The Ignorance of Bourbaki* A. R. D. Mathias
If one looks at the history of mathematics, one sees periods of bursting creativity, when new ideas are being developed in a competitive and therefore very hasty spirit, and periods when people find that the ideas so recently in vogue are inexact, incoherent, and possibly inconsistent. In the latter periods there is an urge to consolidate past achievements. I said "the history of mathematics," but mathematics is a complex sociological organism, and its growth takes place in different branches and in different countries, even different universities, in different ways and at different speeds. Sometimes national groups feel that mathematics in their country is in a bad way: You find an expression of that in the "Introduction" to later editions of Hardy's Pure Mathematics, where he remarks that it was written with an enthusiasm intended to combat the insularity of British mathematics of the turn of the century, which had taken no account of the development of mathematics in France in the 19th century. Indeed, in 1910 France could be proud of her succession of mathematicians such as Legendre, Laplace, Lagrange, Fourier, Cauchy, Galois, Hadamard, Poincar4--a most impressive list of scholars of the highest distinction. But after the first World War, the feeling in France changed, and the young French mathematicians of the day began to consider that the torch of mathematical
research had passed to G e r m a n y - - w h e r e there were many great mathematicians building on the past work of Riemann, Frobenius, Dedekind, Kummer, Kronecker, Minkowski, and Cantor, such as Klein, Hilbert, Weyl, Artin, Noether, Landau, and Hausdorff and that French mathematics had gone into a decline.
* A paper read to an undergraduate mathematical society, the Quintics, in Cambridge on October 29th, 1986, and published in the Cambridge undergraduate magazine Eureka in 1987. This revision has been made in the light of helpful criticisms from Sir Peter Swinnerton-Dyer, Bart, Professor Saunders MacLane, Dr. Francisco CoreUa, Dr. Paolo Mancosu, Dr. G4rard Bricogne, and many others. 4 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3 9 1992SpringerVeflagNew York
So in 1935, a group of young French mathematicians 1 resolved to restore discipline to their subject by writing a series of textbooks, under the joint pseudonym of Nicolas Bourbaki, that aimed to give definitive expositions with full French rigour to w h a t they deemed to be the most important areas of pure mathematicis. Now the question of mathematical rigour was very topical, following the discovery by Russell at the beginning of the twentieth century of a major flaw in Frege's proposed theory of classes. Frege wanted to form for any property ~(y) the class {yl~(y)} of all objects y with the property ~, and at the same time to count all such classes as objects to which such membership tests might be applied. If we write "a ~ b" for "a is a member of b" and "a ~ b" f o r " a is not a member of b," we may express Frege's broad principle as follows. Denote {yl~(y)} by C: then for any object a, aE C if and only if cI)(a). Russell, developing an idea of Cantor, noticed that if ~(y) is taken to be the property y ~ y, of not being a member of oneself, then a contradiction results. Let B be the class of those objects that are not members of themselves; in symbols, B = {YIY ~ Y}: then for any y, y C B iff y ~ y; and so for the particular case when y is B, B E B iff B ~ B. In response to this, there were some w h o wished to ditch all the more speculative areas of mathematics, which made use of the infinite and particularly of Cantor's theories of cardinals and ordinals. Kronecker, Poincar6, Brouwer, and Hermann Weyl should be mentioned here. But there w e r e o t h e r s - - n o t a b l y H i l b e r t - - w h o wished to resist this wholesale amputation, and a programme was proposed aimed at formalising mathematics--the language, the axioms, the modes of reasoning, etc.--and at proving, by means the soundness of which could not possibly be doubted, that the resulting system was free of contradiction, that is, was
consistent. I said "formalise mathematics," but that is vague: how much mathematics can we or should we include? Hilbert certainly would wish to keep Cantor's work on ordinals in his formalisation of mathematics, as it was Cantor who made Hilbert possible. Hilbert leapt to fame with his Basis Theorem, that in m o d e m terminology asserts that if every ideal in the commutative ring R is finitely generated, the same is true of the ring R[X 1. . . . Xn] of polynomials in the indeterminates X 1. . . . . Xn with coefficients in R; and recent studies have shown that the proof of this theorem not only
1 Listed by Chevalley in an interview [M7] as H. Cartan, C. Chevalley, J. Delsarte, J. DieudonnG Sz. Mandelbrojt, R. de Possel, and Andr6 Weil. In a letter cited in the biography [H3] of Cavaill~s by his sister, a further mathematician, Ch. Ehresmann, is mentioned as belonging to the group.
relies on but is in an exact sense 2 equivalent to the wellfoundedness of the order-type co% Thus w h e n Hilbert spoke of Cantor's paradise, it was no idle tribute: he acknowledged the creation of a conceptual framework of transfinite induction within which algebraic geometry could advance. Russell's own ideas on avoiding the paradoxes led to his ramified theory of types; this was cumbersome, and a simpler system was proposed by Zermelo in the first decade of the century. Fraenkel and Skolem in the third decade proposed the axiom of replacement as a strengthening of Zermelo's system; the resulting system is known as Zermelo-Fraenkel. With the addition of the Axiom of Choice, first articulated by Zermelo and of great importance in functional analysis and higher algebra, and the Axiom of Foundation, proposed by von Neumann, ZFC has proved a very serviceable system. 3 There are two elements to Hilbert's programme: the creative side, proposing a system within which to work; and the critical side, testing the adequacy and consistency of the system proposed. Naturally the Bourbaki group, or Bourbachistes, mindful of the possibility of contradiction in mathematics, were determined that their textbooks would be free of such problems, and indeed an early volume in their series, La Th~orie des Ensembles, was devoted to establishing the foundations necessary for the later ones. The other day, I thought I would read it. I was shocked to the core: It appeared to be the work of someone who had read Grundzfige der Mathematik by Hilbert and Ackermann, and Lemons sur les nombres transfinis by Sierpir~ski, which were both published in 1928, but nothing since. Puzzled both by Bourbaki's attitude to foundations and by his attitude to set theory, I started to probe the background and found that the Bourbachistes had published several articles in the 1930s and 1940s expounding the group's position on foundational issues. Henri Cartan and Jean Dieudonn6 wrote essays under their own names on the foundations of mathematics. After the second World War, Nicolas Bourbaki himself addressed the Association for Symbolic Logic in America, and his talk was printed in the Journal of Symbolic Logic. Further, he wrote an essay on "L'Architecture des math6matiques,'" which was translated into English and a p p e a r e d in the American Mathematical
Monthly. There is a uniformity to these essays: on the creative side, the set theory they propose is that of Zermelo--not, let me emphasize, Zermelo-Fraenkel--and they declare it to be adequate for all of mathematics; and on
