l'Histoire, et, on est en droit de l'esp~rer, la justzce immanente s'en souviendra. Par dessus les gouvernements occzdentaux qui vous ont trahzs et nous ont trahis nous vous serrons la main.)

After the French defeat in 1940, Jacques, Louise, and Jacqueline got their American visa and escaped the racial persecutions, thanks to the Joint Jewish Committee and to a very active and efficient young

Jacques Hadamard with his daughter Jacqueline.

Canadian scientist, Louis R a p k i n e - - I already mentioned the tragicomic incident of the lost briefcase. They settled in N e w York. Hadamard lectured at Columbia University and wrote his beautiful book on the psychology of invention in mathematics. A last personal tragedy was the death of Mathieu, his last son, officer in the Free French Forces, in 1942. When he returned to Paris in 1944, Hadamard had no apartment left, none of the books and papers that he had left behind, and had to begin a new life. He got involved more and more in social and political questions. His daughter Jacqueline joined the Communist Party. He became very active in the Peace Movement, then led by Fr~d6ric Joliot-Curie, and he was sometimes considered a communist. This happened to have an effect on the world scene. In 1950, the first International Congress of Mathematicians after the war was held in Cambridge, Massachusetts, and Hadamard was elected honorary president of the Congress. However, it was the time of cold war and McCarthyism, and at first Hadamard was denied an American entrance visa, as was Laurent Schwartz, who was to receive the Fields Medal. A few French mathematicians would have agreed to attend the Congress anyhow, but most said they would not, and the American mathematical community was strong e n o u g h to convince the American g o v e r n m e n t to change its position. THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991 2 5

Hadamard had been involved in mathematical education since his youth. In 1932 he became president of the Commission Internationale de l'Enseignement Math4matique, which had been created by Felix Klein, George Greenhill, and Henri Fehr in 1908. In 1936 he retired from Coll~ge de France. From his scientific Jubil6 in 1936 until his death in 1963, Hadamard received many more h o n o u r s than I could mention in a few pages. He was particularly pleased when he received the FeltrineUi Prize, which had just been created in 1955 in order to replace the Nobel Prize in mathematics; the speech was made by Vito Volterra, the Prize was given by the President of the Republic--it was really a great event in Italy--and Hadamard was angry only about the French ambassador w h o did not find it convenient to attend the ceremony (maybe such anger was part of the pleasure!). Before coming to the second part, let me thank Jacqueline Hadamard* for her help; most unpublished anecdotes are taken from a personal manuscript of hers. Some others are recollected from talks by my teacher, Szolem Mandelbrojt--I am just sorry that I am not up to his warmth and wit. I personally had a few occasions to meet Hadamard in the early 1950s, once at his home, rue Emile Faguet, in one of these university apartments near the Citd Universitaire which were built on his initiative. Otherwise I saw him in different meetings. He was always late, always tiptoed to the floor, asked for a chair, and let his fingers drum until he was invited to say a few words. What I remember best is the expression of his face. From the photographs you can appreciate its sharp and biblical beauty, but more striking was the acuteness of his look and the constant motion of his eyes. More than eighty-five years old, he was not only a living legend, but a truly living mind. Let us come to what remains of him, his written work.

Pearls and Threads There are a b o u t 300 scientific p a p e r s and books written by Jacques Hadamard. I shall select a very few of the results and notions due to him.

The prime number theorem: ~r(x) - r x (1896). It was an old conjecture, known to be derived from the Riemann hypothesis. Hadamard shows that it follows from the nonvanishing of ~(1 + it), and actually proves ~(1 + it) ~ O. This was proved at the same time and independently by Charles de Ia ValiSe Poussin. Hadamard's proof that ~(1 + it) ~ 0 is beautiful. He considers log ~(s) = Xann -s and uses the fact that an I> 0. Since ~(s) has a pole at s = 1, log ~(1 + e) = log 1/e + 0(1) as 9 ~, 0. If ~(s) had a zero at s = 1 + ia, then *Jacquehne Hadamard died shortly after flus article was completed. 26

THE MATHEMATICAL 1NTELLIGENCER VOL 13, NO 1, 1991

log ~(1 + ia + e) = - log lie + 0(1), therefore (using an i> 0) n -~ -~ - 1 for most n's, therefore n -2za ~ 1 for most n's, therefore log ~(1 + 2ia + e) ~- log l/e, impossible since 1 + 2/a is a regular point for ~(s). The upper limit is used in the formula R = (lim sup Ic2,n) -,, giving the radius of convergence R of a Taylor series ECnzn (1892). Though the formula can be found in an 1821 monograph of Cauchy, Hadamard discovered it independently and drew far-reaching conclusions from it. It provided a wealth of results on the location of singular points and was the starting point of a deep investigation of the global behaviour of a function from its Taylor coefficients. Hadamard's booklet on Taylor series and analytic continuation (La s~rie de Taylor et son prolongement analytique, 1901) was the source of most of the 300 papers quoted in Bieberbach's Analytische Fortsetzung (1954).

In 1950, the first International Congress of Mathematicians after the war was held in Cambridge, Massachusetts, and Hadamard was elected honorary president of the Congress. The multiplication theorem (1898) is a pearl of the theory of functions. Roughly speaking, it states that the singular points of E~ anbnz n are contained in the product set AB (= {ot~:ct ~ A, f~ (B}), where A and B are the sets of singular points of ~ anzn and ~ bnzn, respectively. It may remind us of another theorem: the support of a convolution product is contained in the sum of the supports of the factors. Both statements are very similar (and need to be made precise). Hadamard's multiplication theorem is actually an early and excellent e x a m p l e of the p o w e r of c o n v o l u t i o n m e t h o d s - - b e f o r e the notion of convolution was born.

The Hadamard lacunarity condition h n+ l/hn > q > 1 appeared also in connection with analytic continuation. If this condition holds the series E~ anZXn is not continuable across the circle of convergence (1892). This is the prototype of a series of statements of the

type: if a function with a given spectrum enjoys a property on an interval and if the interval is larger than some density of the spectrum, then the property holds everywhere (here, the density is zero and the property is that of being analytic). P61ya, Mandelbrojt, Paley and Wiener, Ingham, and many others contributed theorems of this type. The three circles theorem states that log M(e~) is a convex function of or, where M(r) = suplzl= r If(z)l, f(z) being an analytic function in an annulus r 0 < Iz[ < r 1. Expressed for a strip instead of an annulus, it plays a fundamental role in interpolation theory (theorem of Riesz-Thorin, and the complex method of interpolation).

"Almost convexity" of log M n, where M, = supx [fn(x)l, f bein_g a C| function on ~. Hadamard's theorem is M1 V~0/~, a precise estimate (1914). Precise constants for other inequalities of this type were obtained by A. Kolmogorov, Szolem Mandelbrojt, and Henri Cartan. The problem of quasi-analyticity of a class C(M,) (C| functions g such that Ig(')(x)l ~ M, on a given interval) consists in deciding whether or not the functions in C(Mn) are defined by their germs at a given point. Hadamard asked the question in relation to boundary values of solutions of partial differential equations (1912). The solution was guessed and partially proved by Denjoy, proved completely by Carleman. The linkage between quasi-analyticity and spectral properties was introduced by Mandelbrojt. Here is the general problem of Mandelbrojt: given conditions S on the spectrum, given a class of functions C (such as L1, C, C(M,)" 9 "), consider functions of the class C satisfying S; suppose we know a property of such a function on an interval, or in the neighborhood of a point, or at a point; to what extent does it give information on the whole function? This kind of question appears again and again in partial differential equations, in particular now in control theory. The real part theorem, in its simplest form, says that M(r) ~ 2A(2r), where M(r) = suPlzl= r If(z)l, A(r) = suplzl= r Re f(z), f(z) being an analytic function in Iz < R vanishing at 0, and 2r < R (1892). It plays a key role in the factorization of entire functions.

N o w let m e e x p l a i n h o w H a d a m a r d t u r n e d "traitor," as Hermite said, betraying analysis for geometry. I suppose that Hermite did not have in view the book Lefons de gdorndtrie dldmentaire (1898), which proved quite influential among high school teachers, but Hadamard's works on geodesics, trajectories of differential equations, and analysis situs in the sense of Poincar6 (1896-1910).

Jacques Hadamard Geodesics on surfaces are a beautiful subject and the essential facts were discovered by Hadamard. On surfaces with positive curvature every nonclosed geodesic cuts every closed geodesic infinitely m a n y times (1896). On surfaces with negative curvature any two geodesics have at most one point in common (this was known before Hadamard); given any arc, there exists a unique geodesic arc with the same end points in the same homotopy class. Now comes the most important result, about the asymptotic behaviour. There are four cases: 0) closed geodesics; 1) geodesics tending to infinity; 2) geodesics tending asymptotically to a closed geodesic; 3) erratic geodesics visiting asymptotically neighbourhoods of different closed geodesics. In this last case the sequence of neighbourhoods to visit is quite arbitrary. And Hadamard studies how, given a point P, a geodesic starting from P behaves according to its initial direction 0. Here is a striking use of the notions introduced by Cantor: the set of 0 corresponding to bounded geodesics is perfect and totally disconnected. Therefore the boundedness property is not preserved by an infinitesimal variation of 0. Here is a comment of Hadamard:

It may be that one of the fundamental problems of celestial mechanics, the stability of the solar system, belongs to the category of ill-posed problems. If actually, instead of looking for the stability of the solar system, we consider the analogous question related to geodesics of surfaces with negative curvature, we see that each stable trajectory can be transformed into a totally unstable trajectory going to infinity through an infinitely small change in initial data. Since initial data in astronomical problems are THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

27

known only within a small error, such an error might lead to an absolute perturbation in the result. (Un des probl~mes fondamentaux de la M~camque c~leste, celuz de la stabzht~ du syste~ne solaire, rentre peut-$tre dans la catdgorie des questions mal pos~es. Sz, en effet, on substztue ~ la recherche de la stabiht~ du syst~me solaire la questzon analogue relative aux g~od~szques d'une surface ~ courbure n~gative, on voit que toute trajectoire stable peut ~tre transform~e, par un changement infiniment petzt dans les donn~es mitiales, dans une trajectoire compl~tement instable se perdant ~ l'infini. Or, dans les probl~mes astronomiques, les donndes initiales ne sont ]amms connues qu'avec une certaine erreur. Si petite solt-elle, cette erreur pourrmt amener une perturbation totale et absolue dans le r~sultat cherch~.)

Partial differential equations was a favourite subject of Hadamard, from 1900 until very late in life. The basic book Lefons sur la propagation des ondes et les dquations de l'hydrodynamique appeared in 1903. He considered the Dirichlet problem (one boundary datum at the boundary) and the Cauchy problem (two data on the initial subspace t = 0); for an elliptic operator, the Dirichlet problem is well posed, and for a hyperbolic operator the Cauchy problem is well posed. Well posed in the sense of Hadamard is still a term in use in P. D. E.'s. Hadamard claimed that a wellposed problem is not only one for which the solution exists and is unique for given data; he insisted that the solution should d e p e n d continuously on the data; only such a solution has a physical meaning. In order to elaborate the notion he introduced different types of neighbourhood and continuity. This led to functional spaces, general topology, functional analysis, and to the a priori m e t h o d s u s e d in the modern theory of P. D. E.'s. It is worth noting that the term functional (fonctionnelle) was introduced by Hadamard, inspired by Volterra's fonctions de lignes, and that he gave a general expression for the linear functionals on the class of continuous functions on an interval (equivalent of course to the F. Riesz theorem, but not so simple to write and to use).

Elementary solutions (solutions ~l~mentaires, sometimes translated as fundamental solutions) were also introduced by Hadamard, in a sense slightly different from Laurent Schwartz's, simply because Schwartz's distributions did not exist yet. Clearly, a good part of

28

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

Schwartz's inspiration came from his grand-uncle Hadamard. In the theory of distributions one of the most elaborate notions is finite part (partie finie) of a divergent integral, and it was actually introduced and developed by Hadamard (1908). It is a beautiful and simple calculus of divergent integrals of the form f~x-~f(x)dx (or > 1 and ct non-integer) or generally fv(G(x))-~f(x)dx (x = (xl, 9 "", xn), V included in the hypersurface G = 0). Using this as a tool for the Cauchy problem, Hadamard is able to solve the problem when n is odd, and he introduces the descent method (m~thode de descente) when n is even, solving the problem for n + I and then going d o w n to n. P. D. E.'s were the subject of Hadamard's lectures at Yale University (1920). They resulted in a very inspiring book on the Cauchy problem and hyperbolic P. D. E.'s (English version 1922, French enlarged version 1932). Speaking on Hadamard's work in a ceremony held in 1966 on the occasion of his centenary, Laurent Schwartz expressed a general feeling in saying that Hadamard had a fantastic influence on his time, and that all living analysts were shaped by him, directly or indirectly ("Je crois qu'il a eu une influence ~norme sur son temps, que tousles analystes d"aujourd'hui ont ~t~, directement ou indirectement, formds par Jacques Hadamard."). The few examples I gave are just a very lacunary sample of his mathematical production. I did not enter the realm of mechanics and calculus of variations. Let me go on with three more topics: determinants, set theory, philosophy of mathematics.

The Hadamard inequality on determinants states that a determinant is dominated by the product of the euclidean norms of its columns. In the short paper he wrote on this topic (1893) Hadamard considers determinants whose entries are + 1 or - 1 , and the case when the bound n n/2 is attained; then (except for n = 1 or 2) it is necessary that n = 0 (mod 4) and Hadamard constructs examples for n -- 2k, n = 12, n = 20. Such examples (now obtained up to n = 264) are known as "Hadamard determinants," and they happen to play a role in the theory of error-correcting codes. Hadamard's contribution to the First International Congress of Mathematicians (Ziirich 1897) was on some possible applications of set theory (sur certaines applications possibles de la th~orie des ensembles). Read now, this paper introduces the notion of e-entropy of Kolmogorov; but it was too much in advance and forgotten. In 1905 the Bulletin of the French Mathematical Society published "Five Letters on Set Theory," an exchange of letters b e t w e e n H a d a m a r d , Borel, Lebesgue, and Baire. Hadamard appears as the leading force, not only for carrying out such a correspondence, but for advocating the free use of powerful

methods introduced only a short time earlier (what we now call Zermelo's axiom of choice). I already mentioned his use of Cantor sets in connection with the classification of geodesics. Hadamard's philosophy in mathematics was called idealistic at the time; this means only in opposition to the more constructive approach of Borel. He was influenced by Poincar6 and he has been most influential himself in heuristics. His book on The Psychology of Invention in the Mathematzcal Field (1945) was a constant reference for George P61ya, in particular. This book is an example of what Hadamard was able to write for a general audience. He knew how to speak or write on mathematics, not only in mathematics. He had a very broad view of mathematics. He was able to develop the most abstract parts and at the same time to be inspired by considerations from physics. I already mentioned that he was involved in mathematical education. His book on elementary space geometry has just been published anew (a good sign for the revival of geometry in mathematical training). His papers on scientific education are worth reading: he was more interested in the teaching of experimental sciences than in the teaching of mathematics (again, something to be considered now). Unfortunately, not many of them can be consulted easily. I am sure that the present mini-review will provide the reader with a feeling of frustration. I shall be happy if this frustration leads readers to more substantial articles (I give some in the references) and more than happy if readers go to the inspiring source, the articles and books of Jacques Hadamard.

Jacques Hadamard and E. G. Kogbetliantz at the International Congress of Mathematicians, Harvard, 1950.

8. Szolem Mandelbrojt and Laurent Schwartz, Jacques Hadamard (1865-1963), Bull. Amer. Math. Soc. 71 (1965), 107-129. 9. Francesco Giacomo Tricomi, Commemorazione del Socio straniero Jacques Hadamard, Atti della Accad. Nat. dez Lincei, Rendzcontz classe di sczenze fisiche, matematzche e naturah, 39, 5 (1965), 375-379. By Hadamard

References

On Hadamard 1. M. L. Cartwright, Jacques Hadamard, J. London Math. Soc. 40 (1965), 722-748. 2. Centenaire de Jacques Hadamard, Math6maticien (1865-1963), La jaune et la rouge, n ~ 204, mai 1966. (Ecole Polytechnique). 3. Maurice Fr6chet, Notice n6crologique sur Jacques Hadamard, prononc6e le 23 decembre 1963 devant l'Acad6mie des Sciences (r6sum6: CRAS Paris, tome 257, 4081-4086; version integrale: "Notices et Discours" publi6s par l'Acad6mie des Sciences, Gauthier-Villars 1963). 4. Paul L~vy, Jacques Hadamard, sa vie et son oeuvre. Calcul fonctionnel et questions diverses, L'Enseignement Math~matzque (2)13 (1967), 1-24. 5. Bernard Malgrange, Equations aux d6riv6es partielles, L'Enseignement Math~matique (2)13 (1967), 35-48. 6. Paul Malliavin, Quelques aspects de l'oeuvre de Jacques Hadamard en g6om6trie. L'Enseignement Mathdmatique 13(2~me sdrze), (1967), 49-52. 7. Szolem Mandelbrojt, Th6orie des fonctions et th6orie des hombres dans l'oeuvre de Jacques Hadamard, L'Ensezgnement Math~matzque (2)13 (1967), 25-34.

1. Oeuvres de Jacques Hadamard, 4 volumes. Paris: CNRS (1968). 2. Lemons de gdom~trie ~l~mentazre. Vol. 1: g~om~trze plane. Paris: A. Colin (1898). Vol. 2: gdomdtne dans l'espace. Paris: A. Colin (1901). (Reprinted by J. Gabay, Paris, 1988). 3. La s~rie de Taylor et son prolongement analytzque. Paris, Scientia, 1901. Deuxi~me 6dition compl6t6e, en collaboration avec Sz. Mandelbrojt. Paris: Gauthier-ViUars (1926). 4. Lefons sur le calcul des vamations. Paris: Hermann (1910). 5. Lectures on Cauchy's problem m hnear partial differentzal equations. Cambridge-New Haven (1922). (Traduction fran~aise: Paris: Hermann (1932)). 6. Cours d'analyse de l'Ecole Polytechmque, Pans: Hermann. Volume 1 (1927); Volume 2 (1930). 7. The psychology of inventzon in the mathematzcal field, Princeton: Princeton Univ. Press (1945). (Traduction fran~aise. Paris, Albert Blanchard (1959)). 8. La th~ome des dquatzons aux d&ivdes partielles, P6kin, Editions Scientlfiques (1964). Universitd de Parzs-Sud Math~matzques-Bdtiment 425 91405 Orsay Cedex, France THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

29

Mathematics in Marble and Bronze: The Sculpture of Helaman Rolfe Pratt Ferguson J. w . Cannon

Mighty is geometry; joined with art, resistless.--Euripides At the ACM SIGGRAPH show on computer art, August 1989, a warm, textured bronze sculpture entitled Umbilic Torus NC stole the show. This torus was the creation of mathematician/sculptor H e l a m a n Rolfe Pratt Ferguson, w h o combines mathematical ideas, space-age technology, and ancient artistic tradition to bond art and science. The monthly science magazine Kagaku Asahi, Tokyo, October 1989, featured the Umbilic Torus NC and Ferguson's work. Arts and sciences editor, Itsuo Sakane, wrote, The inseparable relationship between mathematics and art has greatly influenced many cultures from the days of the Egyptian and Greek civilizations to the present. There are people like Roger Penrose of Oxford who wrote essays on the aesthetic appreciation of mathematics. There also is Buckminster Fuller, who imagined polyhedral art on a cosmic scale. Mathematical art up to the present has tended to appear inorganic and cold. I feel relief and salvation in the warmth of Helaman's works. This is surely the result of his continuing haptic dialog between his abstract thoughts and his physical materials, a dialog that has been going on over a period of many years.

will soon be able to make his living from his art alone: Last year he grossed a university professor's salary equivalent in sales. All went back into the purchase of specialized equipment and materials.

Numerically Controlled Umbilic Torus Umbilic Torus NC looks more like a Mesoamerican calendar with its Mayan inscriptions than a space-age creation. This illusion rapidly d i s a p p e a r s as the sculptor describes its conception and creation. Ferg u s o n isolated four separate media in w h i c h he worked to create this sculpture: Mathematical--Transcendental Parametric Equations;

Mathematicians had the opportunity to see more of Ferguson's work at the AMS-MAA meetings in Columbus, Ohio, in August 1990 at an exhibit entitled "Sixteen Theorems in Bronze and Stone." Helaman Ferguson earns his living as a mathematician. He taught mathematics at Brigham Young University for seventeen years. He is a publishing research mathematician, who currently designs new algorithms for scientific visualization and computeraided manufacture. The sculptor creates mathematical art to satisfy a compulsion, his compulsion to view mathematics as a beauty so real that it must be communicated in tangible terms. He is celebrating mathematics. Ferguson 30

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 19 1991 Spnnger-Verlag New York

C o m p u t e r G r a p h i c a l - - P o l a r i z e d Filter Stereo Images; Computer Machining--Three Axis Numerical Control Mill; Lost Wax Casting--Silicon Bronze. One mathematical idea supplies the body of the piece, another supplies the surface texture. The body consists of real binary cubic forms a x 3 + bx2y + cxy 2 + dy a

that may be parametrized by the coefficients (a, b, c, d) in real 4-space or, up to real positive scaling, by projection into the 3-dimensional sphere. The group of 2 x 2 invertible matrices with real entries acts on the variables x and y as a linear change of variables and thereby acts on all binary forms. The 4-dimensional group action naturally partitions 4-space into 4 orbits the hyperbolic umbilics, the elliptic umbilics, the parabolic umbilics, and the exceptionals. The artistic problem is that of viewing the orbits in Euclidean space in their most beautiful form. He explains: I selectively previewed computer generated images of toroids of hypocycloid cross-sections, discarding and eliminating many possibilities to select the current. You may ask, and what was the basis of my selection? I respond, in part, many years of direct carving of tori in many forms. Staring at a computer screen, even in stereo, need mean absolutely nothing three-dimensionally. (Television programs of undersea coral reefs and their inhabitants take on an entirely different meaning after one has actually been down there in scuba.) My viewing computer graphics images as a sculptor brings a unique perspective.

o

What were the viewing tools? Vax 8600 Fortran/C languages, MACSYMA symbolic algebra l a n g u a g e , Movie BYU, 4129 Tektronix Color Stereographical Workstation, a domain-based CAD/CAM, and VTerm for the Vax/PC/NC controller/Kearney-Trekker VB-2 interface. The surface-texture model is formed from Hilbert's version of Peano's surface-filling curve. The classical technique for removing material on a milling machine consists in creating parallel or crosshatched grooves by a back-and-forth pattern. The Peano curve supplies a technologically superior method. Ferguson used the fifth stage of the Hilbert curve. The surface-filling curve can be programmed into a computer-controlled milling machine that can supply the texture in one pass. The texture needs to be fingered to be fully appredated. Surface-filling curves have an interesting visual history. It is difficult to know whether Ferguson's choice of the Hilbert space-filling curve as surface texture was dictated by its artistic merits or by its visual history.

-I-

Top: The Umbilic Torus NC: Hyperbolic points at infinity; center: Computer-generated stereo pair; bottom: Closeup of the Hilbert texture on the toms. THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991

31

1

I

2

I

3

I

4

[

123

[

16

IIIIIII', 7

10

8

A

,,,,,,

',

I

t

11

9

12

3

14

13

2

15

16

B

1. 2 ~1111 I

II I',I1[[I

C

Hilbert's space-filling curve.