2 See [M21]. 3 Zermeloincluded the Axiomof Choice in his list of axiomsin 1908. The present custom is to mention that axiom explicitlyas an extra. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 5
Self-Respect
John de PiUis
the critical side, they all show the influence of Hilbert's formalist programme. None of them mentions G6del. In view of their commitment to Hilbert's programme, that is remarkable; and some comment on the Incompleteness Theorems is in order. There was a meeting at K6nigsberg in September 1930, during which honorary citizenship was conferred on Hilbert, w h o had retired from his Chair at G6ttingen on January 23rd of that year. The famous and powerful address, Naturerkennen und Logik, that he gave on this h a p p y occasion is informed by his credo that there are no insoluble problems 4 and ends with his resolute battlecry Wir miissen wissen; wir werden wissen. - - w e must know, we shall know. With the delicate irony of history, G6del had the very day before, with von Neumann but not Hilbert s in the audience, announced his incompleteness proof, with its applications to any system such as Peano arithmetic or Zermelo-Fraenkel. 6 4 The penultimate sentence of Hilbert's address [M13] runs "'Der wahre Grund, warum es Comte nicht gelang, ein unl6sbares Problem zu finden, besteht meiner Meinung nach darin, class es ein unl6sbares Problem ~iberhaupt nicht gibt.'" 5 who presumably was preparing his talk for the morrow. 6 G6del's announcement at K6nigsberg was followed by the communication of an abstract to the Vienna Academy on October 23rd 1930, and the receipt on November 17th 1930 of the text of his paper for publication. 6 THE MATHEMATICALINTELLIGENCERVOL.14, NO. 3, 1992
One might expect this to have caused a sensation. Hilbert had presented a very positive response to the paradoxes, and disciples such as Herbrand had, in the Hilbertian spirit, established cases of the decision problem. G6del showed that there were serious limitations to Hilbert's proposal. 7 He showed that no system satisfying certain minimal conditions, such as the clearly desirable requirement that there should be an algorithm telling you of any sentence whether it is one of the axioms or not--no system of this kind captures all of mathematics, and proofs of the consistency of such a system can only be given in systems more likely to be inconsistent than the one under discussion. Given the importance of this result for foundational studies, and given the eager response of von Neumann and others to G6del's ideas, it is natural to ask what effect G6del had on the Bourbachistes; and the strange thing is that one searches their publications in vain for mention of his name. One might almost say that they ignored him, except that the tone of certain of their works suggests a conflict between an uneasy awareness that something has happened and a desire to pretend that it has not. It is as though they had discovered that they were on an island with a dragon and in response chose to believe that if the dragon were given no name it would not exist. For instance, Henri Cartan, in a piece entitled "Sur le fondement logique des math6matiques ''8 presents the system of Zermelo, including the Axiom of Choice. Though he says he takes some account of the modifications introduced by Fraenkel, he does not include the main one, the axiom of replacement. He comments that Zermelo's system is inconvenient, lacking as it does suitable definitions of ordered pair, etc., and he reveals ignorance of the distinctions that G 6 d e l stressed by saying "true" where he means "provable," "false" where he means "refutable" and "doubtful" where he means "undecided." Cartan talks of contradictory theories, and says the problem of deciding whether a given theory is contradictory leads to the Entscheidungsproblem, which consists of finding a general method for deciding whether a given relation (i.e., formula) is a logical identity (i.e., theorem). This problem, he says, has only been resolved in particular cases. In general one does not know h o w to do it. He then says, "But these problems, important though they be, are outside our subject." He mentions Herbrand's thesis, Sierpii~ski's "Lemons sur les nombres transfinis," and adopts a view he credits to DieudonnG mentioning that these ideas,
7 For recent appraisals of Hilbert's programme, see, e.g., [Mll], [M18], and [M20]. 8 [M5]: the manuscript was received on January 15, 1942 and published in 1943.
though published in 1939, "remontent a 1938"; and makes this statement: une th~orie mathdmatique n'est pas autre chose qu'une thdorie logique, determin~e par un syst~me d' axiomes . . . les ~tres de la th~orie sont definis ipso facto par le syst~me d'axiomes, qui engendre en quelque sorte le materiel auquel vont pouvoir s'appliquer les propositions vraies; ddfinir ces ftres, les nommer, leur appliquer les propositions et relations, c'est en cela que consiste la partie proprement math~matique de la thdorie logique. 9
He mentions Cantor, Kronecker, Zermelo, Brouwer, Skolem's paradox, PoincarG and Lebesgue, but not G6del!
Clearly Cartan was thinking about foundational questions. Why then does he not mention G6del's results? Among the French speakers I have been able to consult, there is some disagreement, turning on the meaning in 1942 of the phrase est tout iddal, as to whether Cartan's article reveals an awareness of the Incompleteness results and a desire to communicate this awareness, which one presumes he must have possessed, to the reader. The passage in question reads thus: Le probl~me de ddcider si une proposition donnde est vraie dans une thdorie se ram~ne ~ celui-ci: une relation donn~e est-elle une identit~ logique? De m~me pour le probl~me de ddcider si une thdorie est ou n"est pas contradictoire. Ces probl~mes se ram~nent donc, en d~finitive, dz l'Entscheidungsproblem, qui consiste trouver une mdthode gdndrale permettant de ddcider si une relation, explicitement donn~e, est ou n' est pas une identitd logique. Ce probl~'me n'est r~solu que dans des cas particuliers. De sorte que, jusqu'd nouvel ordre, le partage en trois categories dont nous venons de parler (propositions vraies, propositions fausses, propositions douteuses) est tout ideal: dans une thdorie dont on saurait qu'elle n'est pas contradictoire, il y a des propositions dont on a prouv~ qu"elles sont vraies, d"autres dont on a prouv~ qu'elles sont fausses (les n~gations des prdc~dentes), d"autres dont on ignore ?~la fois si elles sont vraies ou si elles sont fausses. Et encore, gdn&alement, ne saura-t-on m&ne pas prouver qu' une th~orie donnde n"est pas contradictoire.
Similarly equivocal attitudes are to be found in the 1939 piece, cited by Cartan, by Jean Dieudonn6: "Les m6thodes axiomatiques modernes et les fondements des math6mafiques." He describes the achievements of Cantor, which Hilbert had found so useful, as "'resultats si choquants pour le bon sens! ''1~ He regards the foundational crisis of the beginning of the century as having been resolved by Hilbert's formalist doctrine that the correctness of a piece of mathematics is a question of its following certain rules, and not a question of its interpretation. He comments that
9 "A mathematical theory is simply a logical theory determined by a system of axioms. The entities of the theory are defined ipso facto by the system of axioms, which generates in some way the material to which true propositions may be applied; the mathematical part proper of the logical theory consists of defining these entities, naming them, and applying propositions and formulas to t h e m . " 1o [M8]: " a n affront to common sense!"
le principal m&ite de la mdthode formaliste sera d'avoir dissip~ ddfinitivement les obscurite~ qui pesaient encore sur la pensde mathdmatique ;11
and says that I1 reste naturellement ?~montrer que la conception de Hilbert est rdalisable. 12
Again, he makes no mention of G6del. Dieudonn6 does, however, hint at a sceptical awareness of G6del's results in these words: En outre, il semble, d'apr~s les travaux les plus r~cents, que, contrairement ~ ce que croyait Hilbert, les r~gles qu'il serait ndcessaire d' adopter en m~tamathdmatique, pour aboutir ~ une ddmonstration de la non-contradiction des mathdmatiques, seraient d"un degr~ d' abstraction aussi ~levd que les r~gles mathdmatiques elles-m~mes, ce qui amoindrit encore la portde que pourrait avoir une telle 'ddmonstration'.13
He confirms this awareness a few years later in his obituary of Hilbert, but still cannot bring himself to mention the dreaded name: Il semble que l'intuition de Hilbert l"ait, pour une fois, entra~n~ des espoirs quelque peu exagdrds, et on a actuellement de bonnes raisons de douter de la possibilit~ de telles 'ddmonstrations'.14
Nicolas Bourbaki, 15 in "The foundations of mathematics for the working 16 mathematician," again presents Zermelo set theory plus the Axiom of Choice, and concludes: On these foundations, I state that I can build up the whole of the mathematics of the present day; and if there is anything original in my procedure, it lies solely in the fact that, instead of being content with such a statement, I proceed to prove it in the same way as Diogenes proved the existence of motion; and my proof will become more and more complete as my treatise grows. As you might by now expect, there is no mention, or even hint in that paper of the existence of G6del's work which in 1948 had been in print for seventeen years. In Bourbaki's other essay, "L'architecture des math~matiques," there is again no mention of G6del, but this time there is a hint of "difficulties." The questions I now want to address are: 11 "The principal merit of the formalist approach will be to have definitively dispelled the obscurities that still clouded mathematical thought.'" 12 "It remains to be proved, naturally, that Hilbert's conception can be realised." 13 "It appears according to very recent work that, contrary to what Hilbert believed, the metamathematical rules that it would be necessary to adopt in order to prove the consistency of mathematics would be of as high a degree of abstraction as the mathematical rules themselves, which much reduces the usefulness or significance of such a proof." 14 [M9]: "It seems that Hilbert's intuition had, for once, led him to slightly exaggerated hopes, and there are today good reasons for doubting the possibility of such [consistency] 'proofs'." 15 See [M4] and [M2] or its translation, [M3]. 16 Is this the first occurrence in history of this odious phrase? THE MATHEMATICAL1NTELLIGENCERVOL. 14, NO. 3, 1992 7
W h y did Bourbaki make no m e n t i o n of G6del?
and W h y did Bourbaki n o t notice that his system of Zermelo set theory w i t h A C w a s inadequate for existing mathematics?
I think these questions are important because the Bourbaki group have had great influence9 I do not dispute the positive worth of their books, nor the magnitude of their achievement; but I suggest that their attitude to logic and to set theory, which has been passed on to y o u n g e r generations of mathematicians, 17 is harmful because it excludes awareness of perceptions of the nature of mathematics that are invigorating; and I almost venture to suggest that if, as some say, Bourbaki is now dead, he was killed by the sterility of his own attitudes. Before attempting necessarily speculative answers to these questions, let us probe a little further the comments of the Bourbachistes on these matters. Bourbaki in "L'architecture des math6matiques" distinguishes carefully between logical formalism, which he is against, and the axiomatic method, of which he approves: What the axiomatic method sets as its essential aim, is exactly that which logical formalism by itself cannot supply, namely the profound intelligibility of mathematics9 So by the axiomatic method, he means not a grand deductive scheme for all of mathematics, but simply the mental discipline of pruning areas to their skeletons, to make similarities clear and theory portable. The unity which [the axiomatic method] gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton9 Many mathematicians have been unwilling to see in axiomatics anything else than futile logical hairsplitting not capable of fructifying any theory whatsoever. Nothing is farther from the axiomatic method than a static conception of the science9We do not want to lead the reader to think that we claim to have traced out a definitive state of the science9 It is quite possible that the future development of mathematics may increase the number of fundamental structures, revealing the fruitfulness of new axioms or of new combinations of axioms9
Mais, si la logique est l'hygi~ne du mathdmaticien, ce n"est pas elle qui lui fournit sa nourriture; le pain quotidien dont il vit, ce sont les grands probl~mes.19
Thus betraying a belief that there are no great problems in logic9 He does, though without mentioning G6del, go on to suggest an awareness that the last word on logic might not have been said: Il se peut sans doute qu'un jour nos successeurs ddsirent introduire en th~orie des ensembles des modes de raisonnement que nous ne nous permettons pas.2~
This vital view, which is reminiscent of the last paragraph, quoted from Bourbaki above, is to be contrasted with the later ossification expressed by Dieudonn6 in his Panorama of Pure Mathematics 21 that "Set theory is well worked out." Bourbaki's general approach is stated quite clearly in his manifesto: The organizing principle will be the concept of a hierarchy of structures, going from the simple to the complex, from the general to the particular9 9. . the theory of groups . . . . the theory of ordered sets (including wellorderings) . . . . the theory of topological structures... but it should be noticed in passing that amid these unobjectionable statements is one which without further comment might mislead: The first axiomatic treatments (Dedekind-Peano arithmetic, Hilbert-Euclid geometry) dealt with univalent theories, i.e., theories which are entirely determined by their complete system of axioms, unlike the theory of groups. It is true that Euclidean geometry both of two and of three dimensions as axiomatised by Hilbert are completely determined, so that a statement of plane geometry provable by use of solid geometry will have a proof in plane geometry; but G6del tells us that arithmetic, as axiomatised by Peano or anyone else, is not; nor, curiously, is projective geometry of two dimensions, though it becomes so on the addition, as a single further axiom, of the statement of Desargues's theorem. 22 In saying that Peano arithmetic is univalent, Bourbaki probably has in mind some second-order characterisation of the standard model of arithmetic, which is, of course, to beg the question. My reading of all these extracts is that Bourbaki had
Andr6 Weil puts the Bourbachist view of logic as the grammar of mathematics more diplomatically:TM
17 Lectures on set theory given to undergraduates of an ancient University in 1988 by a disciple of Bourbaki contained errors, in the form of false proofs of non-theorems, of which the spiritual ancestry may be traced to the Bourbachiste stance of forty-six years previously. 18 in "L'Avenir des math~matiques" [M24]. 8 THE MATHEMATICALINTELL1GENCERVOL.14, NO. 3, 1992
19 "If logic is the hygiene of the mathematician, it is not his source of food: it is the great [mathematical] problems that form his daily bread." 2o "It may well be that one day our successors will want to introduce into set theory modes of reasoning that we do not permit." 21 [M10]: the work that omits the name of Shelah from a list of leading contributors to model theory. Shelah's first two books and first 322 papers are conveniently listed on pages 398--418 of [F1]. 22 For a thorough treatment of these points, see [M1].