The first space-filling curve was of course described by Peano. But Hilbert was the first to draw a picture of a space-filling curve. Here is Hilbert's description in the year 1891 [5], translated from the original German and keyed to the figures above: Recently Peano showed in the MathernatzscheAnnalen by means of an arithmetic argument how the points of a line segment can be mapped continuously onto the points of a surface. The functions necessary for such a mapping can be presented with more clarity if one makes use of the following geometric illustrations. First of all we divide the segment to be mapped--say a segment of length 1--into four equal parts 1,2,3,4 and we divide the surface, which we take in the form of a square of edge length 1, by two perpendicular segments into four equal squares 1,2,3,4 (Figure A). Secondly, we divide each of the subsegments 1,2,3,4 again into four equal parts, so that we obtain on the original segment 16 subsegments 1,2,3. . . . . 16; at the same time each of the four squares 1,2,3,4 is divided into four equal squares, and into the 16 squares thus obtained the numbers 1,2, . . . , 16 are written, whereby the order of the squares is to be chosen in such a way that each successive square is adjacent to its predecessor along an edge (Figure B). If we imagine that this procedure is continued--Figure C illustrates the next step--it is easily seen .how one can associate to each given point of the segment a single definite point of the square. Once the basic form and surface texture had been chosen, precise coordinates needed to be created for the Kearney-Trekker VB-2 three-axis CNC machining center. This space-age machining center drastically reduces the time needed to cut the basic shape of a sculpture. The coordinates of both the basic shape and the texture were produced from the same transcendental parametric equations used in the original computer visualization process. The n u m b e r s became input for generating G-codes to drive the numerically controlled (NC) milling machine which cut the first positive image patterns. The cutting tool was a highspeed spinning ball-end mill. The first positive image was cut in a high-density foam material, the cutting 32 THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

tool followed a three-dimensional space curve. The space curve was the image of the fifth order Hilbert curve on the torus with hypocycloid cross-section. The grooves of the Hilbert curve and the antique green patina coloration were chosen to recall textures from Mesoamerica and the Shang Dynasty. At this point, the sculpture entered the final phase, returning to ancient technique. Ferguson sculpted the original positive image to transcend the industrial processes. The first negative mold was cast from the original first positive in flexible silicon rubber with piaster cloth shells for support. From the first negative mold he obtained a positive wax image, then a negative ceramic image, and a final positive bronze image, in the manner of traditional lost-wax foundry methods. This broad spectrum of knowledge ancient and modem, of technique ancient and modem, of process ancient and modem, of aesthetic intent ancient and m o d e m , slices through our culture along both h u m a n and technological axes. Nature versus Nurture

Ferguson, though very soft-spoken, has a flair. In winter weather he sometimes wraps his 6-foot 3-inch frame in a dramatic cape or cloak, custom-made by his wife Claire. He writes at the blackboard with both hands simultaneously. It is as though he were constructing sentences and descriptions as one would build a house rather than writing them in the usual linear way. Up goes the scaffolding. Then come the floor, ceiling, walls. Next come paint and decoration. He wears the mathematician's uniform with ease: beard and crumpled clothes. He joggles; that is, while juggling, he walks or runs. See the Guiness Book of World Records, 1988, p. 460, also 1989, p. 464, for his 50-mile world distance joggling record. Ferguson wanted to set this record in part because his neighbor at the time, a sports physiologist and orthopaedic sur-

o x

Computer-driven milling machine in action.

geon, said it would be neurologically and physically impossible to sustain that much hand, eye, foot coordination. He accompanied Ferguson in a vehicle, as he said, "to pick Helaman up when Helaman fell over." The Pratt name in Ferguson's pedigree refers to an old Utah pioneer family that includes a large number of the Rocky Mountain scientists and mathematicians. Ferguson's g r e a t - g r e a t - g r a n d u n c l e was M o r m o n apostle Orson Pratt, a self-taught mathematician and scientist, perhaps the first on the Rocky Mountain frontier. Orson Pratt was one of the two scouts who first entered the Salt Lake Valley before Brigham Young. He measured the trek west with an odometer he had constructed from a wagon wheel. Pratt surveyed the trek and set bench marks. He took time out on one of his eleven missions for his church to publish a book in Liverpool, England, entitled Cubic and Biquadratic Equations. It d i s c u s s e d the s t a n d a r d methods for solving polynomial equations of degree greater than two and presented his own numerical methods for finding approximate solutions quickly. Ferguson's earliest olfactory memories were of the smells of turpentine, oil paints, brushes, and canvas. His natural father was a commercial artist who had met his mother in an art school in Los Angeles in the late 1930s. His parents had just moved into a house in Salt Lake City (Helaman was 3, his sister 1), when his mother stepped into the backyard to take clothing from the line and was killed by a lightning bolt. When the children's father was drafted into the armed services in World War II and was sent to the Pacific, their grandmother took the youngsters in hand. The young boy was declared hopelessly uneducable by his kindergarten teacher, who said that he had no future. Mormon relatives were shocked w h e n the grandmother served the young children forbidden coffee. They a r r a n g e d for the two small children to be adopted by distant relatives in Palmyra, New York.

The adoptive parents were a Scotch-Irish immigrant and his wife. For some unknown number of generations the fathers of this new father had been stone masons. At the age of six, Ferguson was "apprenticed" to this charitable cut-and-rubble stone mason. His genetic endowment for higher mathematics and art especially mystified these good folk, who environmentally endowed the boy with the appreciation of the t r a n s f o r m a t i o n of s o m e t h i n g w o r t h l e s s a n d common, like stone, into something treasurable. This dual background merged with a liberal arts undergraduate apprenticeship in art and sculpture and a graduate apprenticeship in science and mathematics. He studied at Hamilton College, the University of Wisconsin-Madison, Brigham Young University, and the University of Washington. He received an undergraduate degree in liberal arts, a doctorate in mathematics. THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

33

The AMS Sculpture.

In 1968 Ferguson and his wife were struggling graduate students at the University of Washington. To put food on the table, he held his first large art show, comprising many landscapes and other paintings, which sold well. Some of the paintings had been commissioned by a Seattle entrepreneur, others had collected in the household over the years, and many were created specifically for the sale. At the same time, he carved a piece of limestone with a carpenter's hammer and a railroad spike; this and a two-dimensional sculpture painting he did were among his first art works that were consciously mathematical. Posters and Postcards

Ferguson is best known to the mathematical community through two posters and two postcards. One of these posters pictures the beautiful bronze Umbilic Torus NC, which we have already discussed. This poster is issued by the Joint Policy Board for Mathematics. The other poster is issued by the American Mathematical Society and celebrates the huge, pure white marble torus with cross-cap and vector field given to the Society by the Mathematical Association of America in commemoration of the AMS Centennial. Visitors can view this sculpture at AMS headquarters in Providence, Rhode Island. The texture represents a vector field on the surface. The effect is strangely that of molded plastic. The viewer should run hands over the textured marble surface to see how the surface turns inside out at passage through the cross-cap, the immersed M6bius band that represents the projective plane summand of this surface. Ferguson carved the Providence piece in an 11-day reduction from 1200 pounds to 550 pounds. Each day required from 10 to 16 hours of hard labor. 34

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

O

Three views of bronze Torus with Cross-Cap.

The two postcards are issued by the Joint Policy Board for Mathematics. The first of the postcards depicts a bronze, highly polished version of the torus with cross-cap. The obvious question posed by the unretouched photograph of this brilliant model is this: where is the photographer? None of the reflections from this golden object reveals a photographer. N e w York photographer, Terry Clough, studied the object and determined to photograph it through a pinhole. The result is striking. The second of the postcards is of Alexander's H o m e d Sphere. The two postcards have rather cryptic titles (see i) and ii) in the quote below). At first glance the titles seem absurd. Simple algebra contradicts the first. The standard notion of identity contradicts the second. The point of the first is that, in the presence of a M6bius band, the nonorientable handle of a Klein bottle can be "untied" and turned into an orientable handle of a torus. Without the MObius band the untying is impossible. The point of the second is that the 2-sphere can be embedded in more than one way in the 3-sphere. The fundamental group of the complement of the first will be trivial, while the fundamental group of the complement of the second need not be so. The reader will enjoy contemplating these important mathematical statements. Ferguson tells the following amusing story about these rifles. On the way back from Boulder, Colorado, I had an interesting conversation with Andrew Gleason, former president of the American Mathematical Society--he thought the titles on the JPBM postcards featuring two of my bronzes were giving mathematics a bad name: i) ~1(S3 \ S2) # ~z(S3 \ $2). I told him this is a very common situation in mathematics and elsewhere. How many Andrews did he know? So Andy # Andy, and Andy is not well defined. ii) 3x = x + h, 2x ~ h. The cross-cap or MObius band acts as a catalyst and allows a transformation that would not be possible otherwise. The Klein bottle is not a torus. The moral is that we must have algebra with meaning, not just dumb rules blindly followed. Ferguson has experimented a great deal with the implications of wild surfaces in dimension three. The standard examples are the Alexander H o m e d Sphere, the Bing Hooked Rug, and the Fox-Artin Feeler. As a first sculpture of this type, we mention the wild singular toms, related to the Fox-Artin Feeler and carved from 2000 pounds of Carrara marble, which appears in the Mary Major Collection in Orem, Utah. As a second example, we consider the Alexander Horned Sphere. The Alexander H o m e d Sphere has a particularly interesting history. In 1921 J. W. Alexander a n n o u n c e d a generalization of the classical Schoenflies theorem. The Schoenflies theorem states that every simple closed curve in the plane bounds a topological disk in the plane. Alexander asserted that e v e r y topological 2-sphere in Euclidean 3-space

@

r\

f13 ")'3

~

..b

] f

/ /

I

s~

Alexander's original drawing. THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991

35

b o u n d s a topological 3-dimensional ball. That Alexander was mistaken was already k n o w n . Even before A l e x a n d e r ' s a n n o u n c e m e n t , A n t o i n e [2] h a d p u b lished a counterexample. Antoine's description was, however, completely verbal; and, as such, it had not become well k n o w n . W h e n the example was called to Alexander's attention, he d e v e l o p e d his o w n example [1], w h i c h has c o m e to be k n o w n as the A l e x a n d e r H o r n e d Sphere. A l e x a n d e r also called attention in p r i n t to the A n t o i n e e x a m p l e s a n d d e d u c e d s o m e properties thereof. Nevertheless, it is Alexander's example, with its a c c o m p a n y i n g picture, that b e c a m e famous. Alexander's figure can hardly be called artistic. Ferguson's sculptures of the Alexander H o r n e d Sphere are beautiful. I h a d t h o u g h t a b o u t the A l e x a n d e r H o r n e d Sphere a n d studied its properties and drawn it on the blackboard for years before I saw Ferguson's sculpture. I t h o u g h t that I really had a feeling for what it l o o k e d like. But s o m e h o w I h a d m i s s e d in m y t h o u g h t some of the essentially 3-dimensional aspects of this beautiful object, some of its potential beauty. I was almost o v e r w h e l m e d to see, touch, a n d heft the bronze. This b r o n z e is the first thing I see w h e n I awake in the morning. Ferguson calls this his Wildfire Bronze, IlL It is fairly small: 12 x 12 x 5 inches, 12 pounds. It is m a d e of silicon bronze, cast, carved, patinaed. This sculpture gives six stages of an infinitely b i f u r c a t e d c o n s t r u c t i o n of a wild i m b e d d i n g of a sphere in three dimensions. This has the appearance of a binary tree in which the branches are linked in a w a y that articulates the t h r e e - d i m e n s i o n a l ambient space and creates a sphere exterior that is not simply c o n n e c t e d . The wild s p h e r e interior is simply connected. The lower branches are the e x t e n d e d bodies of the m o d e r n dancer, the u p p e r b r o n z e arms are the arms of ballerinas stretching u p w a r d , or the supplicating arms of the fearful in Picasso's Guernica. Ferguson's one b r o n z e by no m e a n s exhausts the artistic possibilities of the Alexander H o r n e d Sphere: He has d e v e l o p e d a whole sequence of b r o n z e versions of this sphere. A H o r n e d Sphere with only one stage looks like a seal or dolphin. As the arms branch, the effect changes subtly from stage to stage. My favorite v e r s i o n of the A l e x a n d e r H o r n e d S p h e r e is Whaledream II. The g e n r e form is completely reminiscent of Eskimo sculpture with soft line. The sculpture is large: 30 x 30 x 15 inches, 550 p o u n d s . This sculpture, like the P r o v i d e n c e piece, is of Carrara white marble from the m o u n t a i n s in n o r t h e r n Italy. Whaledream II is o n loan from the artist to the Lee Library on the campus of Brigham Young University. This sculpture gives 23/4 stages of an infinitely bifurcated construction of Alexander's wild i m b e d d i n g of a sphere in three dimensions. There are three stages to account for: from the base the large handles are consciously articulated like the h u m a n neck in its attachment to 36

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

the torso; the handles of the next stage are like the b e a u t i f u l arms of G r e e k m a r b l e s t a t u a r y l i n k e d in friendship; and the final handles grasp one a n o t h e r like h a n d s symbolizing the h a n d s h a k e of peace. The literal surface of this sculptural presentation of a wild s p h e r e is isotopic to a s p h e r e with six handles; the observer will enjoy visualizing this smooth deformation.

Sculptural Beauty That Moves Souls and Mathematical Truth That Moves Minds F e r g u s o n ' s favorite m e d i u m is stone: Canvas and paint deteriorate. A bronze could be melted in war time. No one knows how to incorporate stone in the war machine. Stone is permanent. The most permanent stones we know how to carve are gramte and basalt. In the future I hope to carve in these. I believe that my work has importance beyond our small community of mathematicians. I believe that my work in universities, colleges, even high schools and grade schools, even similar work by someone else, will benefit our society. As you know, mathematics has tended to be largely unknown partly because few mathematicians seem willing to expend the effort to communicate with less prepared souls than themselves. I have seen people change their feelings about mathematics dramatically while viewing one of my sculptures. I have also seen their feelings about art and its possibilitaes change too. The great sacred myths of our day are mathematical equations; their heroes are equal signs. Boats, planes, and satellites are confidently created upon these myths. Do we have cause to celebrate these sublime achievements of the human mind m sculpture as well as function? Yes, and I so celebrate with feeling and knowledge. There was a time when sculpture celebrated the highest and noblest our civilization could muster--many people miss that in contemporary art, and when they see it once again it brings them joy. Beauty and truth: these my sculpture celebrates and incorporates. Sculptural beauty that moves souls and mathematical truth that moves minds. That is my work. F e r g u s o n ' s feelings are reminiscent of those of H e n r i Poincar6, w h o wrote [9]: Truth should not be feared, for it alone is beautiful . . . . The search for truth should be the goal of our activities; it is the sole end worthy of them. Doubtless we should first bend our efforts to assuage human suffering, but why? Not to suffer is a negative ideal more surely attained by the annihilation of the world. If we wish more and more to free man from material cares, it is that he may be able to employ the liberty obtained in the study and contemplation of truth. When I speak here of truth, assuredly I refer first to scientific truth: but I also mean moral truth, of which what we call justice is only one aspect. It may seem that I am misusing words, that I combine thus under the same name two things having nothing in common; that scientific truth, which is demonstrated, can in no way be likened to moral truth, which is felt. And yet I cannot separate them, and whosoever loves the one cannot help loving the other. To find the one, as well as to find the other, it is necessary to free the soul completely from prejudice and from passion; it is necessary to attain absolute sincerity. These two sorts of truth when discovered give

the same joy; each when perceived beams with the same splendor, so that we must see it or close our eyes. Lastly, both attract us and flee from us; they are never fixed: when we think to have reached them, we find that we have still to advance, and he who pursues them is condemned never to know repose. It must be added that those who fear the one will also fear the other; for they are the ones who in everything are concerned above all with consequences. In a word, I liken the two truths, because the same reasons make us love them and because the same reasons make us fear them.

Sculpture, the High Dimensional Art

person w h o m a y have been left cold by photographs of ancient Greek a n d R o m a n sculpture, is startled, e v e n o v e r w h e l m e d , by the lovely translucence, the loving care, the attractive physical presence of a piece of sculpture that can be personally experienced. Most of us seem to be constructed, or trained, or limited in such a way that we make use of only a few of the possible perceptions available to us in our mathematics. Most of us have our favorite points of view, our favorite versions of proofs that can be expressed in m a n y different languages. After we have successfully translated someone's insight into our o w n favorite language, say from geometry to algebra or from algebra

As noted by Morris Kline [6], mathematical analysis has to date successfully dealt primarily with sight and s o u n d a m o n g the senses. A n d as to t h e i n t e r n a l h u m a n interface, even those two senses are not as yet well understood. Touch, smell, and taste have hardly been tapped. All of these senses taken together supply each individual with a high-dimensional view of the physical world. "In examining sculpture," Ferguson says, " o n e s h o u l d use all of the body, not just the eyes. Our feet supply a dimension that brings us in proximity to the object. In order to feel the object we bring into play the joints of feet, ankles, knees, waist, neck, s h o u l d e r s , elbows, wrists, a n d fingers. The nerve impulses from fingertips and palms add to the sensory perception. With truly binocular vision at play with the twelve ocular muscles, with all of the muscles and joints a n d nerves sending their separate messages to our brains, w h e n we view a n d feel the object at close range, we get a h u g e - d i m e n s i o n a l view of a physical s c u l p t u r e . " It is h a r d l y surprising that the

At right: Whaledream H as seen from the front or six o'clock perspective; then, below, as seen from the eight-, ten-, and four-o'clock perspective.

a~

o_

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

37

o

Thurston's Hyperbolic Knotted Wye.

to geometry, we are tempted to say, "'So that is what he meant. If that is what he meant, then w h y didn't he say so?" For example, we have the following delightful expression from Louis Solomon, algebraist at the University of Wisconsin-Madison: After working with Peter Oflik for some time, I finally came to the conclusion that there were certain theorems in topology that would never be proved. (Meaning that the proofs would never be translated into the algebraic language that Solomon would accept as real proof.) Ferguson seems to have retained something that others have lost. Somehow we have lost that synesthetic sense that in very early childhood either coordinated, or perhaps confused, the different senses in the brain. There is great and untapped power in the coordination of our senses. This power does not remain untapped in everyone. The most powerful memory on record was that of "'S" reported by A. R. Luria in The Mind of a Mnemonist [7]. Luria is a highly reputable medical doctor who studied the structure of the brain by studying the effect of brain lesions on behavior and memory. Because of Luria's reputation, when others noted the extremely unusual memory of "S," they sent "S" to Luria. Luria then studied "'S" for a period of more than twenty years. In all of those years, Luria was unable to find any of the sessions spent with "S" that "S" could not reproduce in detail from memory. Luria spent m a n y years trying to fathom the basis of "S'" 's incredible memory. He finally came to believe "'S" 's assertion that his memories coordinated all of the senses. This man's memory combined the synesthetic with the eidetic. Luria quoted from "S": To this day I can't escape from seeing colors when I hear sounds. What first strikes me is the color of someone's voice. Then it fades o f f . . , for it does interfere. If, say, a person says something, I see the word; but should another person's voice break in, blurs appear. These creep into the syllables of the words and I can't make out what is being said. (page 25) 38

THE MATHEMATICAL INTELLIGENCER VOL 13, NO I, 1991

For me 2, 4, 6, 5 are not just numbers. They have forms. 1 is a pointed number--which has nothing to do with the way it's written. It's because it's somehow firm and complete. 2 is flatter, rectangular, whitish in color, sometimes almost a gray. 3 is a pointed segment which rotates. 4 is also square and dull; it looks like 2 but has more substance to it, it's thicker. 5 is absolutely complete and takes the form of a cone or a tower--something substantial. 6, the first number after 5, has a whitish hue; 8 somehow has a naive quality, it's milky blue like l i m e . . . (page 26). William Thurston of Princeton is one of the great geometers of our day. He too uses the senses in ways that are unusual for many of us. Thurston once told me that many of his geometric insights are explicitly kinesthetic. He moves about physically in his mathematical world. He is not always able to express his insights in a language that others can understand. Only after a good deal of practice has he been able to speak his insights in a language that others can accept. He finds it necessary to translate mathematics from body into mind. Ferguson is translating mathematics from mind into body.

"This Beauty, in M y M i n d , Is What I Have Loved" Ferguson's claim is that by translating mathematical ideas into physical realization, many who are unprepared to speak our mathematical language can get a rich feeling for what we as mathematicians "had in mind." My own personal love for Mandelbrot's formalization and computer realization of fractals is that I can look into the wondering eyes of my children and say, "'That is what I have been thinking about. That beauty, in my mind, is what I have loved." I feel the same thankfulness to Ferguson and those like him, who can not only see that beauty but can express it physically in a way that is even more beautiful than I had imagined. I include at this point a bit from a review [3] I wrote of Mandelbrot's book. Much remains to be done to

Helaman Ferguson transform our beautiful world of mathematics into images, objects, feelings, senses that all of the world can appreciate. Benoit Mandelbrot argued [8], "Any natural fluctuation can be processed to be h e a r d - - a s implied by the term noise." More generally, a n y stream of data can be coded into any medium that is sufficiently rich. For example, such a stream can be processed as geometric shapes or pictures. One discovers that algorithms have intrinsic shapes, that those shapes are complex and beautiful, that they have underlying geometries, underlying self-similarity or invariance properties. Mathematicians are in the new position of being able to explore extensively an almost unchartered area, the geometry of algorithms. "What's in a name? That which we call a rose by any other name would smell as sweet . . . . "Shakespeare, Romeo and Juhet, II, ii, 43. Or would it? Let us choose as a name for each individual the algorithm encoded in its genetic structure or in the governing physical or spiritual laws of its creation. If we change the name, then we change the code, change the individual, change the shape, smell, and feel of the individual. If we think in abstract terms, then it amazes us that algorithms have shapes, but when we realize that much of life about us is algorithmic, then we see and smell and hear in every face, bud, flower, and tree, the shape, smell, and sound of an algorithm, a marvelous creation. Ferguson would love to see your favorite theorem in the beauty of stone and bronze. Write to him with your ideas. Perhaps the Riemann mapping and its level sets would be beautiful. Are there other graphs that would be beautiful in stone? the zeta function? Thurston has taught us to understand surface diffeomorphisms through measured foliations or geodesic laminations of surfaces. Show these laminations and foliations in stone and bronze. Bing showed that the identity sewing of two Alexander Horned Spheres yielded S3. How would the spherical interface sit in $3?

How could one show Bing's Sling, or Antoine's Necklace? How would one sculpt algebraic curves? The Peitgen-Richter fractal landscapes are reminiscent of Arches National Monument and Canyonlands. They seem natural for stone. The artist has natural difficulties in our society. Ferguson does his art in his spare time. Few of us imagine the strains put on a life divided between profession and obsession. An old pioneer story describes a large group of men who were discussing a man who was in great financial distress. Great sympathy was being expressed when one man said, "Well, I am sorry fifty dollars' worth," and laid the money down. The other sympathizers disappeared. No one else was sorry enough to do anything. Perhaps our only true appreciation for the artists who interpret us to ourselves and to others is to buy their work!? Stone a n d b r o n z e s e e m so a p p r o p r i a t e to the beauties we deal with day by day. Carl Faith [4] has explained the mathematicians' emotions in a marvelous way. He writes (page IX): Mathematicians have a theory about everything--I have a theory about mathematicians: so intent are we on our scribbles and our equations, we are likely to be adjudged too detached, and therefore too cold and unemotional. My theory is that it is because mathematicians are so emotional that they can become mathematicians. They, at least, do have the capacity to be moved by the austere beauty mathematics possesses--the "white goddess" the poet Robert Graves spoke of as the muse of poetry--the muse of us all. We mathematicians have our own muse. We see our o w n beauty in the mind. It is wonderful to have someone show the rest of the world how beautiful it is.