grasped the positive worth of the work of Hilbert and his school, and welcomed the idea of the reduction of the question of correctness of mathematics to a set of rules, but nevertheless persisted, even after G6del's work showed that Hilbert's programme could never be completed, in thinking of logic and set theory as stuff one settled in Volume One and then forgot about. The later editions of Bourbaki's books shift ground so far as to mention G6del, talk of the independence results, and give the axiom of replacement. But the pre-G6delian attitudes, perception of which started me on this investigation, survive. Thus it appears that this major exposition of mathematics is written by people whose understanding of foundational work is that of 1929. Let me now attack m y first question:
Why did the Bourbachistes not adapt their attitudes to take account of the supremely important contribution of G6del to foundational issues? Why did the foundational understanding of Bourbaki not advance as foundational studies advanced? Answers may be sought at several levels, sociological, psychological, or mathematical. There may, for example, be a nationalist element in Bourbaki's posture. Compare it to Alexander Koyr6's view 23 that among the reasons why Hegel was ignored in France for a hundred years were the obscurity of Hegel's writing, the strength of Cartesian and Kantian philosophical traditions, Hegel's Protestantism, but above all the incredulity of the French towards Hegel's 'strict identity of logical synthesis and historical becoming.' For French rationalists, history was separate from reason or logic, which was eternal, outside time. Examples of intellectual chauvinism are as readily found in France as elsewhere and would include the century-long resistance of the University of Paris to the ideas of Paracelsus, 24 and the resistance, under the influence of Descartes, to Leibniz's ideas concerning infinitesimals. 25 There were, though, in the late 1930s French scholars who were well acquainted with, and actively disseminating, G6del's work. See, for example, Albert Lautman's monograph "Les sch6mas de gen6se," and those by Jean Cavaill6s entitled "La probl6me du fondement des math6matiques" and "La non-contradiction de l'arithm6tique, ''26 so that any nationalist el-
2a See [H4] and [H6]. 24 Documented with all his customary relish in intellectual tussle by the evergreen Lord Dacre of Glanton[H2]. 2s This dispute, though, was settled comparatively quickly: see Mancosu [HS] 26I am grateful to Dr. Mancosu of Wolfson College, Oxford, for putting me on the track of [M6] and [M14].
ement in the anti-G6delian stance may perhaps be local to the Bourbaki group. Interestingly, Cavaill6s sees the year 1929 as marking the transition between two periods, which he calls the naive and the critical, in the development of modern logic. At a psychological level there is an unwillingness of the Bourbachistes to move from the naive conception of logic with which they had grown up; an unwillingness with which not a few European mathematicians are imbued. The Bourbachistes' attitude to logic may derive from Poincar6's mocking attitude to the work of Cantor and Russell; for although in his Last Essays Poincar6 moved towards an understanding of his opponents---in an address, The Moral Alliance, delivered three weeks before his death, he advocated mutual respect among those who, with different ideas and methods, pursue a common ideal these conciliatory gestures may not have undone the harm caused by his earlier sardonic, savagely funny, but ultimately u n s o u n d critical writings on logic. In his "Preface" to the 1968 French edition of Herbrand's ~crits Logiques, van Heijenoort, commenting on the sad state of logic in France, remarks that the harm done by Poincar6 was compounded by the early deaths of many French logicians such as Couturat, killed by a lorry in the mobilisation of 1914, Nicod, who died of tuberculosis in 1924 aged 31, Herbrand, killed mountaineering in 1931 aged 23, and Cavaill6s and Lautman, who were shot, aged 41 and 36, by the Germans in 1944, for their part in the Resistance. These last losses are part of a wider phenomenon: European logicians escaping from Hitler started schools of logic in the United States and in Israel which have flourished, leaving Europe behind. 27 It may be that the Bourbachistes were led astray by Hilbert, whose commitment to his programme made it, at first, very hard for him to accept G6del's work, but as he recovered from the shock more rapidly than his much younger French disciples, some further explanation of their behaviour becomes necessary. It may be that, like many another scientist, they were prevented by their preconceptions from seeing the significance of facts that were k n o w n to them. My guess would be that at whatever level of their psyche the Bourbachistes were disabled, they were not ready to face the possibility, strongly suggested by G6del's work, that there are no foundations of mathematics in the sense proposed by Hilbert and embraced by Bourbaki; that there are no ways of grounding mathematics in logic or classes or whatever so that once a
27 For example, whereas students in four years in Cambridge might hear fifty lectures on logical topics, at Harvard or Princeton they may hear around two hundred and fifty, and at Berkeley, where logic is taken seriously, they may hear about four hundred. THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3, 1992 9
basis has thus b e e n given in some primitive ideas, no further t h o u g h t n e e d be given to them; that t h o u g h there are i n d e e d foundational issues, t h e y cannot be confined to C h a p t e r O n e of the Great Book, for they permeate mathematics. The second q u e s t i o n I p u t above was:
Why did the Bourbaki group not notice the inadequacy of their chosen set theory as a foundation for mathematics? I suggest as a n a n s w e r that they w e r e solely interested in areas of mathematics for w h i c h Zermelo is adequate, and that this area m a y b r o a d l y be described as g e o m e t r y as o p p o s e d to arithmetic. 2s Leibniz w r o t e that there are two f a m o u s labyrinths in which our r e a s o n is often lost. O n e is the problem of f r e e d o m a n d necessity, and the o t h e r is c o n c e r n e d with continuity a n d infinity. Defying this second danger, I wish to explore w h a t I believe to be the underlying dualism of mathematics, n a m e l y the tension bet w e e n these two primitive intuitions, the arithmetical and the geometrical. 29 This tension m a y be amusingly illustrated b y the following c o n u n d r u m : Can y o u describe a spiral staircase w i t h o u t m o v i n g y o u r hands? That question is difficult, perhaps, because words are temporal, h e n c e arithmetical; spirals are spatial. The source of the difficulty m a y be physiological in that there is a m o u n t i n g b o d y of medical evidence 3~ that normally the left half of the brain h a n d l e s temporal concepts while the right half handles spatial ones. 31
28 Dr. Bricogne challenges my suggestion in view of the brilliantly successful cross-pollination between geometry and arithmetic found in the work of Andr4 Well, J. P. Serre, and others; nevertheless he shares Bourbaki's regrettable attitude that "it is unlikely that foundational questions might impinge directly on these areas of mathematics," and I wonder why persons so responsive to one form of cross-poUination should be so resistant to another. For recent instances of problems in algebraic geometry being solved by techniques from logic, see [F2] [F3] and [F5]. 29 Dr. Mancosu draws my attention to Chapter IV, "G4om6trisme cart~sien et arithm6tisme leibnizien" of Belaval's book [H1], in which this dualism is used to interpret the opposition between Descartes and Leibniz. 3o See [P1], [P2], [P3]. I am grateful to John Davis, Professor Emeritus of P~ediatrics at Cambridge, for drawing my attention to this research. 31 Indeed, a friendly critic of an earlier draft of this article writes: "Which half of his brains did Bourbaki use? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the right-brain mathematics of the Italian geometers: significant portions were suspect and might, if one takes "true' and 'false' to be left-brain notions and 'fight' and 'wrong' to be rightbrain ones, be justifiably described as right, but false. "Rather than developing the analytical and topological tools that support the Italian mode of reasoning (Lefschetz, Hodge, et al.) the Bourbachistes chose the route of algebraization (Zariski, Chevalley, Well, Grothendieck). This seems to me to be a revealing choice. In Well's case, I wonder if he wasn't perverting his natural inclinations; 10 THEMATHEMATICALINTELLIGENCERVOL.14, NO. 3, 1992
Bourbaki is aware of the p r o b l e m of the relationship of g e o m e t r y to arithmetic, w h i c h is v e r y ancient, a n d was discussed by the Eleatics; a n d in " T h e architecture of m a t h e m a t i c s " he writes: Indeed, quite apart from applied mathematics, there has always existed a dualism between the origins of geometry and of arithmetic (certainly in their elementary aspects), since the latter was at the start a science of discrete magnitude, while the former has always been a science of continuous extent; these two aspects have brought about two points of view which have been in opposition to each other since the discovery of irrationals. Indeed it is exactly this discovery which defeated the first attempt to unify the science, viz., the arithmetization of the Pythagoreans ("everything is number"). If w e go back a century, w e find A u g u s t u s d e Morgan writing: Geometrical reasoning and arithmetical process have each its own office; to mix the two in elementary instruction, is injurious to the proper acquisition of both. Go back a n o t h e r 1300 years a n d in the quadrivium of Boethius w e find mathematics divided into arithmetic and g e o m e t r y , music and a s t r o n o m y , the second pair being the applied versions of the first pair; this therefore is also a division into two. O n the other h a n d J. J. Sylvester, in " A probationary lecture o n g e o m e t r y ''32 delivered on 4 December 1854, said: There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order. Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties and relations of Space, or of bodies viewed as existing in space . . . . It is the province of the metaphysician to inquire into the nature of space as it exists in itself, or with relation to the human mind. The less aspiring but more satisfactory business of the geometer is to deal with space as an objective reality . . . . But for the discovery of the conic sections, attributed to Plato, the law of universal gravitation might never to this hour have been elicited. Little could Plato himself have imagined that he was writing the grammar of the language in which it would be demonstrated in after ages that the pages of the universe are written. He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer. But Sylvester's three divisions might be r e d u c e d to two b y regarding O r d e r as superstructure of the o t h e r
I always had the impression he thought analytically, but was brilliant enough to adopt uncongenial modes of reasoning. This may be why his Foundations of Algebraic Geometry is generally felt to be awkward." 32 See his Collected Works [M23].
two, and one might wonder whether a further, and final, reduction is possible. 33 I w o u l d speculate, though, that the physiological separation by the brain of the processing of spatial from the processing of temporal thought supports the thesis that a complete unification of mathematics is not possible. Let us therefore consider these two intuitions, the arithmetical and the geometrical. These intuitions are not disjoint. The language of each is sufficiently rich to allow translations of the other: Within set theory one can do a mock-up of the real line by building first the rationals and then (say) Dedekind cuts; and one can mark out equally spaced points as integral points along a line. However, when such translations are made, paradoxes are prone to result, since the translations are of formal properties, not of underlying intuitions. Thus the Pythagoreans wished to believe that all is number, but were dismayed by the demonstration that the diagonal of a square is incommensurable with its side. Here, importing a simple geometric construction generated an arithmetical paradox. Stifel (1487-1567) asked what irrationals are. Geometry suggests they are acceptable, but as lengths, not numbers. He wrote, " a n irrational is not a real number because it lies under some cloud of infinity." He did not believe in X/2. In the other direction there is the Banach-Tarski paradox that a sphere can be decomposed into finitely many parts which can be rearranged by spatial translations and rotations to form two spheres of the same size as the original one. The proof of this is derived from the Schr6der-Bernstein argument, coupled with the Axiom of Choice (in the absence of which the Banach-Tarski theorem might fail). Here arguments that are natural in a set-theoretic context lead to conclusions that are paradoxical geometrically. This is similar in spirit to the result of Fibonacci in the 13th century that the solution of a certain cubic is not one of Euclid's irrationals.
33 O n this topic the same friendly critic writes: "Freeman Dyson expatiates on the subject of unifiers a n d diversifiers in Chapter 3 of his book [F4]. Unifiers revel in unity, diversifiers in diversity. I have long believed that mathematicians tend naturally to be unifiers. On the other hand I believe set-theorists with a strong interest in forcing almost never are. "I think in the case of Bourbaki, Dyson's distinction is more vital than yours. Bourbaki was interested in unity above all. Some degree of unification can be achieved by laying out measure theory carefully, others by converting an analytical theory into an algebraic one, thereby extending its breadth of application and capturing more cases simultaneously. It does seem that algebra is the workhorse of the unifier. Bourbaki tries to make plain the combinatorial content, in a certain limited sense, of those branches of mathematics which were 'ripe' for this treatment." How, I wonder, does Dyson's distinction differ from that made between creation and consolidation in the opening paragraph of this essay?
Split Brain?
John de Pillis
The attitude taken by Bourbaki to the issue of geometry versus arithmetic is still relevant today, for recently the distinguished American mathematician Saunders Mac Lane has called for a revival of discussion of the philosophy of mathematics and has criticised what he calls the Grand Set-Theoretic Foundation of Mathematics in phrases such as the Grand Set-Theoretic Foundation is a mistakenly onesided view of mathematics; set theory is largely irrelevant to the practice of most mathematics; logicism, formalism and Platonism have been too much dominated by the notions of set theory and deductive rigor. There have also been criticisms such as t h a t of Thom: set theory seems to suppress geometry and, before all those, the delicious Schlu~bemerkungof Skolem's 1922 paper, which runs, roughly translated, thus: The most important result above is the relativity of the concepts of set theory. I mentioned this orally to Professor F. Bernstein in Gbttingen in the winter of 1915/16. There are two reasons why I have not published anything about it sooner: first, I have been occupied with other matters since then; the second is that I thought it so clear that this axiomatic set theory was unsatisfying as a final foundation for mathematics that the majority of mathematicians would not bother themselves about it. Recently I have noticed, to my astonishment, that very many mathematicians regard set theory as the ideal foundation of mathematics; it therefore seems to me that the time is ripe for the publication of a critique. 34
34 See [M22]. THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3, 1992 11
I suggest that it is because Bourbaki fossilised mathematicians' knowledge of logic at its 1929 level 35 that this attack is n o w being renewed. Mac Lane, who, having been a pupil of Bernays in G6ttingen from 1930 to 1933, has a much stronger grasp of logic than have the Bourbachistes, is attacking a position from which logicians have been moving for the past 60 years, but which mathematicians are still at. This is not the place to begin a full discussion of the strengths and weaknesses of Mac Lane's foundational views set out in his book Mathematics: Form and Function, but some brief remarks are in order. Against Skolem I would hold that there is no final foundation for mathematics, but that set theory captures a substantial side of mathematics. I would agree with Mac Lane's first comment and with Thom's, and relate them to my idea that set theory is on the arithmetical rather than the geometrical side of mathematics. I would qualify Mac Lane's second comment, by saying set theory is not particularly relevant to the practice of geometry, but is very much relevant to arithmetic in its broadest sense. Though I agree with much of Mac Lane's third criticism, I question his use of the phrase set theory and deductive rigour. He thinks of these as hand-in-hand, and objects to the pair of them masquerading as the final solution for mathematics. I would want to separate the two. Logic is the study of our use of language: 36 set theory is the study of well-foundedness, and not, as Mac Lane thinks, the study of the process of set formation. That is the great difference b e t w e e n Z e r m e l o Fraenkel and Zermelo. Zermelo--more particularly the subsystem of it one may call Mac Lane set theory in view of Mac Lane's support for it in his books and articles---is a system to support set formation, and is adequate for geometrical considerations. ZermeloFraenkel is a system that contains in addition support for definitions by recursion, that is, building structures into the unknown. This element, which is the focal point of Kripke--Platek set theory, is suited to the arithmetical side of mathematics. In Zermelo set theory, one cannot prove that every well-ordering is isomorphic to a v o n N e u m a n n ordinal; one cannot prove the existence of the von Neumann ordinal co + co, though one can prove the existence and well-foundedness of linear orderings of that ordertype; one cannot justify recursion on ordinals or on arbitrary w e l l - f o u n d e d relations. Thus induction,
as Indeed, according to legend, a member of the Bourbaki group said, in a lecture given at Princeton to an audience that included G6del, that nothing had happened in logic since Aristotle. Can any reader tell me who? 36 This statement would be hotly contested by many; but the contest would only reinforce my point that logic is not the same as set theory. 12 THE MATHEMATICALINTELL1GENCERVOL,14, NO. 3, 1992
which is at the heart of arithmetic, is missing from (large parts of) geometry. On the other hand, spatial intuition is missing from arithmetic; so we need both. The geometrical conception of the integers as equally spaced points on a line suggests that all natural numbers are on an equal footing; in Russellian terms they are of the same type. In the arithmetical conception, 0 is the simplest natural number, and larger positive numbers are generated from, and are therefore more complicated than, smaller ones, so that no two natural numbers are of the same type. 37 Violence is done to each of these intuitions by trying to subordinate it to the other, and we should perhaps seek a philosophy of mathematics that allows the two to thrive in healthy interaction. The set theory that Mac Lane proposes in his book 38 as a basis for mathematics is a subsystem of Zermelo set theory plus the Axiom of Choice. His proposals therefore do nothing to answer the criticisms made here, that Bourbaki presents a pre-GOdelian view of mathematics, and of a portion of mathematics biassed towards geometry at that. Let me end on a positive note by recalling a quotation from Jean Dieudonn6: We have not begun to understand the relationship between combinatorics and conceptual mathematics and suggesting that both the philosophy that Mac Lane calls for, and the understanding that Dieudonn6 seeks, will emerge from a renewed study of the interplay between arithmetic and geometry.