References 1. J. W. Alexander, An example of a simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. ScL USA 10 (1924), 8-10. 2. L. A. Antoine, "Sur la possibilit6 d'etendre l'hom6omorphie de deux figures ~ leur voisinages," C. R. Acad. Sci. Paris 171 (1920), 661-664. . J. W. Cannon, review of The Fractal Geometry of Nature, by Benoit Mandelbrot, Freeman: San Francisco (1982). 460 pp., Amer. Math. Monthly 91 (1984), 594-598. 4. Carl Faith, Algebra: Rings, Modules and Categories I, New York: Springer-Verlag, (1973). 5. David Hilbert, Uber die stetige Abbildung einer Linie auf ein Fl/ichenstuck, Math. Ann. 38 (1891), 459-460. 6. Morris Kline, Mathematzcs: A Cultural Approach, Reading, Massachusetts: Addison-Wesley, (1962). 7. A. R. Luria, The Mind of a Mnemomst, New York: Basic Books, Inc. (1968). 8. B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco: Freeman, (1983), 249-250. 9. Henri Poincar6, The Value of Science, New York: Dover, (1958), 11-12. Helaman Ferguson J. W. Cannon 10512 Pilla Terra Court Department of Mathematzcs Laurel, MD 20723 USA Brigham Young University Provo, UT 84602 USA THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991 3 9

David Gale*

This column was originally called The Problem Corner. Under the previous editor the title was changed to Mathematical Entertainments, the idea being to broaden its content to include, for example, contests, hzstorical notes, and the like. It is my intention, starting with this issue, to continue and even accelerate this trend. While problems and puzzles will still be welcome, there will also be emphasis on mathematical games, paradoxes, anecdotes, computer discoveries. In fact, the concept of an entertainment seems sufficiently vague to allow a wide variety of material, provided only that it should not require technical expertise in any particular area of mathematics. I hope readers of the Intelligencer will find this sort of program congenial. Needless to say the success of the endeavor will depend crucially on getting good contributions from you, the readers, which are herewith eagerly solicited.

The following theorem-joke was contributed by Hendrik Lenstra "Perfect squares don't exist. Suppose that n is a perfect square. Look at the odd divisors of n. They all divide the largest of them, which is itself a square, say d2. This shows that the odd divisors of n come in pairs a,b, where a 9b = d 2. Only d is paired to itself. Therefore the number of odd divisors of n is odd. This implies that the sum of all divisors of n is also odd. In particular, it is not 2n. Hence n is not perfect, a contradiction: perfect squares don't exist." Get it? Remark: It seems the joke works only in English. In other languages a square is just a square (the theorem, however, is international).

Computer-Generated Mysteries The heading above describes a feature I would like to incorporate in these columns on a regular basis. Many mathematicians feel that the main impact of comp u t e r s on m a t h e m a t i c s has b e e n not in solving problems, as one might have expected, but rather in posing them. The prime illustration is probably the recent activity in discrete dynamical systems stimulated by the celebrated computer experiments of Mitchell Feigenbaum. Perhaps explorations is a better description of this work, the appropriate analogy being not with physics or biology but with astronomy. The computer is the mathematician's telescope, which when used intelligently helps him/her to find out what is "out there" in the mathematical universe (this whole development should be a source of satisfaction to the Platonists who have been saying all along that, like stars and galaxies, mathematical phenomena are discovered, not invented). Some quite recent work to be described in the next paragraphs gives another striking example of a set of phenomena that would probably never have been observed without the use of computers.

The Strange and Surprising Saga of the Somos Sequences In investigating properties of elliptic theta functions, Michael Somos discovered an infinite sequence whose first 15 terms are 1,1,1,1,1,1,3,5,9,23, 75, 421, 1103, 5047, 41783. The sequence is defined by a z = 1 for 0 ~ i ~ 5 and

*Column editor's address: D e p a r t m e n t of M a t h e m a t i c s , University of California, Berkeley, C A 94720 USA.

40

a, = (an-la,-s + a,-2a,-4 + a2-3)/an-6

THE MATHEMATICALINTELLIGENCERVOL 13, NO 19 1991 Spnnger-Veflag New York

for n > 5.

(1)

The surprising fact was that the recursion generates integers as far as the eye = computer = telescope can see. In fact, for this example a telescope is not required. A good pair of binoculars will do. With a pocket scientific calculator one will easily, for example, verify that the next numerator above is divisible by 23 so that a~s is again an integer. What's going on?* Upon seeing this p h e n o m e n o n it occurred to a number of people to consider the simpler fourth-order recursion arian_ 4 = a n _ l a n _ 3 + a 2 _ 2 , a 0 = a I = a 2 = a 3 =

1

(2)

Once again all entries turn out to be integers, but in this case the situation is manageable and several people have come up with proofs, the first one being given by Janice Malouf. We present here a variant due to George Bergman. First note that because of (2) every four consecutive terms of the sequence are pairwise relatively prime. For suppose this is true up to an. Then an would have a prime factor p in common with a,_l or an_ 3 if and only if p also divided an_ 2, contrary to the induction hypothesis. We now s h o w inductively that if a n _ 4 . . . . . a n, .... an+ 3 are integers (clearly true for n = 4), then so i s an+ 4, and hence all a,. Writing an_ 3 = a, an_ 2 = b, an_ 1 = c we have anan_ 4 = ac + b2, so an divides ac + b2. By the preceding paragraph we may apply (2) to the sequence modulo a n giving c2c3c 3 c5 a, b, c, O, a ' ab' a2' a~n+4 =- a3b2 (ac + b 2) =- O,

(3) so a, divides ana,+ 4. [] Note that although the proof is very simple, it depends on the fortuitous fact that the factor ac + b2 turns up on the fourth iteration. We will return to this point. The same method works for the 5-term recursion anan_ 5 = a n _ l a n _ 4 + an_2an_3 .

(4)

Actually in all of these recursions one may put arbitrary integers as coefficients of the terms a,_ ~an_1 and still get integers, and this can be proved for the recursions (2) and (4).** The next bit of progress came when Dean Hickerson proved that the original Somos sequence gives integers. In fact he showed something more general. Instead of starting with six one's he considered the se*Somos actually &scovered his sequences eight years ago but did not succeed in capturing the attention of the mathematical c o m m u mty untd the s u m m e r of 1989. **If the integer coefficients are allowed to be negative, then it may h a p p e n that s o m e a. = 0, m which case we shall make the convention that the sequence terminates at that point.

quence starting with indeterminates a0, a 1. . . . . as. The recursion then generates rational functions a n = Pn/q, of these a, and the theorem is that the denominators of these functions are always monomials with coefficient 1. This is of course clear for a0. . . . . all but note that to compute a12 one must divide by a6 = (asal + a4a 2 + a~)/a o = p6/ao . One easily sees that h a n d computation of a7, a8, etc. quickly becomes unmanageable. This is of course what symbolic manipulation programs are designed for and using M a c s y m a Hickerson found that as in (3) above P6 occurs as a factor of the numerator of a12 (which when reduced to lowest terms contains 194 terms!). Further M a c s y m a calculations are used to prove that P6 is prime to P7. . . . . P12 and an inductive argument is used to complete the proof. (Richard Stanley has also solved this problem using similar methods.) But what have we learned? As Hickerson puts it, "The thing I dislike about my proof is that it doesn't explain w h y the result is true. It depends primarily on the fact that when you compute a12 there's an unexpected cancellation. But w h y does this happen?" Indeed the proof, rather than illuminating the phenomenon, makes it, if anything, more mysterious. I report this with some embarassment, since I have earlier asserted in this same journal that a proof in mathematics is in some sense equivalent to an explanation. We now see that this clearly need not be the case. Perhaps, if and w h e n we find the "right" proof the situation will become clarified, but must there necessarily be a right proof? One is reminded of the proof of the four-color theorem? One of the most interesting features of the Somos problem, it seems to me, is that it leads to this sort of speculation. Getting back to the question at hand, having found proofs for recursions of order 4, 5, 6, and empirical evidence for 7, it turns out that those of order 8 and above do not give integers. You will easily confirm with your pocket calculator, for example, that for the recursion of order 8, a17 is a fraction. Curiouser and curiouser. The next discovery is due to Raphael Robinson, who found that the integer property of recursions (1), (2), and (4) was (apparently) shared by an infinite family of recursions. For any k I> 6 start with k ones and then use the recursion anan_ k = a n _ l a n _ k + 1 + an_2an_k+2 anan_ k = a n _ l a n _ k + l

(5)

or

+ (ln_2an_k+ 2 + an_3an_k+3 .

(5')

The fact that one is now dealing with an infinite collection of sequences would seem to put the problem out of range of M a c s y m a - t y p e proofs. At this point my pocket calculator convinced me that for any 0 < f < m < k the recursion a,an_k = xan-ean-k+e

+ yan-man-k+m

THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

(6) 41

gives integers, generalizing (5). Further investigations again by Robinson lead to the following Conjecture: For any p, q, r < k the recursion anan_ k = Xan_pan_k+ p + y a n - q a n _ k + q + Zan_ran_k+ r

(7)

generates integers if and only if p, q, r can be chosen so thatp + q + r = k. (Robinson's evidence is only for the case x = y = z = 1. The arbitrary x, y, z are my responsibility.) This would subsume (5') and (6). Namely (6) corresponds to choosingp = e , q = k - m , r = m - f, a n d z = 0 and (5') corresponds to choosing p = 1, q = 2, r = k-3,

x=y=z=l.

The story is not over. Dana Scott set up a program for the simplest case k = 4 but forgot to square the term an_2, yet the recursion still gave integers! In fact, it turns out that recursion (2) can be generalized to ana,_ 4 = aPn_laqn_3 + ar_2

for any p, q, r > 0 (8)

and the Bergman proof goes through as it does for recursion (4) with arbitrary exponents. On the other hand, one cannot choose arbitrary exponents and coefficients. In fact, the recursion a , a , _ 4 = 2 a , _ l a , _ 3 + a,_ 2 does not give integers (although if the righthand side is a , - l a , _ 3 + ya,_2 it can be proved that the recursion gives integers for all y). Recursion (8) is interesting, because in all the other examples the right-hand side was homogeneous. Was this a red herring? Perhaps, but when we go to threeterm sequences, we can no longer throw in arbitrary exponents. In fact, if one forgets to square the term an_ 3 in the original sequence (1), one gets fractions. Perhaps the simplest recursion of all has been discovered by Scott. Namely, for any k anan-k = a2_1 + . . .

2 + a,_k+ 1

(9)

which seems to work for all k. Other "good" recursions seem to be arian_ k = arian_ 2 + . . .

+ an_k+2an_k+l

(10)

and for k odd anan_ k = a n _ l a n _ 2

q- a n _ 3 a n _ 4 q- . . .

q- an_k+2an_k+ 1.

(11)

These recursions break new ground, since the righthand side may have any number of terms, whereas in previous examples three terms seemed to be the maximum. For k = 4 the Bergman proof works for (9) and (11) but not for (10), which (for the moment) remains unsolved. 42

THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

I d o n ' t want to drag this out indefinitely, for it seems new examples of Somos sequences are coming in faster than I can write them down. There is a whole area in which one uses recursions like (1) but starts with sequences other than all ones, e.g., ones and twos or ones and minus ones. Experiments indicate that sometimes one gets integers, other times not, but there seems to be no discernable pattern. On the positive side, using the ideas of Hickerson, Gale, and Robinson have proved integrality for the sequences (5) (but not (5')). I strongly suspect that by the time this appears in print much more will be k n o w n about Somos sequences. Perhaps the problem will even have been solved, but as of this writing the situation remains intriguingly mysterious. [Added in Proof: My suspicions seem to have been justiffed. During the month since the original manuscript was submitted Ben Lotto, using the ideas of Hickerson b u t no c o m p u t e r calculation, has s h o w n that (9) always gives integers. The method doesn't seem to work however for (10) and (11), although Robinson, using an entirely different but elementary argument, has shown that (10) gives integers for k = 4, settling the question raised two paragraphs above. Also conjecture (7) has been proved for k = 7, p = q = r = 1 (using Mathematica rather than Macsyma this time). Finally, Robinson has discovered a whole set of periodicity phenomena which occur when the values of the terms in the sequences are reduced modulo n. Periodicity has been proved for (2), (4) and for (10) with k = 4 and (9) for k = 3, but remains unexplained (so far) otherwise.]

Problems Derivatives eventually zero: Problem 91-1 by E. M. E. Wermuth (Jiilich, Germany) Let f be a C| function defined on some open interval (a,b) such that for every x in (a,b) there is an integer n(x) such that ftn(x))(x) = 0. Show that f is a polynomial. (For multidimensional versions of the problem and its history see MR90e:26040.)

A number of people have noted that the solution to Problem 89-7 by A n d r e w Lenard is incomplete. He concludes that a continuous strictly monotone function from R to R is a homeomorphism but fails to show it is surjective. A correct proof can be given by noting that the fixed points of f are a closed set and then applying Lenard's method to each of the countable sets of open intervals of non-fixed points.

Is there a mathematics gene? The four-year-old niece of a mathematical logician was playing a game in which she was the conductor on a train and her m o t h e r was a passenger. "Wait a minute," said Nancy, "we have to get some paper to make tickets." "Oh", said her mother, who had probably had a long day, "Do we really need them? After all, it's only a pretend game with pretend tickets." " O h no, m o m m y , y o u ' r e w r o n g , " replied Nancy, "They're pretend tickets, but it's a real game."

Archimedes Andrews and the Ball-Bearing Missile Barry Cipra

Archimedes Andrews was juggling golf balls on his roof. I had never seen him do that before. "Hey Einstein! Come on up," he called. Let me stop to explain something. My name is not really Einstein. That's just what Archimedes took to calling me some time ago. Whether it's meant as a compliment or as sarcasm varies, sometimes even within a single conversation. As far as I know, though, "Archimedes" is Archimedes' real name. "Come on u p , " he repeated. Archimedes' h o u s e is a two-story affair with a steeply pitched roof housing an attic where Archimedes sleeps. H o w he was managing to keep from sliding off the roof I didn't u n d e r s t a n d - - b u t I had the sinking feeling that I was going to find out. "Where's the ladder?" I called back. "No ladder," Archimedes replied abstractly, as if he were talking to the golf balls. "Go around in back." So I did. What I found there terrified me. In spite of what Archimedes had said, I was hoping to find a ladder. I was expecting at least a rope or a trellis or a drainpipe. To be honest, I was hoping to find nothing at all. Instead, I found a trampoline. Archimedes appeared over the peak of the roof, still juggling the golf balls, and walked casually down to the edge directly above me. "You're crazy, Archimedes," I said defiantly. "What do you mean? It's the best w a y in the world to get high. It's good exercise, too.'" "It's a good w a y to break your neck, is what it is." Archimedes stopped juggling. Five golf balls rained down on me, one at a time. Fortunately none hit--indicating that Archimedes was not out to hurt me intentionally. Unencumbered by the golf balls, Archimedes took a couple of steps over and then casually 44

stepped off the roof. He plummeted down, landed in the center of the trampoline, and rebounded as high as the second-story windows. On the second bounce he bent his knees on the rebound and only rose a few feet. With one more bounce he was standing happily on the trampoline and I started breathing again. "Do you know how hard it is to juggle on a trampoline?" he asked as he picked up the golf balls. "I don't, and I don't want to know." This was one time I was determined to stand my ground--literally. A few minutes later Archimedes and I were facing each other at arms' length, bouncing higher and higher. Archimedes had the golf balls bulging in his pockets. I had something like one in my throat. "It's all in the timing," Archimedes explained as we rose and fell together. "When you're at the top of the

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 19 1991 Spnnger-Verlag New York

bounce you're really not moving at all, except horizontally when we head for the roof. You want to just barely clear the roof line when you go for it. It's just like jumping off the last step of a stairway. If you need to, there's a pole to grab onto on the roof. I think it's still attached pretty well. You ready?" The answer was "No," but we were already on our way up and angling towards the house. Archimedes, who had been holding me steady by the forearms, let go and landed like a cat on the slanted roof. I reached wildly for the pole, hooked an elbow around it, lost m y footing, a n d w o u n d up s p r a w l e d o u t on the shingles, hugging the pole like a supplicant with my left foot hanging into the abyss. "'Not bad, Einstein," Archimedes said. "Not good, but not bad." As they say about airline pilots, a good landing is one you walk away from. I pulled myself cautiously to my feet, still gripping the pole. "OK, Archimedes, you got me up here. N o w what was it you wanted to show me?" "Isn't the view enough?" Archimedes replied. I looked around for the first time. The view was indeed very beautiful. Archimedes' home was on the edge of a woods next to a golf course. The October trees created a rolling ocean of brilliant red, yellow, and orange, with a few shades of green and brown tossed in as a salad. There was one fairway visible, bordered by rows of gracious oak and maple. Near the green was a sizable water trap, w h e r e a family of ducks was swimming contentedly near the edge. "That's where I get most of the golf balls from," Archimedes said, pointing toward the water trap. "Superballs or ball bearings would be better suited for the project I'm working on, but you can't find them for free. With the golf balls, I don't mind losing them." "Just what are y o u doing with t h e m ? " I asked. "'Seeing how far you can throw them into the woods?" "Hardly," Archimedes replied. "'I'm seeing how far I can shoot them." That set me back to worrying, just when I was getting used to being on the roof. Archimedes Andrews is dangerous enough when he's manipulating equations. The prospect that he had built a gun capable of firing golf balls made me shudder. " C o m e on, I'll s h o w y o u , " A r c h i m e d e s said, heading towards the top of the roof. I did not really want to let go of my pole, but finally, by consciously shutting out the fact that it was a solid twenty-foot drop to the ground if I lost my footing, I was able to abandon myself to fate and followed Archimedes up the shingles. "Isn't friction wonderful?" Archimedes said when I joined him. "It's a good thing that you didn't wear leather-soled street s h o e s - - y o u ' d have been off like a ski jumper by now." As he spoke, Archimedes headed towards the end

of the house, where I finally noticed a jerry-built scaffold set up around the brick chimney, which rose an additional eight feet above the very peak of the house. I remembered this chimney well. Archimedes had once stuffed newspapers up the flue and then set them on fire. Burning ash blew out the top while the house shook as if a subway train were roaring through the basement. Archimedes said at the time that he was just trying to measure the harmonics of the chimney pipe, not set fire to the n e i g h b o r h o o d - - h a d his intention been the latter, he would not have waited for a pouring rain. There was a ladder for getting up the scaffold, for w h i c h I w a s u n a c c o u n t a b l y grateful. H o w e v e r , standing on the platform, which was about two feet wide and four or five feet long, was even worse than standing on the roof: the platform had the advantage of being flat, but the roof was a much bigger surface. Also, the scaffold wobbled. The only thing to hold onto was the chimney and some kind of crane-like contraption that Archimedes had set on top of it. The contraption looked like it was built from a rusted-out erector set. It was bolted to the top of the chimney and rose about three feet above one of the two pipes. There was a pulley attached to it that stuck out away from the chimney but within reach from the platform. Hanging from the pulley and straining the whole setup was a bucket. Inside the bucket were a dozen or so golf-balls-and three shiny shot puts. "Believe it or not, those are actually ball beatings," Archimedes said. "Shot puts are made out of lead, and you don't get the right bounce out of them. These are m a d e of steel, and steel bounces beautifully. Watch this." Archimedes took one of the huge b a l l s - - i t was about four inches in diameter--lined it up in the contraption over the chimney, and threw it down the pipe as hard as he could. The ball rattled on its way down, there was a threatening metallic clank at the bottom, and a moment later the ball reappeared at the top, where Archimedes grabbed it. "I put a heavy steel plate in at the bottom of the fireplace," Archimedes explained. "It's best to bounce steel off steel. Besides, the ball bearing would crack the cement. I also stuck a stove pipe on at the other end, so the cylinder goes all the way d o w n - - I don't need a steel ball ricocheting around in the living room." "Is that what you've been doing? Throwing golf balls and shot puts d o w n your chimney?" " O h no, that was just to show you. What I really do is drop them." I could tell A r c h i m e d e s had s o m e t h i n g u p his s l e e v e - - o r down his pipe, or somewhere. I could tell by the half smile and the stingy way he was parcelling out information. I also knew that the prospects of my THE MATHEMATICAL 1NTELLIGENCER VOL 13, NO 1, 1991

45

outguessing him were lower than the prospects of surviving a fall off the wobbly platform we were occupying. "I see," I said anyway. "You've got some new explosive that goes off w h e n you drop the ball on it. Right?" "Nope, neither new nor explosive," Archimedes replied. "'The energy all comes out of the bucket--or lifting it, to be precise." "All right, Archimedes," I said in exasperation. "I give up. Just show me what you're up to and let me get off your roof." "OK, Einstein," Archimedes replied in a tone that was technically soothing but actually grating, a tone of voice that could easily have resulted in him being pushed to his death in his own backyard or sideyard or wherever. It would be so e a s y - - i f it weren't for a couple of kids who were watching from across the street, and who knows how many adults peering from behind how m a n y curtains. "Usually, as I said, I do this with golf balls, because they're cheap. But they're not as impressive as ball bearings, so I'll start by wasting a steel ball for you." Archimedes still had the shot-put-sized ball in his hand. He rummaged around in the bucket and pulled out a smaller ball bearing, this one about an inch in diameter. "It's best to get down so you're not standing by the top of the f l u e , " A r c h i m e d e s cautioned. He also turned to the kids across the street. "Take cover!" he shouted to t h e m - - a n d surprisingly they did. I guess they've lived near Archimedes long enough. Archimedes then turned back to the chimney. He held the large ball in his right hand over the chimney pipe, and the smaller ball in his left hand directly above it. Then he let go, and both balls disappeared into the hole. A moment later the small ball bearing shot out like it was late for a meeting. It was impossible--for me, at l e a s t ~ t o tell or even estimate how high it went, but I would have guessed a hundred feet or more. I tracked the ball until it crashed into the colored canopy of the woods. " A b o u t 120 f e e t , " Archimedes, w h o had been counting to himself, said unhappily. "I've had golf balls go almost that high. Oh, well, how'd you like it, Einstein?'" "What have you got down there?" I asked accusingly. "Another trampoline?" "Oh no. You drop a dead weight on a trampoline, Einstein, and it won't bounce hardly at all." "Then what in the heck is going on?" "Physics," Archimedes said in his crypto-comment style of explanation. "Elasticity. Conservation of energy and momentum. Gravity." "I suppose you're going to tell me it's all simple." I said, knowing that he was. 46