Bibliography Mathematical Works [M1] K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland, 1960, pp. 277, 435. [M21 N. Bourbaki, "L'Architecture des math4matiques," in [M15], 35-47. [M3] N. Bourbaki, "The architecture of mathematics," American Mathematical Monthly 57 (1950), 221-232. [M4] N. Bourbaki, "Foundations of mathematics for the working mathematician," Journal of Symbolic Logic 14 (1948), 1-14. [M5] H. Cartan, "Sur le fondement logique des math6matiques," Revue Scientifique 81 (1943), 3-11. [M6] J. Cavaill~s, Volumes 608 and 610 of the series Actualit~s scientifiques et industrielles: le Progr~s de I'Esprit, Hermann, Paris 1938. See also Sur la logique et la thdorie de la science, Presses Universitaires de France, 1947. [M7] C. Chevalley, Mathematical Intelligencer 7 (1985), (2), p. 18; see also Mathematical Intelligencer 8 (1986), (2), p. 5.
37 This succession of "arithmetical" integers is beautifully captured by von Neumann's definition of ordinal number; for this reason I cannot accept Professor Mac Lane's view that von Neumann's definition is merely a "gimmick." 3s For a discussion of the logical status of the system of [M16] see
[M17].
[M8] J. DieudonnG "Les m6thodes axiomatiques modemes et les fondements des math6matiques," Revue Scientifique 77 (1939), 224-232. [M9] J. DieudonnG "David Hilbert," in [M15], pp. 291-297. [MIO] J. Dieudonn6, A Panorama of Pure Mathematics, Academic Press, New York, 1982. [Mll] S. Feferman, "Hilbert's program relativized: Prooftheoretical and foundational reductions." Journal of Symbolic Logic 53 (1988), 364-384. [M12] D. Hilbert and W. Ackermann, Grundziige der Mathematik, Springer-Verlag, Berlin, 1928. [M13] D. Hilbert, Gesammelte Abhandlungen, 3. Band, 378387, Berlin 1935; reprinted Chelsea, New York 1965. [M14] A. Lautman, Volume 591 of the series Actualit~s scientifiques et industrielles: le Progr~s de l'Esprit, Hermann, Paris, 1938. See also his Collected Works, 1977. [M151 F. le Lionnais (ed.), Les grands courants de la pensde mathdmatique, Cahiers du Sud, 1948; reviewed by S. Mac Lane in Mathematical Reviews 10, p. 230. [M161 S. Mac Lane, Mathematics: Form and Function, Springer-Veflag, New York, 1986. [M17] A. R. D. Mathias, "Notes on Mac Lane set theory," submitted to the Mathematical Proceedings of the Cambridge Philosophical Society. [M18] W. Sieg, "Hilbert's program sixty years later," Journal of Symbolic Logic 53 (1988), 338-348. [M191 W. Sierpiftski, Lemons sur les hombres transfinis, Collection Borel, Gauthier-ViUars, Paris, 1928. [M20] S. Simpson, "Partial realisations of Hilbert's programme," Journal of Symbolic Logic 53 (1988), 349-363. [M211 S. Simpson, "Ordinal numbers and the Hilbert Basis Theorem," Journal of Symbolic Logic 53 (1988), 961-974. [M22] Th. Skolem, "Einige Bemerkungen zur axiomatischen Begrfindung der Mengenlehre," to be found in Mengenlehre, an anthology of papers written since 1874 on the mathematical, metamathematical and philosophical aspects of set theory, selected and introduced by Ulrich Feigner, Wissenschaftliche Buchgesellschaft, Darmstadt, 1979. [M23] J. J. Sylvester, Collected Works, Volume II, p. 5, Cambridge, 1908; reprinted Chelsea, New York, 1973. [M24] A. Weft, "L'Avenir des math6matiques," in [M15], 307-320; also in his Collected Works, 1947a.
[P3] S. P. Springer and G. Deutsch, Left Brain, Right Brain, W. H. Freeman, New York, 1981, 1985.
Further Reading IF1] J. T. Baldwin (editor), Classification Theory, Springer Lecture Notes in Mathematics 1292 (1987). [F2] J. Denef, "p-adic semi-algebraic sets and cell decomposition," Journal ffir die reine und angewandte Mathematik 369 (1986), 154-166. [F3] J. Denef and L. van den Dries: "p-adic and real subanalytic sets," Annals of Mathematics (2) 128 (1988), 79138. IF4] F. Dyson, Infinite in all Directions. Harper and Row, New York, 1988. [FS] A. MacIntyre, "'On definable subsets of p-adic fields,'" Journal of Symbolic Logic 41 (1976), 605-610.
Instytut Matematyki Uniwersytetu Warszawskiego 00-913 Warszawa, 59 Ulica Stefana Banacha, 2 Poland
Historical Works [H1] Y. Belaval, Leibniz critique de Descartes, Gallimard, Paris, 1960, 1978. [H2] H. R. Trevor-Roper, Baron Dacre of Glanton, "The Paracelsian movement," in Renaissance Essays, Secker and Warburg, 1985. [H3] G. Ferri(}res, Jean Cavaill~s: un philosophe dans la guerre, 1903-1944, t~ditions du Seuft, Paris, 1982. [H4] A. KoyrG "Rapport sur l'6tat des 6tudes hegeliennes en France," Revue d'histoire de la philosophie, 5:2 (AprilJune 1931), p. 147. [HS] P. Mancosu, "The metaphysics of the calculus: A Foundational debate in the Paris Academy of Sciences," 1700-1706, Historia Mathematica 16 (1989), 224-248. [H6] M. Poster, Existential Marxism in Postwar France from Sartre to Althusser, Princeton University Press, 1975.