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

"It has to be simple." Then, amazingly, Archimedes corrected himself. "Actually some of it's complicated because of friction and air resistance and all t h a t - - w e could really get along better if friction would just go a w a y n a n d also if you really want to know where elasticity comes from, that's pretty complicatednlike w h y are ball bearings more elastic than golf balls. But if you ignore friction and think of elastic collisions as a piece of mathematics, then it's really simple. Mathematically, that ball bearing should have gone at least twice as high as it did. Really, fifty percent power efficiency ain't bad." "But how does the ball wind up higher than where you dropped it from? Doesn't that violate conservation of energy or something? Aren't you getting something for nothing?" "'Oh no," Archimedes replied. "This isn't like a lottery. You can never beat the odds in physics. Somebody has to go down and retrieve the big ball bearing." "It really is simple," he continued. "You just have to think like a ball bearing. That shouldn't be too hard for a brain like yours. Keep in mind what happens w h e n you bounce a ball: first it's going down, then it's going up, with the same speed. Actually it loses speed, but let's ignore that. "'It's best to imagine that the balls are slightly separated w h e n the big ball on the bottom hits the ground. If it was going down with speed v, then all of a sudden it's coming up with speed v, while the small ball is still going down with speed v. Here's where you have to think like a ball bearing-the big one, that is. From its point of view, the little ball is coming at it not with speed v, but with speed 2v. Now the big ball is a lot heavier than the little ball--about a hundred times the weight roughly. You can work out the equations exactly, but you might as well just figure that the big ball is infinitely heavy, so that the little ball bounces off it with again the same speed, namely 2v. "But that's 2v from the big ball bearing's point of v i e w - - o r 'reference frame,' if you like fancy terms. From our point of view, we have to add on the big guy's speed, which is another v. So the upshot, if you will, is that the little ball hit with speed v and bounces off with speed 3v. "That means it's got nine times as much kinetic energy, since kinetic energy is 1/~ mv2. But kinetic energy gets converted to potential, mgh, so the ball winds u p - - o r would wind up, if we keep neglecting everything that's really important in the w o r l d - - a t nine times its original height. This ball drops about 35 feet; it 'should' go about 300 feet straight up.'" The reader, who is presumably not crouching on a rickety platform the size of a postage stamp three stories above an early, rather messy death, can presumably make sense of Archimedes' argument. I only pretended to follow it. "'But here's where it gets fun," Archimedes went

on. "I got tired of watching balls go straight up and not knowing where they would l a n d - - I put a dent in the neighbor's car one t i m e - - s o I put this thing on top." Archimedes touched the contraption on top of the chimney. "The main thing is this steel plate"--at the top of the contraption was a rusted rectangular plate about the size of a tabloid newspaper, welded on one side to a rod that connected it to the framework. There was a knob at one end of the rod. I also realized that the base of the contraption was set on some sort of circular track. "I can tilt the plate to any angle by turning the knob, and I can swing it around in any direction. The ball . . . . " "I know. The ball hits the plate and bounces off in the direction you've got the plate pointing. You don't have to tell me everything, Archimedes." "I can never tell what I do or don't have to tell you, Einstein. But what do you mean by the direction the plate is pointing? Actually the ball reflects off the plate." " O f course that's what I mean," I replied. "But doesn't hitting the plate take some of the zing out of the ball?" "'Of course. But what can you do? Besides, I can still get pretty g o o d v e l o c i t y - - g o o d e n o u g h to break w i n d o w s at a h u n d r e d yards. Not that I'd ever do such a thing, of course. In fact another thing the plate's good for is as a shield to keep a straight-up shot from coming out angled at the street. Everything has to go towards the woods. Here, you want to try one?'" Archimedes turned the knob so that the plate angled over the top of the chimney, then handed me a large ball bearing and a golf ball. I held them carefully over the chimney and let go. The ball bearing rattled wonderfully d o w n the chimney, b u t the golf ball, which I'd released a fraction too soon, had rolled off the ball bearing, hit the edge of the pipe and fell unimpressively to the ground. "Not bad, Einstein," Archimedes said once more. "Not good, but not bad. At least you got one down the pipe. You w a n t to try for two this time?" He handed me another golf ball and the last ball bearing. I did it right the second time. The golf ball smacked loudly into the plate when it came back up, and hit a tree branch at the edge of Archimedes' yard. I have to admit, I had a s u d d e n sense of p o w e r - - a s if I had

suddenly sprouted wings and could simply fly from the precarious perch we were on. My shot had disturbed a pair of birds, which wheeled into the open and then resettled in a safer tree. "The only problem is, it's hard to get any kind of precision for targeting because the ball doesn't necessarily come straight out," Archimedes said. "Like, I k n o w I could hit the house across the street, but I couldn't guarantee any particular window.'" "I thought I did pretty well to hit that tree branch," I said. "Sure, Einstein. It's easy to aim at a target that you've already hit." " ' H o w fast do the balls come out, a n y w a y ? " I thought it was best to change the subject. "It's hard to say exactly, because of all the variables. The best thing, I suppose, would be to use a radar gun like in baseball and just measure the speed directly, more or less. But I can tell you the theoretical answer, which gives an upper b o u n d at least. "'From up here it takes about one and a half seconds for the balls to hit the ground. At 32 feet per second per second, they're going about 48 feet per second on impact. So the little ball winds up going three times as fast on the way up, or 144 feet per second. Convert that to miles per hour gives something like 98 miles per hour." "And the higher you start from, the faster it goes," I added. "True, but it's not as good as you think. The velocity goes with the square root of the height, so to get it to go twice as fast, you have to drop it from four times as high. If you want to break the sound barrier - - w h i c h is what I'm aiming at, by the w a y - - y o u ' d need a chimney something like 2000 feet tall." " H o l d on, Archimedes. Did you just say you're aiming at the speed of sound, or did you just say it can't be done?" "Both. It's not practical to build tall towers, but you can still get supersonic speeds. In fact, in principle you could get speeds greater than the speed of light--except for the fact that other principles get in the way. All you need is bigger balls." "I beg your pardon!'" "Think about the collision again. You've got a massive ball coming up at velocity u . . . . " "v," I said. "Call it w h a t e v e r y o u w a n t , " A r c h i m e d e s responded hotly. "The point is, the massive ball and the light ball don't have to be going at the same speed. If one's going up at speed u and the other's coming d o w n at speed v, then after the collision the small one's going up at speed v + 2u, which is v + u from the big bali's point of view, and another u from ours. If u equals v, then you get 3v. "'Now suppose you had a third ball that was even lighter than the second. Then you've got the same sitTHE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991

47

Is information an independent quality of matter, like mass, momentum, or electnc charge? Professor Tom Stomer beheves it m. Information exists independently of anyone being able to understand it. For over two thousand years, nobody could read Egyptian hieroglyphics, but the information was there just the same. At a microscopic level, it is not hard to see that a DNA crystal contains more information than a salt crystal of the same mass. CiXlclally, the DNA crystal also requires more energy to assemble. It takes energy to create information, that is, to increase the degree of organization of the universe, or reduce its entropy. It appears to be possible to calculate an equivalence between energy and information. This may explain mysteries like potential energy. It takes energy to 1~ a brick from the ground to the table, and we say the brick gains potential energy, but there is no test we can perform on the brick to measure it. An explanation may be that the energy has been converted into information, reducing the entropy of the universe. This book sets the scene for a useful theory of information. Tom Stonier suggests that information may even take the form of particles, which might not be limited to travelhng at the speed of light The implications for the physical sciences are profound. Anyone who deals with information in science should read this book. It makes compelling and ~ ' ~ challenging reading.

Springer-Verlag [] Heldelberger Platz 3, D-1000 Berhn 33 [] 175 Fifth Ave, New York, NY 10010,USA [] 8 Alexandra Rd, London SWl9 7JZ, England [] 26, rue des Carmes, F-75005 Pans [] 37-3, Hongo 3-chome, tm 9639/V/2h$ Bunkyo-ku, Tokyo 113,Japan [] CltJcorp Centre, Room 1603,18Whltfield Road, Causeway Bay, Hong Kong [] Avlnguda Diagonal, 468-4~ E-08006 Barcelona

48

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

uation, but with u equals 3v. So the third ball's going to b o u n c e u p with speed 7 v - - t w i c e u plus v. "'Do y o u k n o w h o w high a ball like that w o u l d go? Forty-nine times the original height, about a quarter of a mile. A n d if y o u got a n o t h e r ball on top of that one y o u ' d be u p to twice seven plus one, or 15v. O n e m o r e a n d y o u ' r e u p to 31v, and that's more than e n o u g h to get y o u over the s o u n d barrier. " N o t only that, but a few m o r e balls w o u l d p u t the last o n e into orbit. I estimate that a stack of ten balls is e n o u g h to achieve escape v e l o c i t y - - a n d that's just w o r k i n g from a thirty-foot chimney. " T w o things get in the way, though. Actually three. Friction a n d other garbage eats away at the effect, for one. There's also a big p r o b l e m getting the balls lined up. The best thing to do w o u l d be to p u t t h e m in a v a c u u m and have t h e m d r o p at first t h r o u g h a magnetic field that w o u l d center t h e m in the cylinder. The other a p p r o a c h I'm playing with is to use rods instead of s p h e r e s a n d d r o p t h e m d o w n a pipe. You d o n ' t n e e d balls to have an elastic collision, just the right material. "But the biggest problem, b o t h practically and e v e n mathematically, is the fact that the theoretical velocities are based on each ball being infinitely more massive t h a n the ball above it. Mathematically, of course, that's nonsense. In practice, infinitely massive m e a n s m a y b e a h u n d r e d times the w e i g h t . But i m a g i n e s t a c k i n g e v e n f o u r balls t h a t w a y . If the top o n e w e i g h s a n ounce, the b o t t o m one weighs a b o u t 30 tons. "So to figure out w h a t ' s actually going o n m a t h e m a tically, y o u have to work out the equations that include the masses as well as the velocities. I could talk y o u t h r o u g h it, but if y o u d i d n ' t u n d e r s t a n d w h e r e 3v came from, y o u ' r e going to h a v e a h a r d time following the derivation in y o u r head. So I'll give y o u a break, Einstein. But here's a p r o b l e m to work on. S u p p o s e y o u h a v e a steel rod exactly one meter long, a n d y o u cut off the top centimeter. If y o u w a n t to d r o p the rod d o w n a pipe as I described so as to launch the onecentimeter "nose cone,' w h e r e else should y o u cut the rod so as to get the most height out of it?" With that, Archimedes j u m p e d from the platform, skidded d o w n the r o o f - - e i t h e r in or out of control, it was h a r d to tell w h i c h - - a n d w e n t over the edge. H e w a v e d at m e w h e n h e came u p o n the first r e b o u n d - he h a d l a n d e d o n the trampoline, of c o u r s e - - a n d t h e n I d i d n ' t see him again until he a p p e a r e d in the front yard. " H e y , h o w am I s u p p o s e d to get d o w n from h e r e ? " I shouted. " T h a t ' s a n o t h e r problem for y o u to work o n , " Archimedes replied.

305 Oxford Street Northfield, MN 55057 USA

Organizing a Conference I. M. James

So you're planning a conference! Congratulations-but (as an old hand) I wonder if you fully realise what you may be letting yourself in for! Here is a composite nightmare--all things that have actually happened in my experience, but fortunately not all on the same occasion. On the day of arrival, a key speaker phones to say she won't be coming. A rail strike results in most of the participants arriving in the early hours of the morning. One of the opening speakers is detained at immigration because of not having obtained a visa. The hotel where everyone is staying has just told you that their quotation did not include tourist tax, and that it does not accept credit cards. Some people arrive who have not been invited. The lecture theatre booked for the opening a d d r e s s - - t h e only one suitablemhas been double-booked. The photocopier has broken down. Several people are dissatisfied with their accommodation and want to change hotels. Others thought the last day of lectures was the day of departure and want to change all their travel arrangements. One participant wants to see a doctor and another needs a dentist. The luggage of another participant has gone astray and someone else has had his passport, travellers cheques, etc. stolen. And so on. Would it help if I ran over some of the many points that may need to be considered by someone in your position? I'm assuming that what you have in mind is an average sort of conference--say up to a hundred people, lasting for up to a w e e k - - b u t some of the points I'm going to make apply to conferences of any size or duration. One thing I'd like to say at the outset: You ought to be able to participate fully in the conference yourself. That means very careful advance planning so that, during the conference itself, there is nothing to distract you apart from the unforeseeable. There are sure to be problems, but I hope very much that, not long after the conference opens, you will be able to retreat into the background and, although an organiser, participate in the same w a y as everyone else. No doubt you will have a committee to support you. This is important because if difficult decisions have to

be taken (as is quite likely) it is a great relief to be able to refer them to a committee rather than have to take personal responsibility for them. In the end, however, every conference is mainly d u e to the efforts of a single individual who does almost all the work. That individual, I gather, will be you. Undoubtedly the first thing you need to get clear is what the purpose of the conference is to be. For instance is it to review recent progress in the field? Is it to provide an opportunity for experts to talk about their latest ideas? Is it to give new people a chance to make their debut? Will some effort be made to cater to non-specialists? Is it intended to celebrate some anniversary or to help establish the reputation of some institution as a centre for the subject? Next it is necessary to make an estimate of the desirable numbers and type of participants. There is an obvious difference in character between a small conference of around 50, say, and a large one, of 100 plus. It is also necessary to decide whether the conference should be open to all comers (or at least to all w h o have registered in advance) or should be restricted in some way. For example, attendance may be by invita-

THE MATHEMATICAL INTELLIGENCER VOL 13, N O 19 1991 Spnnger-Verlag New York

49

tion only: this may mean a list fixed at an early stage or may mean that there is discretion to send an invitation to anyone who lets it be known he wishes one. It is not unusual to begin by inviting a few stars--if this is done sufficiently far in advance it will not be easy for them to think of a polite excuse for saying n o - - a n d then make use of their names in the promotional literature for the meeting. Next it is necessary to decide when and where the conference is to be h e l d - - t h e s e t w o decisions are somewhat related, of course. It is important to allow plenty of time for p l a n n i n g - - t h r e e years is not too long. Find out as much as possible about other conferences being planned that might, on the one hand, clash with or, on the other hand, could usefully be combined with yours. (Some information can be obtained from the AMS Notices, the LMS newsletter, the Canberra newsletter, etc., but asking around is useful also.) Then it is necessary to consider w h e n participants are likely to be free to attend (the incidence of university terms varies a lot). Once the location has been chosen, it is necessary to find out whether climatic conditions are likely to be favourable, and w h e t h e r there are any public holidays that might cause problems. Peak holiday p e r i o d s s h o u l d be avoided because of difficulties with travel and accommodation. Even if the precise dates are not fixed until a later stage it is obviously desirable to have some idea of the duration. Generally three or four hours of lectures a day is the limit for most people, and if the programme lasts for more than three or four days, a break of at least half a day in the middle would be a relief. When the time comes to fix dates, it is well to remember that some days may be better for travel than others and that some combinations of dates may fit the rules for cheap plane fares better than others. Confusion often arises between the day of arrival and the first day of lectures, likewise between the last day of lectures and the day of departure. There are various considerations involved in deciding the venue for the meeting, and a balance has to be struck b e t w e e n various desiderata. Usually the choice is between a university or similar institution, a conference centre run or supported by a foundation, and a location such as a hotel with appropriate facili,50

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

ties. Practical considerations are of great importance, of course. Will the place be easy for most participants to reach? Are there lecture rooms of the right capacity and are they properly equipped? Is the accommodation of a suitable standard? Are there enough r o o m s - are there rooms for couples and families? Where do people go for meals? H o w about dietary requirements? Where do participants go between lectures--are there common rooms and rooms for discussions? Where can people congregate in the evenings? If possible, all other requirements need to be combined with an attractive location. It is a good idea to inspect possible places thoroughly before reaching a decision. Turning now to the subject of finance, it is necessary to prepare a realistic budget. Quotations are necessary for use of facilities, for meals, and for accommodation. If local transport may be required, what will the cost be? H o w much should be allowed for secretarial services, for postage, telephone, photocopying, etc.? Should a conference bank account be opened? Will advance money be needed? Then there is the question of whether to offer financial assistance to participants. Should this be offered to everyone or to those who could not attend otherwise? Should grants be offered to invited speakers or to members of the committee? Should they be for a specific sum or a proportion of expenses? Should they be for travel and/or accommodation and/or food? Should they be payable in advance, during the conference, or afterwards? Will the organisers try to find out the funding position of individuals before allocating subsidies from conference funds and if so how? Naturally some of these matters can only be decided once the availability of funds has been investigated. Possible grant-giving bodies should be contacted at an early stage to find out about forms, deadlines, and any special conditions that might affect your plans. H o w long will it be before a decision is given? Publishers could also be worth contacting, particularly if there are going to be promotional opportunities for books or journals. Governmental cultural organisations will sometimes offer grants for participants from particular countries. It is usual to charge a registration fee (which may or may not include social events); perhaps there should be a reduced fee for students. It is also a good idea to get clear in advance what is to happen in the event of a surplus or deficit at the end of it all. It is also necessary to decide whether a proceedings volume should be published and, if so, to interest a publisher and choose an editor. Will contributions be restricted to the invited speakers? What advice/instructions should be given to an author as to format? What will be the deadline for receipt of contributions? Will they be refereed? Will the volume be i n d u d e d in the registration fee or available to participants at a reduced rate? Who will be entitled to royalties? Some publicity will be necessary, unless participa-

Conference on topology and differential structures, Mexico City, July 1971.

tion is strictly by invitation only. In any case, listing in the AMS Notices, LMS newsletter, etc. is helpful to others who might be planning a conference in the same field. Direct mail (i.e., circularising prospective participants) is probably more effective than advertising in journals, but the latter is likely to attract the attention of people the committee might not have thought of. As regards the programme, it is necessary to decide whether speakers will be required to give titles (or even abstracts) in advance. Will there be parallel sessions? Will short talks by non-invited speakers be enJcouraged/permitted? Will there be poster sessions? Should some "star" speakers be kept until the end to help prevent people drifting away? Should the whole programme be fixed up in advance or should some gaps be left to be filled at the last minute in the light of recent developments and/or offers of talks? Bear in mind that people invited to speak without advance notice may not have brought notes, slides, etc. Participants will need advance information about arrival/departure procedures, and at least a provisional programme. The correct address for mail and telephone number (if only for emergencies) will also be needed, and a local map with the registration desk and other important locations marked. Drivers would no doubt appreciate a semi-local map as well. Some participants may need visas, which may take time to obtain and require a formal letter of invitation; these and other difficulties individuals may experience should be foreseen as far as possible. What methods of payment are acceptable to (i) the conference office, (ii)

hotels etc? Should individuals be recommended to take out personal medical insurance? On the whole, the more general information sent out in advance the less correspondence with individuals. On arrival participants will need further information on, for example, collection of grants, cashing of cheques, use of library facilities, photocopying, and will need as well advice about restaurants, etc., an updated programme and a list of participants. Don't forget about identity labels? Be prepared for late arrivals. Finally there is the social programme. Will there be a r e c e p t i o n a n d / o r b a n q u e t - - i f so, will there be speeches? Will there be excursions? How about sporting facilities? What plays/concerts/films/exhibitions will be on during the conference? Does the local tourist office have any useful literature? Will any local firms offer concessions to participants--or gifts? Will there be special arrangements for spouses, children, etc.? Will there be a group photograph and, if so, should it be professional or amateur? Of course there are a hundred and one other things you will need to think about, depending on the actual venue of the conference and so forth. I began with a nightmare so let me end with a dream. Towards the end of the banquet the senior participant proposes a vote of thanks. He says how much the lectures have helped him to understand the subject better and how exciting the new theories and results announced in the course of the conference have been. He says that the informal discussions have been equally valuable and that some very interesting new ideas have emerged in the course of these. He says how very agreeable it is to meet old friends and how glad he is to have had the opportunity to meet some people he hitherto k n e w only by reputation. He thanks various sponsors for their financial support. Finally he congratulates the committee, and especially you, on the excellence of the arrangements, and now everyone is on his feet drinking your health. I hope that at that point, if not before, you will feel that all the effort has been worthwhile.

Mathematical Institute Oxford Unzverszty Oxford, England OX1 3LB THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991 5 1

Recent Developments in Braid and Link Theory Joan S. Birman

This article is about the theory of braids and the geometry of links in the 3-sphere and new interconnections between them. We are particularly interested in the family of polynomial invariants of link type discovered by Vaughan Jones in 1984 and its recent generalizations. The discovery of new powerful and easily computed invariants of oriented links in oriented 3-space in 1984 was a huge surprise. The hard work had been done already in what seemed to be totally unrelated studies of Von Neumann algebras. The proofs of the topological invariance of the new polynomials, first by Jones [14], [15] and later by others [12], [21], [17] gave essentially no insight into the geometric meaning of the new tools. The surprises deepened when it was realized that Jones's link polynomials were related in a bewildering variety of ways to areas of physics where there had been no previous hints that knotting or linking might be involved [3], [23], [27]. Physicists who had studied the (quantum) Yang-Baxter equations seemed to have a machine ready to grind out link polynomials in such profusion that it seemed as if one needed invariants to distinguish the invariants, yet they had not known that the Yang-Baxter equation was related in any way to link theory. None of the new polynomials was previously known to topologists. More recently it has even become likely that the new invariants of links in Sa generalize to new invariants of closed 3manifolds and of link complements in 3-manifolds [271, [10]. Braid groups had been introduced into mathematical literature by Emil Artin in 1925 [2]. Artin's motivation in studying them was the interesting relationship between braiding and knotting. While braid groups 52

were of clear interest in their own right, they had contributed little that was new to knot theory until 1984, w h e n it soon became clear that the existence of some kind of braiding was a connecting thread between knot theory, operator algebras, and the various areas of physics involved in the discoveries. That is the theme we wish to explore in this article. The full story, whose implications are certain to be deep and farreaching, is only beginning to be understood.

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 19 1991 Spnnger-Verlag New York

Links and C l o s e d Braids

A link K is a finite collection of pairwise disjoint oriented circles embedded in oriented 3-space R3 or the 3-sphere S3. If K consists of only one embedded circle it is called a knot. We restrict ourselves to piecewise linear or (equivalently) smooth embeddings to avoid links that have pathological local behavior. Links K and K' are said to determine the same link type if the oriented link K can be deformed onto the oriented link K' by an isotopy of S3. The equivalence class of all links with the same type as K is denoted by K. The nine examples depicted in Figure 1 determine fewer than nine distinct link types. H o w can we tell? Our problem is known as the knot problem. That it is a non-triviality, even in the special case of recognizing the "unknot," will be obvious to anyone who has attempted to restore order in a box used to save leftover bits of string. There seems to be no systematic way to do it. Near the turn of this century there was a flurry of interest in our problem on the part of physicists, notably Lord Kelvin and Peter Guthrie Tait, w h o thought that the arrangement of distinct elements in the peri'odic table might be related to knotting in the ether. Their ideas led to the gathering of a mass of experimental data in the form of tables of presumably distinct knot types, organized according to the number of double points in a planar projection. An enormous amount of patience and the liberal use of the eraser evidently dominated that early work. The expressed hope [25] was that computable invariants of knot type would be revealed in the natural course of assembling such data, but to the disappointment of those early workers that did not happen. The tables, instead, served two other equally important purposes. First, they gave convincing evidence of the non-triviality of the link problem. Second, they provided a rich set of examples for the more sophisticated studies that followed. The tables are used to this day and have had a strong influence on all of the work in the area. They can be seen in all their beauty and with surprisingly few corrections in current graduate-level textbooks on the subject, for example [24]. A sample page from the tables of 10-crossing knots is given in Figure 2. Faced with the problem that there are just too many different ways to depict a single link type, one might try to reduce the number of representatives by adding extra structure. James Alexander did just that in 1923 [1]. His contribution is of fundamental importance to our story, so we pause to describe it. Let K be a knot or link type, and let K be a representative parametrized by cylindrical coordinates (r,O,z), r # O, in R3. Let t denote arclength on K. Our representative K is called a closed n-braid if dO/dt >0 at all points of K. The integer n is the number of times K meets a half-plane through the z-axis. (This number is necessarily inde-

(iI)

(nl)

(v)

(vO

( (vii)

(IX)

(viii)

Figure 1. Examples of knots and links.

1047 5,21,2 [7--7+6-3+1

~

1052 311,3,2 [15--13+7--2

[11-9+6-3+1

[25-18+6

[13-12+8-3

[11-10+6-2

[13-11+7-2

[21-15+5

1051 32,21,2 ~ [19-15+7-2

1056 221,3,2 [17--14+8-2

Figure 2. A sample page from the knot tables.

THEMATHEMATICAL INTELLIGENCER VOL13,NO1,1991 53

I

Z l lL'~x I\

m F3

I

r

m F2

i

~

m F 1

', '

r, Z1

0.1 (0-2 )-1 01 (0. 2 )-1

z4

,r>tj i

' i

Figure 3. An open braid.

Figure 4. Factorization of a pure 4-braid.

p e n d e n t of the choice of the half-plane). The z-axis A is the braid axis. Examples can be seen in Figures 1 (i)-(iii), (v), a n d (ix). The braid axis is orthogonal to the plane of the paper and its intersection with that plane is indicated by a black dot.

describes K as an oriented closed braid. This word, c o m m u n i c a t e d by telephone, will give our friend precise instructions for reconstructing our picture of K as a closed braid. It also s h o w s that the set of all link types is countable.

THEOREM 1. [1] Every link type can be represented by a closed braid.

A G r o u p of Braids

Proof: If K is not already a closed braid, we show h o w to change it to one. It will be convenient to assume that (possibly after a small deformation) K is polygonal, with edges el . . . . . era, and that dO/dt ~ 0 on the interior of each edge. Call an edge e, bad if dO/dt < 0 on ei. By subdividing the bad edges if necessary we can arrange that each bad edge e, is an edge of a planar triangle "r, such that K N "q = e, and A N -q contains exactly one point. We can then replace e, by O~z - ei to remove the b a d edge. After finitely m a n y such replacements we'll have a closed braid representative ofK. 9 Different proofs of Theorem 1 are given in [20], [28]; each brings n e w insights to the geometry. Closed braids provide an i m m e d i a t e a n s w e r to a question that m a y have occurred to the reader. H o w can we describe a favorite link so that, for example, the description could be communicated by telephone to a colleague in another city? To obtain such a description from a closed braid representative K, let "ff:B3 ~2 be orthogonal projection onto the plane z = 0. We m a y assume (if necessary after a small isotopy of K) that the singularities of ~[K are at m o s t a finite n u m b e r of transverse double points at, say, 0 = 01 < 02 < 9 9 9 < 0k. The intersections of K with the halfplanes H(0) defined by 0 = 0s - e, e small a n d S0, will t h e n be n points which have distinct r-coordinates rl(0) < r2(0) < . . . < r,(0). Thus there will be a unique pair, say r~(0) and rt+l(0), that exchange r-order at the jth double point 0s. We can then associate to the jth double point the symbol ~, (respectively, ~-1) according as the z-coordinate of r.(0s) is bigger (or smaller) than that of r, + 1(01). In this w a y we obtain a cyclic w o r d of length k in the symbols or1. . . . . % - 1 and their inverses, which 54

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

Closed braids lead naturally to open braids, as follows: Cut R3NA open along any half-plane H(0) to obtain an open solid cylinder D x I, where D is an open 2-disc and I an interval. The closed braid K then goes over to a u n i o n of n disjoint arcs in D x I, which intersect each disc D x {t} in n points. The union of these arcs is an (open) braid. There is an equivalence relation on o p e n braids i n d u c e d by link equivalence. Figure 3 shows an open braid associated to the closed braid of Figure 1 (ix). A further standardization is n o w in order: we can choose n distinguished points z ~ = (z1. . . . . z,) on D, a n d a s s u m e without loss of generality that the initial (resp., terminal) endpoints of our braided arcs are at z ~ x {0} (resp., z ~ x {1}). This has an immediate bonus, because n o w we have a w a y to " m u l t i p l y " two nbraids, namely by pasting together the associated D x I's, matching the D x {1} face of the first with the D x {0} face of the second, and rescaling. This multiplication can be seen to be associative. Moreover z ~ x I represents the identity element, and each braid has an inverse, obtained by reflection t h r o u g h the disc D x {1}. In short, the n-braids form a group, the n-strand braid group B n, discovered by Artin in 1925 [2]. We n o w show that there is a beautiful and simple w a y to redefine the group Bn, which will at one a n d the same time make precise all we have just said, reveal the structure of Bn, a n d lead to generalizations and applications. With all those goals in mind, define the configuration space ~,(M) of a manifold M to be the space {(zl . . . . , zn)[z, E M and z, ~ zj if i ~a j} of all n-tuples of distinct points of M. Note that even t h o u g h E,(M) has n times the dimension of M, we can think of its points as being a set of n distinct points on a single copy of M. Now, the permutation group S, acts freely

group F._I of rank n - 1. A deeper study of the long exact homotopy sequence of the fibration shows that groups that precede and follow the ones we have identified, i.e., F,_I, Pn, and Pn-1, are trivial. Thus one obtains a short exact sequence:

X

{1} ~ F._ 1 ~ Pn ~ P,-x --* {1}. .1) o,5 = 50, If li --11 ~ 2

(3.2) o, ojo, = olo, o1 ff Iz --11 = 1

We can say more. The group P . - i clearly occurs as a subgroup of Pn. It is the subgroup of pure braids on the first n - 1 strands. In fact, the short exact sequence of (2) splits, i.e., P, is a semi-direct product of F,_ 1 and

Figure 5. Defining relations in the braid group.

Pn-l"

on E.(M), permuting coordinates, and we define a second and closely related configuration space to be the orbit space f~(M) = E(M)/Sn. The new way to look at the braid group B,, due to FadeU and Neuwirth [11], is to t h i n k of B n as the f u n d a m e n t a l group ~l(12n(D),z~ There is also a related colored braid group P, = wl(~n(D),z~ so called because there is a well-defined assignment of colors to each of the n strands, which is preserved by group multiplication. We can recover our intuitive definition of Bn as follows: An element 13 E "rq(lq,(D),z ~ is represented by a z~ loop in the space 12,(D), i.e., by n coordinate functions [31. . . . . 13~, whose graphs are the n arcs that join z ~ x {0} to z ~ x {1} in D x I. These arcs intersect each intermediate plane D x {t} in exactly n distinct points. The fact that we are using 12~ rather than ~, allows the ith braid strand to begin at z, and end at some other zr The equivalence relation on the various geometric representatives of an element 13 E B, is h o m o t o p y in the configuration space, which means that braid strands can be deformed by levelpreserving deformations, arbitrary except that two strands cannot pass through one another. This is of course the essence of the p h e n o m e n o n of knotting and linking. T h e A l g e b r a i c Structure o f B n We promised to uncover the structure of Bn, and shall do so. The first observation is that E, is a (regular) covering space of 1~, with S, as its group of covering translations. This reveals immediately that Pn is a normal subgroup of B,, with quotient group S,; i.e., we have a short-exact sequence: {1} ---~ Pn--~ B. ---~ Sn ---* {1}.

(2)

(1)

We can say more. There is a natural map f.:En ~ E,-1, defined by forgetting the last coordinate. A straightforward construction (try it for an exercise!) proves that E, is a fiber-space over the base E.-1, with projection fn, the fiber being D \(z 1. . . . . Zn_x), the (n - 1)times punctured plane D. The latter group is a free

The decomposition we just described can be repeated for P,_ 1, and so on down to P2, which is isomorphic to the infinite cyclic group F1. In this way one sees that P, is built up from the free groups F,_ 1, Fn_2, . . . . F 1 by a sequence of semi-direct products. An example of a pure 4-braid that is a product of braids in F3, F2, and F1 is given in Figure 4. The point is that every pure braid admits a unique factorization of this type. For details of the proof, see [11] or [5]. The approach just described yields, with a little more work, a presentation for the group B,, with generators the elementary braids r . . . . . r described earlier. Defining relations can be shown to be: (3.1) (3.2)

r r

= %
if [i - j[ > 1 if [i - j [ = 1.

(3)

These relations are illustrated in Figure 5. We make one final remark. There are two symmetries of special interest in knot and link theory, and both have simple meanings from the braid point of view. The first is a change in the orientation of K, and the second in the orientation of the ambient space S3. Assume that K is the closure of an n-braid 13, and that [3 is expressed as a word in the generators r . . . . . r Changing the orientation of K corresponds to reading the braid word backwards, and changing the orientation of S3 corresponds to replacing each r by its inverse. Replacing a braid word by its inverse thus corresponds to simultaneously changing the orientations on K and S3. Markov's Theorem Markov's Theorem concerns the relationship between the various (open) braids whose closures define the same oriented link type. We would like to precede our discussion of this important theorem by backing up a little bit and returning to the looser concept of a link diagram, defined above. Let D and D' be oriented link diagrams. Call D and D' Reidemeister-equivalent if they define the same oriented link type. See Figure 6. Notice that relation (3.2) is a special case of R-III, whereas "free reduction" in Bn is essentially R-II. THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991 5 5

= fl~+(f~)) = flp.-(~)) for all 13 E B.. The following corollary of Markov's theorem is immediate: Corollary 4. Any Markov trace is a link-type invariant.

R-II

Indeed, most of the k n o w n link invariants have b e e n interpreted as Markov traces on B..

R-I

The Symmetric Group and the Braid Group

R-III

Figure 6. Reidemeister's moves. T H E O R E M 2. [22] Reidemeister equivalence is generated by the three moves R-I, R-II, R-III that are illustrated in Figure 6. A small a m o u n t of e x p e r i m e n t a t i o n (or a proof b a s e d u p o n the use of p o l y g o n a l r e p r e s e n t a t i v e s ) should convince the reader of the reasonableness of Theorem 2. A proof can be found in [9], but the reader w h o has succeeded in working out our earlier exercises might wish to construct one with the help of polygonal representatives. The proof p r o c e e d s in m u c h the same w a y as the proof we sketched for Theorem 1, beginning with t w o polygonal representatives of a link, and assumes they are equivalent u n d e r a finite sequence of moves, each of which is an interchange that replaces one side of a planar triangle with the other two, or two sides with one. N o w , Markov's Theorem will be seen to be similar to Reidemeister's, except that the diagrams in question come from closed braids, and the equivalence is via a sequence of closed braid diagrams. The proof is harder because of these requirements. To state the theorem, let B~ denote the disjoint union of braid groups B~, B2, B3. . . . . . Call braids 13 E Bn and 13' E B m Markov equivalent if the closed braids they determine have the same oriented link type. T H E O R E M 3. Markov equivalence is generated by: (i) Conjugacy (ii) The maps p,+: Bk---) Bk+l, with f3---* ~cr~ 1. Theorem 3 was announced by Markov in 1935 with a working outline for a proof. Proofs are in [5], [4], and [20]. Versions of Theorem 3 that replace the "stabilization" move (li) with braid index-preserving moves are given in [6], [7]. Markov's Theorem has an immediate consequence. A function f:B, ~ K, where K is a ring, is a Markov trace if it is a class invariant in each B k and satisfies f(~) 56

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

A h e a d - o n attack on M a r k o v equivalence in B~ is hopelessly difficult, but one might hope that something could be d o n e by passing to quotients of the Bn's. In view of the short exact sequence (1) above, a logical place to begin might be with the symmetric groups S v $2, $3. . . . . Accordingly, let "rr:B, ~ S, be the h o m o m o r p h i s m defined by the exact sequence (1). There is an o b v i o u s c a n d i d a t e for a M a r k o v trace which factors through "rr, i.e., set f([3) = the n u m b e r of cycles in ~r(f~). The link invariant it determines is the n u m b e r of components. Encouraged by success, w e look to representation theory for more subtle invariants. A remarkable phen o m e n o n emerges. To illustrate this in a non-trivial case, w e first note that the group S, is generated by the transpositions s, = (i,i + 1), i = 1 . . . . . n - 1, and in terms of these generators it has defining relations: (4.1) (4.2) (4.3)

s,s1 = sls , s,sfi, = sts,sj s2 = 1

if li - jl > 1 if li - jl = 1 (4) i = 1. . . . . n - 1.

In view of the similarity b e t w e e n relations (3) and (4) one might w o n d e r h o w the representations of Bn are related to those of Sn. We investigate an example. There is an (n + 1) x (n + 1)-dimensional matrix representation of S, over the integers, defined by mapping s, to a matrix S, that differs from the identity matrix only in the (i - 1) st, i th, and (i + 1)st entries in the i th row, which are (1, - 1,1) instead of (0,1,0). Let t be a real n u m b e r close to 1. We can "deform" this representation by changing these three entries to (t, - t , 1 ) or ( 1 , - t - l , t - 1 ) , the two choices giving a deformed matrix S.(t) or its inverse. A short calculation s h o w s that the matrices Si(t) satisfy relations (4.1) and (4.2), but not (4.3); in fact, S,(t) has infinite order if t ~ 1. In view of the d e f i n i n g relations for B n g i v e n b y (3), the d e f o r m e d matrices yield a one-parameter family of (n + 1)-dimensional representations of Bm Alternatively, w e can think of t h e m as representations of B, by matrices with entries in the ring 77[t,t-1]. Label the rows and columns of the matrices 0, 1, . . . . n. Our representations are obviously reducible, because the 0 th row and the n th row of each S,(t) are unit vectors. Therefore, the submatrix obtained by de-

leting the 0 th and n th rows and columns multiplies independently of the remaining entries and so yields an (n - 1)-dimensional representation pn:Bn ~ M n (27[t,t-x]). It is irreducible, because the representation of S n obtained from it by setting t = 1 is known to be irreducible. The representations Pn were discovered by Werner Burau in 1938 and have been the object of intensive investigations ever since. They yield Markov traces A: B n ~ Z[t,t-1], defined by the formula: tn-X-'~ a.(t)

= 1 + t + t2 +

- Pn(f~)) .

.

.

(5)

+ tn-1

where A,(t) is the Laurent polynomial in t determined by the image Pn(~) of f~ E B n and co(B) is the sum of the exponents of ~ when written as a product of the cr,'s. The invariant &~(t) is the Alexander polynomial of the link defined by the closed braid. (Alexander's original methods, however, had nothing to do with braids.) To see that this is indeed a Markov trace, one must verify that it is invariant under the two changes described in Theorem 3. Invariance under (3.1) is immediate, because the characteristic polynomial of Pn(~) is a class invariant; also co(B) is invariant under the braid relations (3) and conjugacy. See [19] for a braid-theoretic proof of invariance under (3.2), or see Burau's proof, presented as Theorem 3.11 in [5]. The fact that we could deform one particular matrix representation of Sn to a parametrized family of representations of B n was not an isolated phenomenon. Much more is true. The irreducible representations of S n are well known. They are classified by Young diagrams, and can be given by matrices whose entries are all O's and l's. Every irreducible representation of Sn deforms to a parametrized family of irreducible representation of Bn. In fact the entire group algebra of Sn deforms to an algebra H n (the Hecke algebra of the symmetric group) that is a quotient of the group algebra of B n. See [15] for details. The algebras H n support a Markov trace that is a weighted sum of matrix traces on their irreducible summands. The link inv a r i a n t d e t e r m i n e d b y this M a r k o v trace is the "Homily" or 2-variable Jones polynomial of [12]. The picture does not end there. Kauffman has discovered in [17] yet another 2-variable polynomial-invariant of oriented link type, which is independent of all of the ones just described. Unlike the one-variable Jones polynomial, the Kauffman polynomial was discovered by purely combinatorial techniques and seemed at first glance to be completely unrelated to braids. However, algebras we denote by Wn were constructed in [8]. They are quotients of the complex group algebra CBn of the braid group, and support a 2-parameter family of Markov traces w h o s e associated link invariant is the Kauffman polynomial. Each Wn contains H n as a direct summand, and the Markov trace that

defines the Homily polynomial is the restriction to H n of the Markov trace on W n that defines the Kauffman polynomial. The algebras W n are, moreover, deformations of a generalization of the complex group algebra CSn, much as Hn is a deformation of CSn. If the irreducible representations of S n are restricted to ones belonging to Young diagrams with at most 2 rows, the deformed algebra is the Jones algebra A n, studied in [14]. That algebra is shown in [15] to support a Markov trace. The link invariant it determines is the Jones polynomial. We will learn how to compute it in the next section. We summarize the situation. Let R n denote the algebra (over the complex numbers) generated by the Burau matrices. Because R n occurs as an irreducible summand of A n, we have a sequence of algebra homomorphisms: C B n - . Wn--~ Hn--~ A n - - , Rn--~ Sn.

(6)

Each algebra supports a Markov trace and so determines a link-type invariant. In this way a uniform picture of the new and old link invariants is emerging, with the representation theory of B n an important part of the picture.

Combinatorics and Link Theory Among the new polynomials, the 1-variable Jones polynomial plays a very special role as the simplest example, after the connectivity, of a link invariant that arises as a Markov trace. We turn to Louis Kauffman's work in [16] for an exceptionally simple proof that the Jones polynomial is an invariant of oriented link type in oriented S3. At the same time it will also show us h o w to compute the Jones polynomial from a link diagram.

Physicists who had studied the (quantum) Yang-Baxter equations seemed to have a machine ready to grind out link polynomials in such profusion that it seemed as if one needed invariants to distinguish the invariants. Kauffman's work begins with Reidemeister's theorem, stated earlier as Theorem 2. The changes in a link diagram, which we depicted in Figure 6, are the Reidemeister moves of types I, II, and III. Kauffman's method is to use Reidemeister's moves to deduce the existence and invariance of the Jones polynomial. We begin with an oriented link defined by a diagram denoted K. Our diagram is not, in general, a braid diagram. The diagram determines an algebraic crossing number co(K), the sign conventions being given in THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

57

Figure 7a. Kauffman's version of the Jones polynomial takes the form:

FK(a) = (-a)-3,,(m(K),

(7)

where (K) is a polynomial in the variable a which will be computed from the link diagram, ignoring the orientation. It is known as the "bracket polynomial." We describe Kauffman's method for computing (K). It rests on known properties of the Jones polynomial. Let O denote a simple closed curve in the plane. Let K U O be the disjoint union of a nonempty diagram K and the diagram O. Consider four links, all defined by unoriented diagrams, the diagrams being identical except near a single crossing, where they differ in the manner illustrated in Figure 7b. We call our four diagrams K 1, K 2, K 3, and K4. In general, they determine four distinct link types. The properties that will be seen to characterize FK (a) are: (O) = 1, (K U O) = ( - a 2 - a-2)(K), (K1) = a-l(K3) + a(K4), (K2) = a(K3) + a-l(K4).

(P1) (P2) (P3) (P3')

Note that (P3) implies (P3') because if we rotate our pictures clockwise by 90~ we interchange K 1 and K 2, and K3 and K 4. This would not be true if the diagrams were oriented. Because repeated applications of (P3) and (P3') yield diagrams that have no crossings, and which therefore can only be a collection of disjoint circles, (K) is determined unambiguously on all link diagrams and is a Laurent polynomial over the integers in the indeterminate a. Even more, if K had r crossings, it would follow that (K) would be a sum of 2r terms. We illustrate this with two examples:

Example 1. Let K be the r-component "unlink", i.e., a link that can be represented as the disjoint union of r planar circles. Applying (P2) r - 1 times we find that FK(a) = ( - a 2 -- a-2)r-l{O). It then follows from (P1) that the polynomial of the r-component unlink is ( - a 2 -- a-2)r-

(a)

/NN~ p~

<x> <><><• K 1

K2

K3

K4

Figure 7. (a) Signed crossings; (b) Four related link diagrams.

+a -1 < ( ~ >

+a

<%(~>

+a -1 <

+a

<~9~>

~

>

+a < ( ~ >

F K (a) = - a 9 ( - a - 5 - a

+a 3

<%(~)>

3 +a 7 ) = a 4 +a 12 - a 16

Figure 8. Kauffman's version of the Jones polynomial of the trefoil.

the Jones polynomial. See [16] for a proof of this assertion and other related ideas. It does a very good job of telling knots and links apart (although there are distinct links with the same Jones polynomial), and would have enabled Tait and his co-workers to reduce years of work into a few days of calculation.

1.

The Yang-Baxter E q u a t i o n

Example 2 is more complicated. We compute the polynomial of the trefoil knot shown earlier in Figure l(ii). See Figure 8 for the steps in the calculation, which involve the repeated use of properties (P3) and (P3'). If we can prove that FK(a) depends only on K we will have proved that this trefoil is not equivalent to the u n k n o t . Fortunately, there is a very easy method, based upon Theorem 2. See Figure 9 for a picture proof that FK(a) is invariant under R-II. We leave it as a simple exercise for the tireless reader to check invariance under R-I and R-III. The polynomial FK(a) is, after a change in variables, 58

THE MATHEMATICAL INTELLIGENCER VOL. 13, NO. 1, 1991

A head-on attack on the problem of finding invariants of Markov equivalence would be hopelessly difficult, but there are presents from the physicists. The YangBaxter equation and its solutions play a fundamental role in two physical problems: the theory of exactly solvable models in statistical mechanics (see [3]) and the theory of completely integrable systems. In statistical mechanics one studies systems of interacting particles, and attempts to predict properties of the system that depend upon averages over all possible configurations or states of the system. As an example one might

study a 2-dimensional array of atoms located at the vertices of a lattice, and interpret a "state" of the lattice to mean the assignment of a spin (which can take on q/> 2 possible values) at each vertex. The total energy E(cr) d e p e n d s u p o n the state (r; the partition function, which is the object of ultimate interest, is a sum over all possible or's of the function exp(-kE(~)), where k is an appropriate constant. The algebraic difficulties in computing the partition function are formidable; however, under certain conditions the problem is in fact solvable. The conditions are that certain matrices that describe the states of the system satisfy what is k n o w n as the Yang-Baxter equation. This has a curious geometric meaning: the matrices satisfy the Yang-Baxter equation if and only if they define a representation of the braid group B,. To make this precise, we follow the development in [26], and let V be a free module with free basis v 1. . . . . v m over a commutative ring K. Let V| be the n-fold tensor product of V with itself. We define elements {R, i = 1. . . . . n - 1} of Aut(V ~ ) by setting R, equal to the identity on all but the zth and (z + 1) st factors, while R, restricted to those two factors is a fixed Klinear isomorphism: R: V |

V-* V|

V.

Note that it follows immediately that R 1. . . . . satisfy the condition:

(8.0)

determines, for each n, a representation of the braid group Bn in Aut(V~n), with or~~ R,. This representation in turn determines a finite-dimensional matrix representation of B,. See [18] for explicit examples. Before 1984 it was not understood that the Yang-Baxter equation had anything to do with braids, and at this writing it is still unclear how braiding arises in the physical problem. There are general methods [13] for finding solutions to the Yang-Baxter equations. (Solutions are actually associated to each representation of a simple Lie algebra.) In [26], further conditions are placed on the automorphism R, which ensure that the solutions so obtained support a Markov trace with values in the ring K. It turns out that every solution to the YangBaxter equation satisfies Turaev's extra conditions. Thus, there is a machine ready to go, to produce further link invariants, all initially quite mysterious. Reshetiken has now shown in [23] that they can in fact all be obtained from the more basic polynomials described earlier via the algebra homomorphisms of (6), if one replaces a link by an appropriate "(p,q) cable" on it (see [24] or [9]). Thus order is emerging from chaos. The order, however, appears to be part of an even larger order, which involves the physics of conformal field theory and leads to further invariants, now in arbitrary 3-manifolds ([27],[10]).

R,_I Concluding

R,R I = RIR ,

if Iz - ]1 > 1.

The automorphism R satisfies the Yang-Baxter equation if, in addition, R1R2R 1 = R2R1R 2. In view of the way in which we defined the R,'s, this implies that R,R1R , = R1R~R1

if l i - Jl = 1.

Remarks

(8.1)

(8.2)

Comparing equations (3.1) and (3.2) with (8.1) and (8.2), we see that each solution to the Yang-Baxter equation

Figure 9. The invariance of FK(a) under Reidemeister's move R-II.