Physiological Works [P1] M. Annett, Left, Right, Hand and Brain: The Right Shift Theory, Erlbaum, New Jersey, 1985. [P2] A. Beaton, Left Side, Right Side: A Review of Laterality Research, Batsford 1985. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
13
Magic Cubes and the 3-Adic Zeta Function Allan Adler
Introduction In this paper, we make some general observations about the construction of magic N-cubes. It then appears, in light of these observations, that the 3-adic zeta function can be used to construct an infinite class of magic cubes. As there is no need to rush into a mass of technical details w h e n treating a subject that has always been regarded as fun, in the first two sections I will attempt to motivate the general discussion by describing two known methods of constructing magic N-cubes. In doing so, I will enlarge upon methods appearing in Kraitchik's book [3], in recreational paperbacks I read as a child, including [4], and in the article [1]. In the third section, I will show how to combine these methods with another technique from I1] to obtain a more general perspective of the problem of constructing magic N-cubes. Finally, in the last section I will illustrate this more general point of view using the 3-adic zeta function. The results of this article were obtained in 1984 while preparing a colloquium lecture on magic N-cubes at SUNY Stony Brook. I would like to thank the mathematics department at Stony Brook for its hospitality during the period 1983-1986. I would also like to thank Bob Messer and other correspondents from the electronic mailing list TEXhax for their help in typesetting the magic squares and Denys Duchier for his help typesetting the tessaract in Figure 10. I would also like to thank Larry Washington for carefully checking the proof of the main result of this paper and pointing out an error. Finally, I would like to thank Roger Howe and Walter Feit for their help in making the facilities of Yale University available to me while I typeset this article.
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T h e M e t h o d o f de la L o u b ~ r e This is an old method (cf. [3], w167 pp. 18-19) for constructing magic squares of odd order. One begins with an empty square of odd order M and has to count out the numbers from 1 to M 2 and place them in the boxes. Basically, one counts diagonally upwards to the right, with the following modifications. (1) The top edge is pasted to the bottom edge. So after reaching the top row, one continues by placing the next number in the bottom row and one column to the fight. (2) The fight edge is pasted to the left edge. So after reaching the right edge, one continues by placing the next number in the leftmost column and one row higher. (3) If the box one would want to place the number k + 1 in is already occupied, then one places the number k + 1 in the box directly below that containing the number k. (4) The number 1 is placed in the middle of the top row.
To illustrate we will lead the reader through the construction of a 5 x 5 magic square. In Figure 1, we have shown how to count out the first six numbers. The number 1 is placed in the middle of the top row, according to rule (4). The number 2 is placed, according to rule (1), in the bottom row and one column to the right of the 1. The number 3 is placed diagonally upwards to the fight, according to the basic method of counting out the numbers. The number 4 is placed, according to rule (2), in the leftmost column and one row higher than the 3. The hum-
3 2
ber 5 is placed diagonally upwards from the 4. If the box containing the I were empty, we would now want to place the 6 in it. But since that box is already occupied, we place the 6 directly below the 5 in accordance with rule (3). If one continues in this way, one eventually arrives at the magic square shown in Figure 2. One step might require some explanation, namely the placing of number 16 below 15. According to rules (1) and (2), it would seem that the 16 should be placed where the 11 is. Because the box is occupied, the 16 is placed below the 15 in accordance with rule (3). To see w h y this method of making magic squares works, note first of all that if we had followed the same procedure but counted from 0 to 24 instead of 1 to 25, it would not affect the fact that we get (or don't get) a magic square. In our constructions, it is generally more convenient to count from 0 to M 2 - 1 (or to M N - I for N-cubes), but the reader who prefers the traditional form of magic squares is welcome to add 1 to all the entries of the square we obtain. If we count from 0 to 24, we get the square illustrated in Figure 3. Next, we write all of the entries of the square as two-digit numbers in the base 5 and we get the square illustrated in Figure 4. Having done so, we now observe that in every row and column, the set of units digits is 0, 1, 2, 3, 4 in some order and the set of "fives" digits is also 0, 1, 2, 3, 4 in some order. It is therefore obvious that all rows and columns have the same sum. The same applies to the main diagonal of the square. For the other diagonal, a separate argument is required for the "fives" digits. If we view the boxes of the empty 5 x 5 frame as the vector space F2 of dimension 2 over the field F5 with 5 16
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Figure 1. Starting a 5 x 5 magic square.
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Figure 2. The completed 5 x 5 magic square.
Figure 4. The preceding 5 x 5 magic square, in base 5. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 1 5
elements with the origin in the center of the frame, t h e n the square in Figure 4 defines a function ~b from this vector space to itself. That function will be given by 6(x,y) = (2 + x + y, 2 + 2x + y) as one can easily verify. Note that it is an affine transformation. O n e can t h e n ask w h e t h e r one could make other magic squares using affine transformations. Consider an o d d integer M and let R d e n o t e the ring of integers m o d u l o M. The boxes of an e m p t y M x M square can be v i e w e d as points of the free m o d u l e R 2 of rank 2 over R, the center of the square being the origin. Consider an affine transformation 6 of R 2 of the form 6(x,y) = (ax + by + e, cx + dy + f). We require, of course, that 6 be a bijection, so ad - bc must be a unit of R. We w o u l d also like each of the digits 0, 1. . . . . M - 1 to occur a m o n g the units digits of any row or c o l u m n and similarly for the " M ' s " digits. This is the same as saying that each of the n u m b e r s a, b, c, and d m u s t be a unit of R. This guarantees that all rows and c o l u m n s have the same sum. Furthermore, the diagonals will have this s u m p r o v i d e d that in R we have that a + b and c +- d are units of R. H o w e v e r , as the example above of the 5 x 5 magic square shows, w i t h a suitable choice of e a n d f one can sometimes obtain a magic square e v e n if a = +b or c = - d . If a = • w e must take e to be (M - 1)/2 and if c = • w e m u s t take f = (M - 1)/2. O n e can also examine the case w h e r e M is even, b u t one discovers that a, b, c, d, a n d ad - bc cannot all be units of R simultaneously, a n d the m e t h o d fails. More generally, one can construct magic N-cubes of o d d order M b y considering affine transformations of the free m o d u l e a N of rank N over R. If w e write such a transformation in the form ~(v) = u + A 9 v, w h e r e u is some fixed e l e m e n t of R N and A is an invertible N x N matrix with entries in R, t h e n 6 defines a magic N-cube p r o v i d e d all of the entries of A are units and provided that for e v e r y sequence et . . . . . eN of --I's, each entry of the vector 2u + I is a multiple in R of the corresponding e n t r y of the vector
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In the examples given above, w e chose the origin to be the center of the magic square (or more generally of the magic N-cube). This is satisfying from the standpoint of s y m m e t r y , and w i t h o u t this choice the conditions o n the affine transformation b e c o m e m u c h m o r e complicated to state. H o w e v e r , in the following sections, w e will consider o t h e r m e t h o d s of making magic N-cubes, a n d in case the o r d e r of the N-cube is even, there is n o central square. T h e r e f o r e in order to h a v e a u n i f o r m treatment of magic N-cubes of arbitrary orders, w e will a d o p t a different c o n v e n t i o n in the rest of this article and take the origin to be one of the corners of the N-cube. By choosing the corner containing the e n t r y I as origin, the 3 x 3 x 3 magic cube in Figure 11 will be defined b y the affine transformation with the same matrix A as above b u t w e will have u = 0.