One of the great puzzles about the Jones polynomial and its various relatives is that, at this writing, we do not have any real understanding of their topological meaning. We know that they are invanants of oriented link type in oriented 3-space, but our proofs do not yield interpretations in terms of the link complement or link group, or in terms of covering spaces or surfaces that the link bounds or, indeed, in terms of any of the familiar machinery of geometric and algebraic topology. There also appears to be little understanding, at this writing, of the "braiding" that occurs in the various areas of mathematics and physics that have been related to our story. These are hints of the enormous amount of work still to be done. In conclusion, we challenge the reader to use the methods described above to distinguish the links in Figure 1, by computing their Jones and or Alexander p o l y n o m i a l s , w i t h the help of relations (5), or (P1)-(P3'). Examples were chosen to include a link w h o s e type changes and one whose type does not change when one reverses the orientation of S3, also to illustrate that a link type can change when one reverses the orientation on one of the components. Note that, by the derivation we gave in w the Jones polynomial is necessarily invariant under a reversal of orientation of all of the components. THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991 5 9

If two d i a g r a m s d e t e r m i n e links w i t h distinct invariants, their link types will have b e e n p r o v e d to be distinct. H o w e v e r , if they have the same invariants, they m a y or m a y not be distinct. O u r examples illustrate this too. If one suspects they are not, one m a y then attempt to d e f o r m one diagram into the other to complete the proof.

14. 15. 16. 17.

This paper was partially supported by National Science Foundation Grant #DMS-88-05672

18. 19.

References 1. J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sciences USA 9 (1923), 93-95. 2. E. Artin, Theorie der Zopfe, Hamburg. Abh. 4 (1925), 47-72. 3. R. J. Baxter, Exactly Solved Models in Statistical Mechanics. London: Academic Press (1982). 4. D. Bennequin, Entrelacements et equations de Pfaff, Astdrisque 107-108 (1983), 87-161. 5. J. S. Birman, Braids, links and mapping class groups. Ann. of Math. Studies No. 82, Princeton Univ. Press (1974). 6. J. Birman and W. Menasco, Closed braid representatives of the unlink, preprint, 1989. 7. _ _ , On the classification of links that are closed 3braids, preprint (1989). 8. J. S. Birman and H. Wenzl, Braids, link polynomials and a new algebra. Trans. AMS, to appear, preprint, New York: Columbia Univ. 9. G. Burde and H. Zieschgang, Knots. Berlin: de Gruyter (1986). 10. L. Crane, Topology of 3-manifolds and conformal field theories, preprint, Yale Univ. (1989). 11. E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. 12. P. Freyd, J. Hoste, W. Lickorish, K. Millett, A. Ocneanu and D. Yetter, A new polynomial invariant of knots and links, Bull Amer. Math. Soc. (2)12 (1985), 257-267. 13. M. Jimbo, Quantum R-matrix related to the generalized

20. 21. 22. 23.

24. 25. 26. 27. 28.

Toda system: an algebraic approach, Lecture Notes m Physics 246 (1986), 335-361. V. Jones, Braid groups, Hecke algebras and type II1 factors, Proc. US Japan Seminar Kyoto (Araki and Effros, eds.) New York: John Wiley (1973). __, Hecke algebra representation of braid groups and link polynomials, Ann. of Math. (2)126 (1987), 335-388. L. Kauffman, States models and the Jones polynomial, Topology 26 (1987), 395-407. __, An invariant of regular isotopy, Trans. Amer. Math. Soc., to appear. T. Kohno, Linear representations of braid groups and classical Yang-Baxter equatmns, BRAIDS, Contemp. Math. 78, 339-364, Amer. Math. Soc. (1988). A. King and M. Rocek, The Burau representation and the Alexander polynomial, preprint, Stony Brook: SUNY (1988). H. Morton, Threading knot diagrams, Math Proc. Camb. Phil. Soc. 99 (1986), 246-260. J. Przytycki and P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987), 115-139. K. Reidemeister, Knotentheone. New York: Chelsea Pub. Co. (1948). English translation. Knot Theory, BSC Associates, Moscow: ID (1983). N. Reshetiken, Quantized universal enveloping algebras, the Yang-Baxter equation, and mvariants of links I and II, preprint, Leningrad. Steklov Institute of Math. (1987). D. Rolfsen, Knots and Links, Berkeley: Publish or Perish (1976). P. G. Tait, On Knots I, II, III. Sczentific papers I, London: Camb. Univ Press (1898). V. Turaev, The Yang-Baxter equation and invariants of links, preprmt, Leningrad: Steklov Institute of Math (1987). E. Witten, Quantum field theory and the Jones polynomial, preprint, Institute for Advanced Study (1988). S. Yamada. The minimum number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), 347-356.

Department of Mathematzcs Columbia Unwersity and Barnard College New York, N Y 10027 USA

'::'""""........MOVING? We need your ~ww address so that you do not m~ss any zssues of The Mathematical Intelhgencer. Please fill out the ]orm below and send ~t to:

Spnnger-VerlagNew York, Inc. Journal FulfillmentServmes 44 Hart.zWay Secaucus, NJ07094

;.'%, eeee

~e

Old Address (or label) Name

eo e Address

0@ @O

City/State/Zip

New Address Name

60

Please g w e us

Address

sLx w e e k s notice.

City/S~te/Zlp

THE MATHEMATICAL INTELL1GENCER VOL 13, N O 1, 1991

Ian Stewart* The catapult that Archimedes budt, the gambhng-houses that Descartes frequented in h~s dissolute youth, the field where Galols fought his duel, the bridge where Hamzlton carved quaternions-not all of these monuments to mathematical hzstory surwve today, but the mathematician on vacation can stUl find many reminders of our subject's glorious and mglorzous past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the

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

Sir Isaac N e w t o n Radoslav Dimitri4 Fierce fighting between two boys in a churchyard intensifies as a fair-haired youth rages with excessive e n e r g y and d e t e r m i n a t i o n matching those of the eternal forces of nature. The other boy gives up, is pulled up by his ears and has his nose rubbed in humiliation against the church wall. The event took place in about 1655 at St. Wulfram's church, the crowning glory of Grantham, a small town in Lincolnshire. The victor was Isaac Newton, a pupil last but one in academic achievement in his class; the loser was Arthur Storer, one of the better students in the class. We know of the event from Isaac Newton's list of sins, w h i c h contains an e n t r y that r e a d s "Beating Arthur Storer." The son of a Lincolnshire yeoman farmer, Isaac Newton was born prematurely on Christmas Day 1642 (on 4 January 1643 by the Gregorian calendar), three months after his father died, pathetically weak and s o tiny that he could fit into a quart pot. He was born in Woolsthorpe-by-Colsterworth, in the heart of England (7 miles south west of Grantham), the same year that Galileo Galilei died and England embarked on the great Civil War. People in Lincolnshire believed at that time that posthumous children had unique life forces, which they were able to use and pass on to others. This force certainly helped y o u n g Newton to survive the o b -

* Column Editor's address: Mathematics Institute, University of Warwick, Coventry CV4 7AL England

stacles of his early childhood and, moreover, there is evidence of his strong analytical and mechanical abilities at that time, realized by making models of windmills (driven alternatively by air and mouse-power), water clocks, etc. I s a a c Newton was not satisfied with beating up Arthur Storer--this catalytic event was the bridge he so much needed for the transition from the weak, inferior period into the heroic stage of his life--he also organized his intellectual forces and ended up first in his c l a s s at Grantham Grammar School (today King's School). On the recommendation of his uncle, William Ayscough, he was rescued from his mother's ambitions for him to become a farmer and entered Trinity College, Cambridge, becoming a subsizar (a poor student earning his subsidy and education by serving more senior students and teachers) on 5 June 1661. Newton began serious studies of Aristotle (Organon and Ethics) and then Copernicus (De revolutionibus orbium coelestium), Kepler (Dioptrice, Astronomia Nova, Harmonices Mundi), Galileo (Dialogo), and Descartes (G~orn~trie and Principia Philosophiae) and soon found himself in the promised land of discoveries. He obtained his bachelor of arts degree on 14 January 1665, the same year that Cambridge University closed because it was hit by plague. Newton then spent most of the next two years in his birthplace Woolsthorpe and, by his own account, this was his most fertile period, with 1665/1666 termed the "'Anni Mirabiles.'" According to him, the method of fluxions and its applications to tangents and curvatures and works on series date

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 19 1991 Sprmger-Verlag New York

61

from this period, as well as work on mechanics, early notions of force, the centripetal force on the Moon's orbit, the theory of colours, and initial thoughts on gravity. Newton was appointed Lucasian Professor at Cambridge, at the age of twenty-six, succeeding Isaac Barrow, who is believed to have stood down for him. In this professional period of his life, Newton deposited with the University Library his annual lectures on optics (1670-1672) and algebra and arithmetic (16731683); the great part of the first book of the future Principia (1684-1685); and The System of the World (1687). Membership in the Royal Society came four years later in 1672 after he impressed the society with his development of the reflecting telescope; he used his k n o w l e d g e of alchemy to make n e w kinds of mirrors from a mixture of tin and copper. The crux of Newton's breakthrough into the world of mathematical discovery was the discovery in 1664-1665 of the infinite binomial formula, u p o n reading John Wallis's Arithmetica Infinitorum, a landmark in the h i s t o r y of m a t h e m a t i c s , w h e r e the squaring of the circle (finding its area) is done through evaluation of the integral f~ X/T ~ -fidx. In his letters of 13 June and 24 October 1676 to Oldenburg (which were meant for transmission to Leibniz) Newton gives the expansion (p + pQ)m/,, = pm/n + m AQ + m - n BQ n 2n m - 2n + - CQ 3n m

+

3n

-

4~

DQ + &c.,

where in the second summand A stands for the first term and in the third term B stands for the second term and so on . . . . By the appropriate substitution of the parameters it is possible to derive the following formula: X/c2 + x2 = c +

x2 2c

x* x6 +-8c3 16cs

5x8 7xX0 - 128c----~+ ~ + etc. This crucial invention and freeing the upper bound in the integral enables him to get an infinite series X

_~X3 3

IX5 -- ~X7 _~-~8 5 x9 _ 5 7 9 "'"

containing the sequence of rational binomial coefficients. By squaring the hyperbola: f~(1 + t)-ldt, Newton also found the series 62

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

x2 x3 ln(1 + x) = x - - ~ + 3

x4 xs 4 +-5-

x6 6 + ....

At the same time he developed the general differentiation techniques using an infinitely small quantity o, which he replaced by "fluxion" of a variable, which in modern terminology can be interpreted as speed dx/dt (Newton introduced the notation 5cin 1691 and a notation for partial derivatives in 1665). Nicolaus Mercator's Logarithmotechnia, published in September 1668, aroused Newton's concern as he saw there "his (own) reduction of log (1 + a) to an infinite series by continuing division of 1 + a into 1 and successive integration of the quotient term by term." In order to protect his discoveries, Newton submitted the tract De analysi per aequationes infinitas to Barrow and this was incorporated into another manuscript, Methodus fluxionum et serierum infinitarum, whose original Latin text was printed in 1779 and the English version in 1736. It is believed that among the scholars w h o saw the manuscript of De analysi was Leibniz, while on a visit to London in October of 1676. The Methodus fluxionum displays general techniques for differentiation and the inverse operation of integration. Newton did not explicitly discuss or obtain any results on the convergence of his series. Other of Newton's essential contributions were in optics, work published as late as 1704 under the title Opticks, where he presented his important discoveries on light and colour, such as refractions of white light through prisms, reflections, and interference effects (the ring phenomenon, nowadays called N e w t o n ' s rings). Newton was the main proponent of the corpuscular theory of light, although he also incorporated the wave theory. The first edition of Opticks ends with sixteen queries. Newton suggested in the first query that bodies act on light at a distance to bend the rays - - a fact proved only within the framework of relativity theory. With all the splendid discoveries we have mentioned, Newton was ready to give an all-embracing philosophy of N a t u r e - - a n explanation of absolute laws based on his discoveries in mathematics, to be recorded in 1685, in Philosophiae Naturalis Principia Mathematica. Principia was written in a short time and was subsidized by Halley's money for publication. It can be v i e w e d as a " s y n t h e s i s " of Kepler's three laws of planetary motion and Galileo's laws of falling bodies and projectile motions, after these have been appropriately revised. Book I of Pnncipia is a mathematical treatment of motion under the action of forces in spaces without resistance, as well as some aspects of motion according to Kepler's laws. Newton introduces methods of approximation to find the future position of a body on an ellipse using the law of areas and presents a

THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991

63

truly general resolution of the inverse problem of finding the orbit from a given force law. In proposition 59 Newton states that Kepler's third law should not be written as ~ : ~ = a13..a2,3and a corollary can be drawn that the law should be corrected as (M + ml)~:(M + m2)~ = am3..a2,3where M is the mass of the sun and m, are masses corresponding to the distances a,. From this expression we immediately obtain the modern version of the law. Proposition 66 is a restricted three-body-problem--a major aspect of celestial mechanics inaugurated by Newton. Book II is on the motion of bodies in fluids and is of greater interest for mathematics than for physics. He discusses general principles of hydrostatics and the motion of pendulums in resisting mediums.

Several psychoanalysts attribute Newton's meticulousness in sending a number of counterfeiters to the gallows to his traumatic childhood. In Book III, in the Newtonian system of the world, the motions of planets and their satellites, the motions of comets, and the phenomena of tides are all embraced within a single mode of explanation. Newton stated that the force that causes the observed celestial motions and the tides and the force that causes weight and centrally directed acceleration of freely falling bodies are one and the same; for this reason he gave the name "gravity" to the centripetal force of universal attraction. He shows that the earth must be a flattened spheroid and computes the magnitude of the equatorial bulge in relation to the pull of the moon producing the long-known constant of precession; he also gave an explanation of variation in weight as a function of latitude on such a rotating non-spherical earth, a problem treated by his predecessors. Above all, Newton stated the law of universal gravitation, i.e., that the force of gravity varies as the inverse square of the distance and that the power of gravity pertinent to all bodies is proportional to the quantities of matter they contain. The story that Newton had received his ideas on gravity from a tree, after observing a falling apple in his orchard in Woolsthorpe, had been invented by Voltaire, who claimed to have heard it from Newton's niece during his visit to London (he, however, never saw Newton). Although Lincolnshire indeed seems to be an apple county and although Newton might have used the story in his late London years (as reported by some other sources), it can be seen as nothing more than a simple (and inappropriate) trivialisation of his discoveries. For it is certain that his inventions came about only by dint of concentrated meditations, after 64

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

years of painstaking study and effort to understand the works of his predecessors. Newton, however, hardly ever acknowledged this fact and only after Hooke's accusations of plagiarism did he state: "If I have seen further it is by standing on the shoulders of Giants." The contributions and results from Wallis's Arithmetica Infinitorum are immeasurable in Newton's invention (perfection) of the method of infinite series, the quadrature of curves, and the generalization of the binomial theorem. Influence in calculus was no doubt exerted by Isaac Barrow, Newton's teacher, whose Lectiones Geometricae were an organized and complete treatise on calculus cast in a then fashionable but awkward geometrical form. Newton helped prepare these Lectiones for publication. That book contains all the derivatives and integrals of ordinary functions, rules for differentiation, including the quotient and chain rules, the proof that integration and differentiation are inverse operators, and the basic theorem for rectification of arcs. Newton disliked geometry, and the plague of 1665 forced him away from Barrow's geometrical influence. This was enough for him to give the more acceptable analytical theory of fluxions that caught on. Kepler, Leonardo da Vinci, Archimedes, Galileo-all had played a part in moulding Newton, the genius. There would not have been a Newton without Galileo, Kepler, and Wallis, just as there would not have been Einstein without Newton. Newton's real genius lay in patient study and thinking about the works of "the Giants," seeing the weaknesses of their theories, strengthening them, reformulating and putting them in an appropriate context, thus seeing wider contexts and building a noble edifice to serve and impress generations to come. Falling apples could not have helped much in this difficult and lucid process of invention. After 35 years at Cambridge Newton had become tired and bored with the academic atmosphere and the third phase of his life began when, on 19 March 1696, he received a letter from Charles Montagu informing him of his appointment as the Warden of the Mint in London. Newton's last thirty years were marked by his more philosophical thoughts and writings on alchemy, theology, and prophecy (Observations upon the

Prophecies of Daniel and the Apocalypse of St. John, London 1733). This was the period of highest social recognition: a foreign member of the Paris Academy of Sciences, president of the Royal Society, Member of Parliament for Cambridge University, and a knighthood in 1705. The Principia, composed in the 1680s, was the last great exertion of his immeasurable mathematical genius. However, in his later years Newton solved the problem of finding the curve of swiftest descent (a cycloid), which proved to be an early example of what was to become the calculus of variations. This was also the period of Newton's bitter priority disputes, first his incredible accusations of Locke and Pepys; then those of Huygens and Hooke (who ac-

cused him of plagiarism connected with the inverse square law and some results in optics), as well as Flamsteed (observations on the motions of the Moon). Newton's duties as Warden (and later as Master) of the Mint were practical--he was the first to propose gold as a basis for the monetary system, supervision of the recoinage, and the capture, interrogation, and prosecution of counterfeiters. Several psychoanalysts attribute N e w t o n ' s meticulousness in sending a number of counterfeiters to the gallows to his traumatic childhood (he never saw his father, he barely survived after birth, and his mother remarried when he was two). An especially bitter quarrel was with Leibniz about philosophy or theology in relation to science and over the invention of calculus, with both sides having their own supporters. Leibniz appealed to the Royal Society for a fair hearing and Newton quickly appointed a committee of his own supporters and wrote the committee's report, the (in)famous Commercium epistolicum. He has been reported to have said with pleasure that he had broken Leibniz's heart with his reply. Newton died, at the age of eighty-five, on Monday, 20 March 1727. N u m e r o u s places are directly or indirectly connected with Newton; we mention some: Newton's tomb and statue (executed by Rysbrack) are in the most conspicuous place in Westminster Abbey in London. In the Tower of London Newton interrogated and imprisoned counterfeiters. The British Museum, The Royal Society, and the Royal Mint Library Public Record Office contain pictures as well as letters and other documents of and related to Newton. During his Cambridge period Newton lived in a monastic-like cell, thought to have been the room marked E4, a m o n g the first-floor chambers in the Great Gate of Trinity College, directly above what is now a porter's lodge. Trinity College Ante-Chapel houses Roubiliac's fine statue of Newton in a prominent place. Trinity College (or Wren) Library has a stained-glass window depicting the "Muse of the College" presenting Newton to George III, with Francis Bacon taking notes of the proceedings. The library contains original manuscripts by Newton (such as a copy of Principia with comments in his handwriting) and a great number of books from his private library. The Fitzwilliam Museum, the University Library, and King's College Library contain a great number of items of interest for Newton scholars and admirers. The recently founded mathematical institute in Cambridge, which will open its doors in 1992, is to be called the Sir Isaac Newton Institute. Grantham is situated 105 miles north of London on the A1 highway. Newton's Grammar School (now expanded to King's School) and St. Wulfram's Church are today separated by Church Street. The inside (and outside) wails of the old school have a crowd of past

Roubiliac's statue of Newton, Ante-Chapel, Trinity College, Cambridge. pupils' names carved on them, with Newton's on one of the window ledges. Theed's large bronze statue of N e w t o n s t a n d s in f r o n t of t h e T o w n Hall a n d Grantham Museum, which also contains Newtonia. During his school years Newton lived in a building on the left-hand side of what is now George Hotel in High Street. The hamlet of Woolsthorpe, Newton's birthplace, lies seven miles south of Grantham and half a mile northwest of Colsterworth. The Manor House, where he was born, is well preserved and is a place of pilgrimage. There are two apple trees in the yard in front of the house; the gnarled old apple tree ("Flower of Kent," a cooking-apple tree still bearing fruit) may have been grafted from the proverbial original that blew down. St. John the Baptist Church in nearby Colsterworth is the church where Newton was baptized. A photocopy of the entry of his christening (the original is kept in the Lincolnshire Archives) in the records reads: "ISAAC SO NNE OF ISAAC & HANNA NEWTON, 1 JAN." This church also houses a sundial (a carved semicircle divided into twelve segments) that is supposed to have been found on one of the wails of the Manor House. Before Grantham Newton went briefly to schools in neighbouring Skillington and Stoke, where there is an obelisk set up in his honour. Additional documents are to be found in the Bodleian Library and Corpus Christi College in Oxford, the Jewish National and University Library in Jerusalem, the Dibner Collection in the Smithsonian Institute Libraries, the Pierpont Morgan Library in New York, and the Babson College Library, Babson Park, in Massachusetts.

Department of Mathematzcs Stanford University Stanford, CA 94305 USA THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991 6 5

Abelian and Nonabelian Mathematics I. R. Shafarevich

Many theories that are part of the most beautiful chapters in the mathematics of the nineteenth and twentieth centuries have a common algebraic nucleus: the duality theory of abelian groups and their groups of characters. It is natural to assume that a series of basic difficulties in mathematics can be explained by the lack of a generalization of this duality to nonabelian groups. Such a point of view has been presented in A. Weil's paper [6]. The "nonabelian mathematics of the future" philosophy also inspired me when I was just starting to work on mathematics. Now, half a century later, I think it would be interesting to sum up the achievements in this direction, the more so since, just as before, the u n s o l v e d problems o u t n u m b e r the solved ones, so that the old program may apparently inspire yet another generation. The goal of the following survey is to give a general idea of the intertwining of ideas that arise in this area.

The most important example of a pairing is provided by the integration of differential forms. If, for example, 00 is a 1-form, and r is a curve on a manifold X, then f~00 can be considered as a pairing (r More precisely, the pairing is defined between the space H~(X;R) of one-dimensional homologies and the quotient space {o~ : d00 = 0}/{00 = df}. The nondegeneracy of this pairing and its generalization to m-dimensional chains is guaranteed by the de Rham theorem. (One could also substitute C for R everywhere.) A nondegenerate symmetric form on V determines an isomorphism V* -~ V. If the real vector space V is oriented, then the operation W ~ W • carries over to

A b e l i a n Mathematics

1. The duality between a finite-dimensional vector space V and its dual V* can be given by a pairing (x,y) R, x ( V, y ( V*, which is bilinear. This pairing is nondegenerate, i.e., if (x,y) = 0 for all y ( V*, then x = 0, and similarly for x and y interchanged. For each subspace W C V there is an orthogonal subspace W • = {,!/ ( V* 9 (x,y) = 0 for x E W~. The correspondence W ~ W • is the object of the theory of linear equations. Passing to projective spaces P(V) and P(V*) leads to duality in projective and algebraic geometries. If V is a Banach space, then so is V* and their unit balls are dual convex sets (e.g., the icosahedron and dodecahedron, the cube and octahedron are mutually dual). THE MATHEMATICAL INTELLIGENCER VOL 13, NO 19 1991 Spnnger-Verlag New York

67

the corresponding decomposable polyvectors and determines an isomorphism * : A m v ---> A n - m V *, n = dim V. This operation carries over to differential forms on manifolds. For example, there is a 2-form in 4-dimensional space-time R4, which corresponds to the electric and magnetic fields given by the vectors E = (E1, E2, E3) and H = (HI, H 2, H3): 3

to = ~ E , d x , /Xdt + H~dx 2 A d x 3 + H2dx 3 A d x 1 1

+ H3dx I A d x 2.

If the metric in R 4 has the form dt 2 - ~,3 dx 2, then %0 = - ~

Hzdx , /k dt + Eldx 2 /k dx 3 + E2dx 3 A dx 1

+ E3dx I / k dx 2.

The operation * reflects the duality between the electric and magnetic fields. Maxwell's equations have the following form in the absence of current: do = 0, d(*oo) = 0. Analogous relations on an arbitrary Riemannian manifold lead to the theory of harmonic differential forms and to Hodge theory. 2. The characters of a finite abelian group A are the homomorphisms • A ~ U(1) into the 1-dimensional unitary group U(1) = {z E C : ] z I = 1}. The duality between a group A and the group of characters X(A) is also given by a nonsingular pairing (a,x) = x(a) E U(1), a E A, X E X(A). A homomorphism f : A ~ B d e t e r m i n e s a h o m o m o r p h i s m X(f) : X ( B ) --~ X ( A ) which sends X E X(B) to the composition of f and X. If f is the inclusion of A into B, then X(f) is a map onto all of X(A). The kernel of the homomorphism X(f) is denoted A • in this case and it has all the properties of an orthogonal subspace. In particular, X(B/A) = A •

(1)

The characters X E X(A) form a basis for the space ~(A) of complex-valued functions on A, dim ~ = ]A I, the order of A. The decomposition r = E• for r E ~(A) is called the Fourier decomposition of the function r If for all a E A we have a m -- 1, then the values of the characters of the group A are m-th roots of unity. Assume they all belong to a field K. They form a group, denoted p.m" In what follows, in order to shorten the statements we will always assume that the fields have characteristic 0. Let V be a finite-dimensional vector field over a field K, on which a group A acts (i.e., there is a representation of A on V). Then there is a decomposition V = Ge,K,

(2)

in which the 1-dimensional subspaces e,K are invariant 68

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

under the action of A, i.e., ae, = x,(a)e,, where X, E X(A). The notions above generalize to locally compact abelian g r o u p s - - t h a t is Pontryagin duality. In such generality they contain the theory of Fourier series and integrals, and the mathematical description of the particle-wave duality in quantum mechanics. Here are the simplest examples. (a) The groups Z and U(1) are mutually dual; the pairing n E Z and o~ E U(1) is (n,o0 = o~n. (b) The additive group R is self-dual: the pairing is (x,y) = e2~'~. (c) The same is true for the group C : (z1, z2) = e2"m'Rezlz2,

3. Galois theory of abelian extensions. The field C of complex numbers has two automorphisms that leave the reals invariant: the identity and "r(z) = ~. In other words, G = {1,7} is the Galois group of the extension C/R (denoted Gal(C/R) = G). In this sense C is a representation of the group G and the decomposition (2) has the following form: C = R + iR. This is connected with the fact that R U iR are the only complex numbers for which z2 E R. Let, in general, ~ / K be a Galois extension, G = GaI(~/K), and p.m C K for some integer m. If ~ contains the root mVa for a E K, a # 0, then for each cr E G, (cr(mVa)) m = a and therefore (r(mv~)/mVa E P'm" This defines a pairing between the group G and the group 3~ of those a E K, a # 0, for which mVa E ~s : (r = cr(mVa)/mVa. The group {a E K : a # 0} (under multiplication) is denoted K*. It is easy to see that (or,a) is a pairing between G and ~ / K *m, where K*m = {am : a E K*}. If the field ~ is g_enerated by the roots m V ~ , a E ~ , we write ~ = K(mV3r In this case the pairing (or,a) is nonsingular, so that G = X ( ~ / K *m) and, in particular, the group G is commutative. The extensions with commutative Galois group are said to be abelian. The extension (1) then has the form = a~:~l~wmmX/'aK.

Vice versa, if G is an abelian group such that g " = 1 for all g E G and p.m C K, then any extension ~./K with Galois group G has the form ~ = K ( m v ~ ) for some group 3E, K* D 3~ D K*m. This is the Galois theory of solving in radicals e q u a t i o n s with " c o m m u t a t i v e Galois g r o u p , " t h o u g h n o w it is called K u m m e r theory. It can be given a more elegant form by considering the union K~b of all extensions of the field K with Galois groups of the kind described, for a given m. As a rule these are infinite extensions: for exam_~ple, for K = Q, m = 2, it is obtained by adjoining V - 1 and V~, for all prime p, to Q. The Galois group of an infinite extension, being a transformation group on an infinite set, has a natural topology: the neighbourhoods of the identity are the stabilizers of finite sets. It is compact

with this topology, and Kummer theory claims that the group Gal(K~b/K) is dual to the discrete group K*/K*".

4. The theory of abelian functions. This is an analogue of Kummer theory for the case when K is the field of rational functions C(S) on an algebraic curve S. For example, if S is given in affine coordinates by the equation F(x,y) = 0, then C(S) = C(x,y), where x and y are connected by this relation. The curve S is a subspace of some projective space pN, actually an oriented surface: a sphere with g handles; g is called the genus of the curve S and the field K. If K = C(S) and K' D K is a finite extension, then K' = C(S'), -where S' is another algebraic curve and the inclusion K C K' is determined by a rational map q~: S' --> S which is a finitesheeted cover. We consider the case w h e n q~ is an unramified cover, i.e., all points s ( S have the same number of preimages. This number equals the degree [K':K] of the extension K'/K. Then K'/K is also said to be an unramified extension. In this case the problem is essentially a topological one. Indeed, it follows from basic results of the theory , of algebraic curves and Riemann surfaces that the topological type of the covering S' ~ S uniquely determines the curve S' and the field K' = C(S') (up to isomorphism), and any unramified finite-sheeted topological cover is determined by some curve S'. So, the set of all unramified extensions K'/K is "isomorphic" to the set of topological finite-sheeted covers. The latter is well known to be described by the sub-groups of finite index of the fundamental group ~1(S) of the curve S, which is completely known, and which has 2g generators a I . . . . . as, bI . . . . . b~ and one relation a l b l a l - l b l - 1 . , agbgag-lbg -1 = 1. This description is based on topological and analytical considerations, and a natural question arises: to find an algebraic equivalent. The answer is known for the case of abelian unramified covers. The unramified topological coverings with abelian automorphism group correspond to subgroups of finite index of the quotient group of ~r~(S) by its commutant, which is isomorphic to the group HI(S;Z ). This group is isomorphic to Z 2g. We shall denote it in what follows by H. The algebraic interpretation of the subgroups I~ C H of finite index is connected with the inclusion H C Cx, which uses the homology and forms duality from Section 1. More precisely, there are g linearly independent holomorphic 1-forms ~o1. . . . . o~ on S ("differentials of the first kind"). The correspondence or ~ (J'~oh . . . . . .f~%), or E H, determines an inclusion He--* Cg, and Cx = H . R. The quotient group Cg/H is a g-dimensional complex toms. It is called the Jacobian of the curve S and is denoted by ~(S). The curve S can be embedded in it (for g > 0). Namely, choose a point So ~ S and connect each point s ( S by a path cr to So. The map

j(s) = (f~o~1. . . . .

J'r

mod H

(3)

does not depend on the choice of path or and determines an embedding of S into ~(S). The Jacobian is a projective algebraic variety and, via the map (3), may be constructed purely algebraicaUy. We are going to say more about this later, and for the time being will give a description of the subgroups of finite index F C H, [H:F] < % in terms of Jacobians. The intersection index determines a skew-symmetric integer-valued unimodular form (a,b), which we extend by linearity to C~ = H " R. The function (x,y) = e2~"(x,Y~, x,y ~ Cg, determines a pairing of Cg with itself, under which H" = H. This implies, along with (1) in Section 2, that ~(S) -~ X(H).

(4)

If F C H, then F • D H, [F':H] = [H:F]. If [H:F] < % then the group F• C Cg/H is a finite subgroup of the torus ~(S), and any of its finite subgroups can be obtained in this way. It follows that there is a one-to-one correspondence between the finite subgroups F' C ~(S) and the abelian unramified extensions K'/K. It is easy to see that GaI(K'/K) = X(F'). If we introduce as in Section 3 the maximal abelian unramified extension K~ of the field K, this result can be expressed by the formula GaI(K~/K) = X(~(S)tors), where, in general, l,ttors denotes the subgroup of elements of finite order of the abelian group II. Of course, the subgroup F' may be characterized as the kernel of the homomorphism of tori Cg/H ~ C~/F" and is uniquely determined by the homomorphism C~/F -~ Cg/H. In the latter case, the covering S' ~ S itself m a y be recovered as the inverse-image of S (more precisely, j(S) C Cx/H) under this homomorphism. Here is the algebraic description of the Jacobian. It is

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991 6 9

based on the following notions. A formal combination = m l s I + . . . + m r S r , m~ E Z, s~ ff S, is called a divisor on the curve S. The divisors form a group Div(S). The number X m~ = deg 5~ is called the degree of the divisor 5~. The divisors of degree 0 form a subgroup Div~ C Div(S). If a function f E C(S), f # 0, has zeroes and poles at s~. . . . . Sr on S of degree rnx. . . . . m r (m, > 0 for the zeroes and m, < 0 for the poles), then (f) = ~ re,s, is called the divisor of the function. All the divisors (f), f E C(S), f # 0, form a group P(S) C Div(S). It is always the case that deg(f) = 0 (the number of zeroes is equal to the number of poles), so P(S) C Div~ The Abel-Jacobi theorem states that Div~

= ~(S).

(5)

The isomorphism in (5) is given by the condition q~(X m,s,) = Y, md(s,), where the right-hand side ~ denotes summation in

~(s). This is the algebraic characterization of the Jacobian ~(S). It affords a description of the Jacobian in purely algebraic terms as a g-dimensional algebraic variety in projective space. There is another important interpretation of the Jacobian. It is based on the fact that the divisor @ = Xm,s z can be given by exhibiting functions f, having zeroes or poles of degree m, at the points s~ E S. Here, one allows a chm.ge of functions f, ~ f~u,, where u,(s,) 0, ~. One can also assume that, at each point s E S, a function fs # 0, fs(s) ~ 0, ~, is given, for almost all s E S. "Almost all" means all but a finite number. Then Div(S) = {fs, f~ # 0, fs(s) ~ O, oo for almost all s E S} ! {us, u~(s) ~ O, ~ for all s E S}. Let a divisor @ be given by the points Sl . . . . . Sr and the functions h ..... fr" Consider neighborhoods G 3 s, such that f, 0, oo in U~Ns,, and introduce one more neighborhood U0 such that U0 U (U U~) = S; we shall assume that for it f0 = 1. Using this, one can construct a fiber bundle E~ over S with fibre C. Over U, it has the form U, x C, and the points (s,z) E U, x C and (s,z') E U~ x C are identified provided s E U, n U~, f,z' = f~z. Since the function f71f, ~ 0, ~ on U, n /./i, this map is an isomorphism, and E~ is a locally trivial complex fiber bundle. It depends only on the divisor @: a change of variables f, ~ f,u v u, ~ 0, ~, in U, does not change its type. But it also does not change under the substitution f, ~ ff,, i.e., it depends only on the image of @ in the group Div(S)/P(S). The topological type of the fiber bundle E~ is determined by the number deg @ (i.e., its Chern class c~(E~)). So, the group )(S) = Div~ describes the set of locally trivial analytical (or algebraic) fiber bundles over S with fiber C, which are topologically trivial. 70

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

The whole theory generalizes to ramified coverings; to describe a ramified covering over a finite set X C S, one has to use the group HI(SNX;Z ). It also generalizes to algebraic varieties of arbitrary dimension n. 5. Class field theory is the theory of abelian extensions of fields of algebraic numbers, i.e., finite extensions of the field Q of rational numbers. We start with the field Q itself. An example of an abelian extension of Q is the field K, = Q(~n) obtained by adjoining all n-th roots of 1. Its automorphisms have the form % ( 0 = ~a, ~ E ~n, where (a,n) = 1 and a is determined uniquely (mod n). So, Gal(K,,Q) = (Z/nZ)* is the group of invertible elements of the ring Z/nZ. The Kronecker-Weber fundamental theorem states that any abelian extension K/Q is contained in some Kn. So, the union Qab of all abelian extensions coincides with U, Kn. We can thus find its Galois group. It is easy to see that U , K~ = Up K(P), where for a prime p, K(P) = CJm>0 Kpm and Gal(U'K,/Q) = lip GaI(K(P)/Q). The automorphisms of the field K(P)are determined by the sequence of residues a m (mod pro) satisfying the conditions (1) am+1 =- am (mod pm), (2) (am,p) = 1. The sequence of residues am determines by (1) a padic integer tx E Zp, and (2) means that oLis invertible, i.e., oL E Zp. So, Gal(K(P)/Q) - Z~, and Gal(Qab/Q) - lipZp.

(6)

If K is an arbitrary algebraic number field, then its prime divisors play the role of prime numbers. Denote the set of prime divisors by S. To each ~ E S there corresponds a V-adic completion Kp of K. Let ~ C K~ be the set of its integers. Consider the set IIpK~ of sequences {% E K~ : 10 E S}. Its element {% : ~ E S} is called an idele provided % E (~ for almost all I0 E S. The group of ideles is denoted I(K). Each element o~ E K* determines an idele because of the inclusion K C K~ for all 10 E S. The idele class group is the group C(K) = I(K)/K+, where K+ consists of those a E K* for which all conjugates are positive (if real). Class field theory establishes an isomorphism of the groups C(K) and Gal(Kab/K). For K = Q it is easy to see that I(Q) = Q+ lip Zp and that isomorphism becomes (6). Notice the analogy between this description and the one give in Section 4. The prime divisors in K are analogous to the points s on the curve S; the ideles to the sets of functions {fs : s E S, fs(s) ~ 0,~ for almost all s E S}, and the elements a E K+ to the divisors (f) E P(S), f E K(S)*. The remaining difference is that w h e n defining divisors @ E Div(S) we factor sets of functions {fs} by sets {u x : us(s ) ~ 0,~ for s E S}. But this difference is connected with the fact that in Section 4 we considered only unramified abelian extensions K'/K. The description of analogous extensions of the field K also necessitates factoring the group C(K) by a corn-

pletely analogous subgroup {% ~ ~; : 10 ~ S}. It is even easier to describe the "local" analogue of class field theory w h e n K is a finite extension of the field of p-adic n u m b e r s Qp. In this case the group K* plays the role of the group C(K): there is a 1-1 correspondence b e t w e e n the finite abelian extensions ~ / K a n d the s u b g r o u p s of finite index of that group. In particular, if ~m C K, then Gal(K~,b) = K~/K~m. Comparing this with K u m m e r theory (Section 3) we see that K*/K *m = Gal(K~,b) = X(K*/K*m), i.e., there is a nondegenerate pairing (a,b) E I~m of the group K*/K*m with itself. For example, for K = Qp, m = 2, p # 2, (a,b) = (asb-r/p) if a = pra', b = pSb', a',b' ~ Zp and is not divisible by p, a n d (x/p) denotes the Legendre symbol. If, on the other hand, p = 2, a a n d b are odd, then (a,b) = ( - 1) (a-1)(b-1)/4. In the case of algebraic n u m b e r fields, class field theory describes not only the group Gal(K~b/K) but also, because of the arithmetic character of this description, allows us to study in detail the arithmetic of the abelian extensions K ' / K - - t h e decomposition laws for prime divisors of the field K into the field K' and reciprocity laws. Assume that K D I~m; in this case we m a y again relate class field theory and K u m m e r theory. We see that again there is a pairing (a,[3) ~ ~m, where a ( C ( K ) , [3 ~ K*, or equivalently, a pairing (a,~), a E I(K), [3 ~ K+, equal to 1 w h e n a ~ K+ or [3 (K*)m. The explicit calculation of this pairing leads to the expression (a,[3) = II~s(%,~), if a = {% : p ~ S} a n d (%,[3) is the pairing of the group K'dlC~m with itself, which was derived from local class field theory. The relation (a,[3) = 1 for a = o~ E K+ gives the classical reciprocity laws. For example, one can easily see that for K = Q and m = 2 the relation IIp(a,b)p = 1 gives the Gauss quadratic reciprocity law.

N o n a b e l i a n Mathematics 6. The algebra of representations of a finite group. If G is a finite not necessarily commutative group, a natural analogue of characters are the irreducible repre-

s e n t a t i o n s ~ : G --* GL(V) into the g r o u p of linear transformations of a complex finite-dimensional space V. There are a finite n u m b e r of such representations, q01. . . . . %, up to equivalence determined by an isom o r p h i s m of the space V. A tensor product can be defined on the set di = {@1 . . . . . ~h} : for ~, : G ~ GL(Vz), i = 1,2, (q01 q~2)(g) = %(g) | q~ ( GL(V1 | V2)" The representation qh | q~2 is, in general, reducible and therefore a direct s u m of irreducible representations with some multiplicities: h

% | cp, = ~

n~k)%,

n~k) ~ Z, n~) t> 0.

(7)

k=l

The trivial representation ~01(g ) = 1 ( G L ( C ) plays the role of unit element for this operation: qh | q0, = %. The analogue of the inverse is the representation cpx (g) = ~(g-1)T ~ GL(V*). It is inverse to q0 in the sense that this is the unique irreducible representation for which in the decomposition ~0 | q0x = E nA0, the coefficient n 1 in front of the trivial representation % is positive. Thus, the set A = {% . . . . . q~h} retains, to a weak degree, some group properties. The n u m b e r h of irreducible representations is equal to the n u m b e r of conjugacy classes in the group G, and the coefficients n}k) in (7) can be f o u n d from the multiplication of these classes. Namely, if Ci and CI are two conjugacy classes, then, in the set CzC1 = {ab : a Cv b ~ Cj}, all elements of the class Ck enter the same n u m b e r of times. Denote that n u m b e r by m~), a n d write

c,q =

(8)

It follows from ba = (bab-1)b that C,Cj = CIC,, i.e., m~k) = m~k). So, even a noncommutative group can be decomposed into smaller subsets, on which the multiplication is still commutative, t h o u g h t h e y no longer form a group, but only a multiplication table (8). Here again, the trivial class C 1 = {e} plays the role of unit element, and the class C -1 = {c -1 : c ~ C} plays the role of the inverse in the same sense as in the table (7). It follows easily that the commutative algebra A with multiplication table (8) is semisimple and decomposes as a s u m ~ C. The projection onto the i-th s u m m a n d determines a h o m o m o r p h i s m • : A --> C. This is the character of the irreducible r e p r e s e n t a t i o n %, i = 1. . . . . h. That is exactly h o w Frobenius introduced characters. Their multiplication as functions on the set F of conjugacy classes (after " a v e r a g i n g " over the class) gives the multiplication table (7). We see that the sets F a n d A form tables that are mutually dual, just as an abelian group and its character group. Thus the description of the set A of irreducible representations of the group G and even their multiplicaTHE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991 71

tion table (7) is still connected with a certain commutative algebra (though not a group). We are dealing with only "semi-noncommutative" mathematics. Even so, when applying this to specific situations, analogous to those considered in Sections 4 and 5, substantial results are obtained along the way: cf. Sections 7 and 8. But first two remarks are in order. Remark 1. This situation (duality between the sets A and F) is analogous for compact groups as well. For compact Lie groups the sets of irreducible representations can be described by Hermann Weyl's formula for characters. The multiplication law in A can be derived from it. For example, for G = SU(2) or SO(3) it is the so-called Clebsch-Gordan formula: the representations % ~ A are given by an integer index n > 0 and for n >~ m, r

~

r

=

En_m~k~n+m"

k~--n+m (2) r

It seems one could describe the set F along these lines; a natural measure on it and a multiplication law in the form of an integral over it. As far as I know, this has not been done. Moreover, the description of a measure and multiplication law on F for the case of an interesting compact p-adic Lie group (the group of elements x in a division algebra over Qp, for which ,Y(x) = 1) is unknown, though set-theoretically F is easy to describe. This would be interesting in connection with the p-adic Langlands conjectures (cf. Section 8). Remark 2. The duality between the sets A and F can be defined for arbitrary "multiplication tables" ab = ~P(d,~ c

(9)

under the assumption there is a unit element e and inverse a* (p(ae,), > 0). If the classes C, are replaced by their "averages" zr = (1/[C,[)C, ([C,[ is the number of elements in the class Cz), then ~ k p~k) = 1, p~k) >I 0 for the corresponding table. The set {zI . . . . . Zh} may be considered as a "probability g r o u p " in which the product zrzl is the element z k with probability p~k). Such a reformulation is possible for any table (9) with p~(~ t> 0. Many classical systems of orthogonal polynomials form similar, though infinite, tables: {cos nx}, the Legendre, Laguerre, Chebyshev, Hermite, and Gegenbauer polynomials. For example, for the Legendre polynomials, P.Pm =

E

C~)m P~+m- Z~

n - m~k~n + m k=--m + n (2)

for m ~ n, where c(~ = ~ n + m + l

~.+m--k~.--,~+k~m--.+d~+m+k,

o.=?:] It is natural to ask in this connection whether the condition N.b ~> 0 implies an analogous condition for 7 '~ THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

the dual table. In the early 1950s, S. P. Demushkin (then a student) constructed simple examples for which this is not the case. It would be interesting to find a condition for the relation P(d,~~> 0 to be preserved in the dual table, as is the case for A and F. It would be even more interesting to understand along the w a y the duality of the two theorems "The number of elements in a conjugacy class [resp., the degree of an irreducible representation] divides the order of the group." Clearly, the question is reasonable for multiplication tables (9) with p(aC,~ ~ Z. It is probably connected with the operations C ~ C h, C h = {Elh} if C = {a} ("Adams operations").

The m o s t natural w a y to a "'noncommutative'" generalization of some theory is to replace numbers by matrices. 7. Vector bundles. In Section 4 we established a correspondence between a) the 1-dimensional unitary representations of the group HI(S;Z ) (or, equivalently, 9rl(S)), b) the dasses of degree 0 divisors on S (i.e., ~(S); cf. Section 4), and c) bundles over S with fiber C that are topologically trivial. We are now going to describe a generalization of this theory to arbitrary irreducible unitary representations of ~q(S). The most natural way to a "noncommutative" generalization of some theory is to replace numbers by matrices. So, in analogy with Section 4, an n-dimensional matrix divisor consists in giving, in r points s~. . . . . Sr, invertible matrices M1. . . . . M r ~ GL(n,K) with coefficients in the field K = C(S). Here the substitution M, - ~ M,P, is allowed, where the elements of the matrix P, belong to the ring r of functions that are regular at the point s, and (det P,)(sz) ~ O, i.e., Pz GL(n,~s, ). In analogy with Section 4, the set Divn(S) of n-dimensional matrix divisors m a y be written as Divn(S) = {M s : s ~ S, M s ~ GL(n,K), M s E GL(n,f?s) for almost all s} / {Ps ~ GL(n,f?s)}. Two divisors {Ms : s ( S} and {M's : s ( S} are said to be equivalent provided Mx = C M x, C = GL(n,K). The equivalent divisors form one class. Just as in Section 4, a bundle E over S with fiber C n is associated with the class of n-dimensional matrix divisors. If {Ms1. . . . . Msr} is a divisor in that class, then over sufficiently small neighborhoods U, ~ s,, E -~ U, x C", and over U, N U1, U, x C" and U1 x C" are glued using the matrix Ms, Ms~.1 The correspondence between analytical (or, equivalently, algebraic) vector bundles (up to isomorphism) and classes of matrix divisors is one-to-one. Finally, to any representation ~p : ,rrl(S) ~ GL(n,C) there corresponds a vector bundle E: if U is the universal cover of the curve S and S = U/G, G ~- "rrl(S), then E = (U x C)/,rrl(S), where "rrl(S) acts on C" via ~, and on U as does G. But this correspondence is neither

an embedding, nor a map onto. We shall first describe its image. By taking the determinant of each matrix M s we thus associate to each matrix divisor {Ms1. . . . . ~/sr}, and hence to each bundle E, an ordinary divisor, denoted det E. The number deg(det E), denoted deg E, is the Chern class Cl(E) of the bundle E. It is easy to see that deg E = 0 for the bundle E corresponding to the group "rrl(S). The converse is also true provided E is not decomposable into a direct sum of representations. In general, the condition is deg Ez = 0 if E = | is a direct sum decomposition into irreducible bundles.

It looks as though a particularly great future is in store for vector bundles of dimension greater than 1 in the theory of algebraic varieties of dimension greater than 1. We are interested, though, in irreducible unitary representations ~p : ~rl(S) ~ U(n) C GL(n,C). Two such give rise to isomorphic bundles only if they are equivalent. A bundle E (necessarily with deg E = 0) corre"sponding to an irreducible unitary representation is stable. We do not give their definition, though it is not complicated. Their importance lies in the fact that the set of stable bundles E of a given dimension n and with given value deg E = d has the structure of a finite-dimensional algebraic variety. For simplicity we consider bundles with fixed dass det E = 8. They form a variety S(n,8). As all varieties S(n,8) with deg 8 = d are isomorphic, one simply writes S(n,d). If (n,d) = 1, then the variety S(n,d) is smooth and projective. Its dimension is equal to (g - 1)(n 2 - 1), where g is the genus of the curve S. For example, for a genus 2 curve given by the equation y2 = li5(x _ Ks), the variety S(2,1) is determined in p5 by the equations Eo5 x 2 = 0, E05 k,x2 = 0. And the variety S(2,0) has the form p3 ~ V , where V is the so-called Kummer surface. It looks as though a particularly great future is in store for vector bundles of dimension greater than I in the theory of algebraic varieties of dimension greater than 1. While the set of 1-dimensional bundles ~(S) over a curve S basically determines S, on an algebraic surface the set of 1-dimensional bundles may be very poor. In many questions the bundles of dimension greater than 1 are the natural generalization of classes of divisors, reflecting the geometry of the variety. In mathematical physics, the vector bundles over S4 (considered as the compactification of 4-dimensional space-time) with a compact Lie group G as structure group, lead to a "nonabelian" generalization of the Maxwell equations: the Yang-Mills equations. The Maxwell equations are a special case with G = U(1). An analogous theory exists for manifolds other than S4

and gives a "nonabelian" analogue of Hodge theory (Section 1). An essential characteristic of this generalization is that the "nonabelianness" implies the nonlinearity of the corresponding differential equations.

8. The Langlands conjectures. This is a whole system of conjectures with the goal of constructing a generalization of class field theory (Section 5) analogous to the generalization of abelian functions (Section 4), described in Section 7. These conjectures are proved only in very special cases, but their plausibility is based on their mutual compatibility and relations with other important questions. It is simpler to describe the "local" version, which concerns extensions of a local field K (a finite extension of the field of p-adic numbers Qp). In this case the "division algebra" @n (this means that every element x @,, x # 0, has an inverse) with center K and rank n 2 over this field, plays the basic role. The general theory claims that the rank of any division algebra with center K is a square. Such algebras of rank n 2 correspond naturally to the ~(n) residues mod n that are prime to n. The algebra @n corresponds to the residue 1. The elements x s @,, x # 0, form a group @,*. In this case the conjecture claims that there is a one-to-one correspondence between the irreducible representations of the group Gal(K/K), where K is the algebraic closure of K, whose degree divides a given number n, and the irreducible representations of the group @n* with open subgroups of finite index as kernel. The conjectures are proved for (n,p) = 1, n = p. Here, I find it hard to refrain from a personal reminiscence. In 1944-45 1 was crazy about the conjecture that the group GaI(K/K)/,Y, where d~ denotes the intersection of the kernels of all irreducible representations of Gal(K/K) of degree dividing n, is isomorphic to a certain quotient group of the group @,*, and thought that this isomorphism would shed light on the mysterious conductor introduced by Artin. But__after having calculated sufficiently far the groups GaI(K/K) and @n*, I became convinced that there is no such isomorphism. If the Langlands conjectures are true (and there are many indications that is the case), then we have an interesting example of nonisomorphic groups with the same algebra A of irreducible representations (cf. Section 6). The set F of conjugacy classes of @n* is very simple: it is the set of irreducible polynomials f(t) K[t] with I as leading coefficient and degree dividing n. The calculation of the algebra F could turn out to be useful for clarifying the whole situation. There is another version of the same conjecture in which the same representations of the group GaI(K/K) are associated with irreducible unitary infinite-dimensional representations of the group GL(n,K). This is the version that has an analogue for the case K = Q. In this case, the basic role is played by the adele group GL(n,A) C GL(n,R) x lip GL(n, Qp), determined by the THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991 7 3

condition that in ot = oto~ x 11 Otp, OLoo ~ GL(n,R), Otp GL(n,Qv ), Otp ~ GL(n,Zp), for a l m o s t all p. Clearly GL(n,Q) C GL(n,A). The set GL(n,A) / GL(n, Q) is not a group but only a h o m o g e n e o u s space with respect to GL(n,F). The analogue of an "irreducible representation" for it is an irreducible unitary infinite-dimensional representation of the group GL(n,A), contained in the natural representation in the function space L2(GL(n,A)/GL(n,Q)). The precise formulation of the conjecture in this case w o u l d require too great a number of technical details. The conjecture extends to arbitrary finite extensions K of the field Q. Its interest increases, as it w o u l d give exact information on some very interesting arithmetic invariants of the field K, namely on its ~-series. 9. Nilextensions of number fields. In Sections 7 and 8 we saw h o w rich in contents the set of irreducible representations of the group "rrl(S), as well as its analogue Gal(K/K) for n u m b e r fields, is (and w e didn't even draw on the operation | in the set of irreducible representations). We h a v e n ' t yet left the confines of "semi-nonabelian" mathematics. What can be said, in the spirit of properly nonabelian mathematics, about the group "rrl(S) or GaI(-K/K)? As of now, something is k n o w n only about nilpotent quotients of these groups. As a finite nilpotent group is the product of finite pgroups, we will consider the union K(P) of all Galois extensions K'/K for which Gal(K'/K) is a p-group. Let first K = Qv' the field of p-adic numbers. If q p, then Gal(Q(pq)/Qp) has a simple description, which we omit. If p # 2, then Gal(Qp(p),Qp) has two generators and no relations (i.e., is a free p-group). Thus, the extensions K/Qp for which Gal(K/Qp) is a p-group are the same as the analogous unramified covers of the figure eight. The most curious is the group Gal(Q~2)/Q2): it has three generators x, y, z, connected by the relation X 2 y 4 ( y , z ) = 1, w h e r e (y,z) = yzy-lz -1. It w o u l d be interesting to have some geometric description of it. Analogous results are k n o w n for finite extensions of the field Qp. In the case of the field Q or a finite extension K of Q,

74 THE MATHEMATICAL INTELL1GENCERVOL 13, NO 1, 1991

it is natural to consider the extension K~), the union of the extensions K'/K for which Gal(K'/K) is a p-group and which are ramified only over the finite set X of simple divisors of the field K. The group Gal(K~)/K) is finitely generated, and sometimes its relations can be determined. It is k n o w n that Q~)= Q provided E is empty. If E (p) h as two generators x, y, and a = {2} then Ga 1(Qx/Q) single relation x 2 = 1. If E = {Pl,P2}, P # 2, p, -= 1 (mod p), p, ~ 1 (mod p2), i = 1,2, Pl ~ xP (modp2), then Gal(Q~)/Q) is a noncommutative group of order p3 and exponent p2. If K = Q ( p v ~ ) , the_prime n u m b e r p is regular and X = {'rr}, "rr = 1 - PV1, then Gal(K~)/K) is a free group with (p + 1)/2 generators. If E is empty, K = 0 ( % / - 5), then K~)= O(:k/-z5,X/-U-1). If X is e m p t y and the ideal class group of the field K has sufficiently m a n y generators compared to the degree [K:Q], then K~) is_ infinite f__orsome p. For example, for p -- 2, K = Q ( V - 3 9 7" 17. ~ ) . Such fields cannot be e m b e d d e d into fields with unique decomposition of divisors into prime factors. We see that there is often a beautiful description of the group Gal(K~)/K) via generators and relations, similar to, for example, the description of the f u n d a mental group of a 3-manifold. What is clearly lacking, though, is a specific description of these groups like the one using Jacobians and class field theory in the commutative case. 10. Iterated integrals. In the case of a function field C(S) on a Riemann surface S there is a beautiful approach to the study of coverings (we restrict ourselves to unramified ones) with nilpotent Galois group. As explained in Section 4, the theory of abelian covers is b a s e d on integration of differential forms; here this approach is connected with a "nonabelian" generalization of integration. Let 001. . . . . cor be 1-dimensional differential forms on S, and let ~/: (q~ : I ---> X, I = [0,1]) be a path. An iterated integral f~co1 . . " 00r is defined inductively: letting ~l(t) = f~0*co 1, ~2(t) = y~lq0~'OJ2 ..... w e define y ( D I " ' ' O r = fl~r_l~*O0 r. Such a construction arises, for example, w h e n constructing solutions of ordinary differential equations via the m e t h o d of successive approximations. Let [P denote the i-th tensor p o w e r of the space of 1-dimensional differential forms. An iterated integral f~ d e t e r m i n e s a linear f u n c t i o n on the space T = (~ : f~(.O 1 ~ ' ' " ~ COr = f ~ f ( O l ' ' ' ( O r. One can write d o w n explicitly a condition for f~co, for a given co, to d e p e n d only on the h o m o t o p y class of % For example, if o = W + E V~,| W E T 1, E V~, | E T2, then the condition is that d~ + Z ~,/~ W', = 0, E oqW', = E od,W, = 0, if d~, = oqfl, dw', = od,,Q, and fl is some standard 2-form on S. A tensor form 00 ( T satisfying these conditions determines a function on the group "rrx(S). There are 00 ~ T such that f~co = 0 for all ~/ ( ~rx(S): for example, for ~o = f dr. + V. | df, f ( T O, V- ~ TL

Such elements can also be described. As a result, we obtain an explicit description of a space of functions o n the group ~rl(S), a n d hence on its group ring C[~I(S)]. Let ~ = J" co be one of the functions that arise in this way, and let co ~ ( ~ , ~ r z~. It turns out that t h e n ~(x) = 0 for x ~ 1 (~r+l), where ~ C C['rrl(S)] is the ideal of elements of the form E o%% E o% = 0. Thus, ~(x) is a linear function on the algebra C[~rl(S)]/~ r+l as well as a function on the group F, = ']TI(S)/']TI(S)(r), where ~rl(S)(r) = {~/ E ~rl(S), ~/ -= 1 (o~r+l)}. The groups "rrl(S)(r) coincide w i t h the l o w e r central series for the g r o u p -rrl(S ) (because of t h e specific p r o p e r t i e s of these groups). The nilpotent group F r embeds in the algebra C['rrl(S)]/o~r+~ a n d has in it an "algebraic e n v e l o p e "

In the near future one can hope for a construction, if not of "'nonabelian mathematics" in general, then, as a first step, of "'nilpotent mathematics.'" which coincides w i t h its so-called Mal'tsev completion. T h i s is the analogue of associating A --> A | C for an abelian group A. The Mal'tsev completion of a g r o u p F will also be denoted by G = F | C; it is an algebraic group. Clearly, F C G(C) and one can identify F with G(Z). In the case of the group F r and Gr = F r | C, Gr C C[~rl(S)]/~ r+x and therefore, if gr denotes the algebra of the Lie group Gr, then gr C C[~rl(S)]/~ r+l (the Lie operation in C[~rl(S)/~ r+l is defined by [x,y] = xy - yx). For r = 1, G 1 = HI(X;Z ) | C = H 1 = (X,C). The system of groups Gr, r = 1, 2 . . . . . (and their discrete subgroups Gr(Z)), defines a tower of algebraic groups, which contains the description of the unramifled covers S' ~ S with nilpotent Galois group, and the iterated integrals play the same role as de Rham theory for r = 1. Up until now, we have talked about a purely topological theory. If S is a Riemann surface, this is reflected in the decomposition of each 1-form ~ E T~ as a s u m ~ = ~' + ~", where, locally, ~' = u'dz, ~" = u"d-~, z a local parameter. Similarly, a form co E Tr m a y be d e c o m p o s e d into h o m o g e n e o u s s u m m a n d s co = Y~ coP, w h e r e cop c o n t a i n s dz as a f a c t o r p t i m e s . The description we m e n t i o n e d earlier of the space C[~r1(S)]/o~r+l using iterated integrals makes it possible to extend this decomposition to that space. In particular, there is a subspace ~f C C[~rl(S)]/o~r+l that is the kernel of all functions fco, co ( O,_
a "nilpotent" generalization of the Jacobian of a manifold from Section 4. The integration J'~0 co, co ( O/~' defines a map S --> Gr(Z ) \ Gr / GOwhich generalizes the map j (cf. (2) in Section 4). The theory we have sketched is far from having a final character. The "nonabelian Jacobians" are, as a rule, not algebraic and not even K/ihler manifolds, a n d their natural maps are not always analytic, so that it is hard to even pose the question about their algebraic interpretation. Nevertheless, it is clear that there is here a substantial progress in the theory of nonabelian nilpotent covers. Combining the facts e x p o u n d e d in the last two sections leads us to the t h o u g h t that in the near future o n e can hope for a construction, if not of "nonabelian mathematics" in general, then, as a first step, of "nilpotent mathematics." To draw attention to this newly emerging field was the main stimulus for writing this survey.

Translated by Smilka Zdravkovska wzth the heartfelt gratitude of the author.

Bibliography In connection with vector spaces, group representations, a n d K u m m e r theory: 1. S. Lang, Algebra, Reading, MA: Addison-Wesley (1965). In a less algebraic spirit: 2. H. Weyl, Gruppentheorie und Quantenmechanzk, Leipzig: S. Hirzel (1928). The theory of abelian functions: 3. H. Wehyl, Die Idee der Rzemannzschen Fldchen, Leipzig: B. G. Teubner (1923). The algebraic aspect: 4. J. P. Serre, Groupes algdbraiques et corps des classes, Pans: Hermann (1959). Class field theory: 5. J. W. S. Cassels and A. Frohlich, Algebrazc number theory, New York: Academic Press (1967). Vector bundles, fundamental work: 6. A. Weil, G~n&ahsatwn des fonctzons abdliennes, collected papers, Vol. I, New York: Sprlnger-Verlag (1979). A survey of the current state: 7. S. Ramanan, Vector bundles on algebraic curves, Proc. Int. Cong. Math., Vol. II, pp. 543-547. Helsinki (1978). Langland's conjectures: 8. S. Gelbart, An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. (2)10 (1984), 177-219. Extensions of algebraic number fields: 9. H. Koch, Galoissche Theorze der p-Erwezterungen, Berlin. Spnnger-Verlag (1970). Iterated integrals, a survey: 10. P. Cartier, Seminaire Bourbakl, Exp. No. 687, Astdrisque, N161-162 (1989). I. R. Shafarevich Steklov Institute of Mathematics ul. Vavilova, GSP-1 117333 Moscow, USSR

Smilka Zdravkovska Mathematical Reviews 416 Fourth Street P.O. Box 8604 Ann Arbor, MI 48107 USA

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991 7 5

Chandler Davis*

Women and Mathematics: Balancing the E q u a t i o n edited by Susan F. Chipman, Lorelei R. Brush, and Donna M. Wilson

Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1985, xi + 381 pp.

Reviewed by Ann Hibner K o b l i t z In the past ten or fifteen years, much has been written about the fact that relatively few young women study mathematics b e y o n d m i n i m u m requirements, and that consequently women are much underrepresented in the mathematics and engineering professions. Mathematicians, educators, feminists, policy-makers, psychologists, and social theorists of various types have all tried to address the phenomenon in their own fashion. In this extensive literature, many ideological viewpoints and pedagogical schools are represented. 1 Consensus has not been reached even on whether intervention (by parents, teachers, counselors, etc.) is feasible, let alone whether it is necessary. And, of course, debates on the nature of intervention programs--at w h a t age they s h o u l d begin, w h o m t h e y should target, what strategies are most effective with girls and young women, and so on--continue to rage. W o m e n and Mathematics contains studies of individual programs, gender difference in mathematical achievement, differing self-perceptions of mathematical ability, the importance to mathematics of visualspatial skills, and the influence of sex-role stereotyping inside and outside the classroom, to name a few of the many topics addressed. In general, the authors concentrate on the junior- and senior-highschool level, although there is some information about course and career choices of older s t u d e n t s and women professionals. For the most part, the authors represented in this collection appear to have been trained in pedagogy, psychology, and social studies rather than mathe-

matics or specifically m a t h education, and their writing runs to jargon and opaqueness. Even the most carefully argued and (to my mind) interesting of the articles suffer from this. Mathematicians of both sexes are likely to find some of the verbiage taxing and some of the attempts at quantification dubious at best. (See, for example, Chapter 7 on "The Influence of Sex-Role Stereotyping on Women's Attitudes and Achievement in Mathematics," which presents a fair amount of annoyingly obvious observation in the form of graphs and tables with questionable statistical methodology.) M a n y of the situations described in W o m e n and Mathematics will not be surprising to those Intelligencer readers who are familiar with the Association for Women in Mathematics Newsletter or the newsletter of the International Organization of Women in Mathematics Education. Most of us already have at least some cognizance of factors militating against equal access to scientific careers for women; what might be slightly surprising are the conclusions reached by some of the authors in this collection. Take the wellk n o w n fact that adolescent girls (and often their parents and teachers) tend to perceive mathematics as " u n f e m i n i n e . ' " This m a y make y o u n g w o m e n of proven mathematical ability opt out of taking advanced mathematics courses rather early, thereby automatically cutting themselves off from various mathrelated career alternatives (see, for example, chapters by Jane M. Armstrong, Patricia L. Casserly a n d Donald Rock, and Jacquelynne Eccles, et al.). Curiously, however, not all commentators feel that the 1 See, for example, C. P. Benbowand J. Stanley,"Sex differencesm mathematical ablhty: Fact or artifact?" Sczence 210 (1980), 1262-1264; L Brush, Encouraging gzrls m mathematics, Cambridge. ABT Books (1980); E. Fennema, "Mathematicslearning and the sexes A review," Journal for Research m Mathematzcs Educatwn 5 (1974), 126-139; L. H Fox, L Brody, D Tobln, Women and the Matheraat~cal Mystique, Baltimore Johns Hopkins UniversityPress (1980); A. S. Rossl, "Womenin science Why so few~"Science 148 (1965), 1196-1202; S Tobias, Overcoming math anx2ety, New York. Norton (1978); plus discussionsm The Arithmetic Teacher, The Mathemat~cs Teacher, and the Association for Women m Mathematics Newsletter.

* Column editor's address: MathematicsDepartment, Universityof Toronto, Toronto, Ontario MSS IAI Canada

For a critique of mnatlstexplanabonsof women'slowerachievement in the sciences, see Anne Fausto-Sterlmg,Myths of Gender, New York' BasicBooks(1985).

THE MATHEMATICAL INTELL1GENCER VOL 13, NO 1 9 1991 Spnnger-Verlag New York

77

s e x - s t e r e o t y p i n g of m a t h e m a t i c s as m a l e w o r k s against y o u n g women: [P]erceiving math as very useful for males does not necessarily have a negative consequence for girls, perhaps especially when the stereotype reflects an awareness of the high status jobs that are both male-dominated and math-related. In this case, it may be the status of the job rather than its male domination that elevates the perceived usefulness of advanced math courses for both high-ability boys and girls. 2 This is just one example of the intriguing and possibly counter-intuitive conclusions suggested by the contributors to Women and Mathematics. Some of the information presented in Women and Mathematics is probably out of date. Virtually all of the studies used were done before 1982, and there have been considerable changes in our school systems since then, as well as considerable shifts in societal perceptions of w o m e n ' s roles and abilities. Not all of this shift has been in a positive direction, unfortunately. Today's media blare out a n n o u n c e m e n t s that we live in a "post-feminist age." Meanwhile, television ads e x h o r t little girls to b u y s t e r e o t y p i c a l l y f e m i n i n e "Pretty Ponies" and "Miss Lees Press-On Nails," and their mothers are finding themselves relegated to the "Mommie Track" in their careers. I would be curious to see a similar collection of more recent intervention programs, with longitudinal studies tracing the subsequent career patterns of early participants in such programs. Most disturbing for me, however, are the disregard of social and political context by m a n y of the authors a n d their obliviousness to possible methodological problems. Several of the studies presented contradict one another, even on such presumably basic points as the value of role models, the extent of discrimination against y o u n g w o m e n in the classroom, the possibility of tutoring to improve performance on tests of visualspatial skills, a n d the i m p a c t of the a t t i t u d e s of parents, teachers, guidance counselors, and peers on decisions to continue studying mathematics. Only one chapter in the book, the well-constructed overview of Susan F. C h i p m a n and Donna M. Wilson, attempts to come to grips with the uncertainties inherent in attitudinal surveys and classroom observation (see, for example, p. 295). I n d e e d , after reading their cautions about the subjective nature of surveys and the virtual impossibility of accurately quantifying or even consistently defining societal variables, one can reasonably have doubts about the essential validity of the studies cited in the book. Women and Mathematics is not a source of definitive commentary on the question of equal access to mathematical careers; I do not k n o w that there is such a source. Nor is it a manual for action-oriented professionals with a c o m m i t m e n t to social justice. It is, if anything, apt to leave you more confused about the 78 THE MATHEMATICALINTELLIGENCERVOL 13, NO 1, 1991

possibilities for change than you were before. There are, however, interesting data in several of the articles, a n d the book raises s o m e valid questions. Perhaps the foremost lesson to be learned from Women and Mathematics is that mathematicians need to concern themselves with h o w their field is being presented in primary and secondary school to potential mathematicians of both sexes, but especially to girls. According to Casserly and Rock, successful programs are those with enthusiastic, professionally committed teachers w h o convey the beauty and elegance of mathematics as well as its utility: Just as the teachers knew where their students' various abilities might lead them, so too they knew to put mathematics in a full array of contexts for all their students. Its innate beauty, its connections with artistic and aesthetic endeavors, as well as its utility in a broad array of other fields, were all brought into the classroom. This strategy was particularly beneficial to girls. For In this study girls were found to have a greater variety of reasons for and satisfactions from studying mathematics than did boys. They also seem to have broader or less channeled interests. These mathematics teachers felt that, while girls certainly need to be as aware as boys of the utility of mathematics in other fields, to emphasize only its utility is to sell short both the students and the field. 3 U n f o r t u n a t e l y , j u d g i n g f r o m the i n f o r m a t i o n presented in Women and Mathematics, such programs and teachers are rare.

History Department Hartwzck College Oneonta, NY 13820 USA

2 j. Eccles (Parsons) et a l , "Self-Perceptions, Task Perceptions, Soaalizlng Influences, and the Decision to Enroll m Mathematics,'" Women and Mathematzcs p. 106 3 p Casserly and D. Rock, "Factors Related to Young Women's Perslstence and Aduevement in Advanced Placement Mathemabcs," Women and Mathematzcs p. 238

Robin Wilson*

Impossible Figures Impossible figures have proved to be irresistibly fascinating to artists and mathematicians alike. In 1934, the Swedish artist Oscar Reutersv/ird drew the first impossible triangle, an arrangement of nine cubes. Three of his impossible figures were later featured in a set of Swedish stamps, issued in 1982 to commemorate his work. In the 1950s, the Dutch graphic artist Maurits Escher began to incorporate impossible objects into his lithographs, and shortly afterwards a seminal article on their construction and features was published by Lionel and Roger Penrose.

An impossible triangle is featured on the Israeli stamp, issued in 1973 to commemorate the jubilee of the Technion in Haifa. An impossible cube appears on the Austrian stamp, issued to celebrate the tenth International Austrian Mathematical Congress in 1981.

If you are interested in mathematical stamps, you are invited to subscribe to Philamath. Details may be obtained from the secretary, Estelle A. Buccino, 135 Witherspoon Court, Athens, GA 30606 USA.

* C o l u m n editor's address: Faculty of M a t h e m a t i c s , T h e O p e n Uruverslty, Milton K e y n e s MK7 6 A A E n g l a n d 80 THE MATHEMATICALINTELLIGENCERVOL 13, NO 1 9 1991Spnnger-VerlagNew York