T h e M e t h o d of P r o u h e t S e q u e n c e s A m e t h o d of making magic N-cubes using the MorseH e d l u n d sequence was d i s c o v e r e d by the a u t h o r a n d g e n e r a l i z e d w i t h the h e l p of S.-Y. Robert Li to a m e t h o d using Prouhet sequences. A full discussion of these a n d other results m a y be f o u n d in [1]. W e will s u m m a r i z e the ideas and results of [1] that we n e e d for our discussion. Again, w e will take a leisurely approach, beginning with a simple example. O n e can make a 4 x 4 magic square (as I learned to do as a child from [4], p. 51) b y marking the diagonals of an e m p t y 4 x 4 frame (see Figure 5), counting the boxes from left to right, one r o w at a time, and writing d o w n the n u m b e r s of the boxes that lie on the diagonals. The result is s h o w n in Figure 6. One t h e n completes the square by counting d o w n from 16 to 1 a n d
Figure 5. The diagonals of a 4 x 4 magic square. E
For example, if A is the 3 x 3 matrix
t!2 tl and u the c o l u m n vector t(1,1,1), we obtain the 3 x 3 x 3 magic cube s h o w n in Figure 11 near the e n d of this article. 16
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Figure 7. The completed 4 x 4 magic square.
filling in all the e m p t y boxes along the way. O n e thereby obtains the magic square s h o w n in Figure 7. One night m a n y years later, w h e n I was in graduate school, I was p l a y i n g with the M o r s e - H e d l u n d sequence 10010110011010010110100110010110011 . . . given b y al = 1 a n d a k + j = 1 -- aj for 1 ~< j ~ k = 2 n. I h a p p e n e d to write d o w n the values of n for which a n = 1 and obtained the sequence 1, 4, 6, 7, 10, 11, 13, 16, 18, 19 . . . . It s e e m e d to me that the beginning of this sequence looked awfully familiar and after a little t h o u g h t I rem e m b e r e d that I h a d seen it on the diagonals of the 4 x 4 magic square I u s e d to make as a child (compare this with Figure 6). That suggested that the sequence m i g h t c o n t a i n t h e s e c r e t of m a k i n g o t h e r magic squares. I had tried, e v e n as a child, to imitate the m e t h o d in order to m a k e a 6 x 6 magic square and was unsuccessful. But the fact that the M o r s e - H e d l u n d sequence is so closely related to powers of 2 suggested that the next size to try might be 8 x 8. Instead of writing d o w n the n u m b e r s o n the diagonal, o n e w o u l d write d o w n the n u m b e r s of those boxes that corres p o n d to l's in the M o r s e - H e d l u n d sequence. That results in the partially c o m p l e t e d square 1 of Figure 8a, and if one t h e n c o u n t s d o w n backwards f r o m 64 to 1 and fills in the e m p t y squares, one obtains the 8 x 8 magic square s h o w n in Figure 8b. In general, w e obtain 2 n x 2 n magic squares for all n ~ 2 b y this m e t h o d . F u r t h e r m o r e , t h e y h a v e remarkable properties r e g a r d i n g sums of p o w e r s of their entries, generalizing the properties of the 4 x 4 case square (see Figure 7) described in [4], pp. 47-51. The M o r s e - H e d l u n d s e q u e n c e can also be u s e d to obtain N-dimensional magic cubes of order 2 n p r o v i d e d that the p r o d u c t of N a n d n is an even n u m b e r . For example, one can construct the 4 x 4 x 4 magic cube of Figure 9b in this way, the partially c o m p l e t e d cube being s h o w n in Figure 9a.
1 Simpler patterns for counting up and down are used in [3],w Fig. 213--215, p. 68, to make magic squares of order n = 4k. In general, these do not have the power sum properties enjoyed by the magic N-cubes made using Prouhet sequences.
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Figure 8a. A partially completed 8 • 8 magic square.
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Figure 8b. An 8 x 8 magic square.
These N-dimensional magic cubes also have remarkable properties involving sums of p o w e r s of their entries, as do the m o r e g e n e r a l magic N-cubes constructed using P r o u h e t sequences. These properties are discussed in [1] and will not concern us here. Magic N-cubes are c o n s t r u c t e d f r o m P r o u h e t seq u e n c e s in the following w a y . Let t be an i n t e g e r greater t h a n I and suppose that M is a p o w e r of t, say, M = t K. We then want to place the n u m b e r s O, 1 . . . . .
tKN -
1
in an N-dimensional cubic array. Each of these n u m bers can be r e p r e s e n t e d as a n u m b e r with KN digits in the base t, and m a y therefore be identified with an element of a free m o d u l e S K N of rank KN over the ring S of integers m o d u l o t. The individual boxes m a y also be r e g a r d e d as points of SKN if w e take one of the corners to be the origin a n d the diagonally opposite corner to be the vector w h o s e coordinates are all equal to t - 1. We can then view a magic N-cube of o r d e r M as a function from S K N to itself with certain properties. THE MATHEMATICALINTELLIGENCERVOL.14, NO. 3, 1992 17
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Figure 9a. Half-completed 4 x 4 x 4 magic cube. 1
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Figure 9b. The completed 4 x 4 x 4 magic cube.
The function qb used to construct magic N-cubes of order M = t K by means of Prouhet sequences is a linear mapping given by 4~(v) = A 9v in which the (i,j)-th entry of A is 1 if i # j and is equal to 2 if i = j. This construction gives rise to magic N-cubes of order t K provided that t divides KN and if M 2 in case t is even. Furthermore, they have the remarkable power sum properties mentioned above. D e c o m p o s i n g M a g i c N - c u b e s and Reassembling Them The reader may already have noticed (cf. [1],w that the 8 x 8 magic square of Figure 8b may be obtained from the 4 x 4 x 4 magic cube of Figure 9b by removing the rows of the magic cube and placing them in the rows of the magic square. Thus, for example, the first two rows of the magic cube are 1, 63, 62, 4 and 60, 6, 7, 57 and together they constitute the first row 1, 63, 62, 4, 60, 6, 7, 57 of the magic square. The next two rows of the cube give the next row of the magic square, and so on. One could obtain a different 8 x 8 magic square by using the columns of the cube instead or by using the orthogonals in the remaining direction. We refer to the above process of decomposing a magic N-cube into 18
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
orthogonals and reassembling them to form a magic cube of a possibly different dimension as packing. Thus, the 8 x 8 magic square of Figure 8(b) is obtained by packing the rows of the magic cube of Figure 9(b) into an 8 x 8 frame. One can also reverse the process and remove the rows or columns of the 8 x 8 magic square, break them into smaller units and place them in a 4 x 4 x 4 cubical array to obtain a 4 x 4 x 4 magic cube. We refer to this reverse process as unpacking. Thus the 4 x 4 x 4 magic cube of Figure 9(b) is obtained by unpacking the rows of the 8 x 8 magic square of Figure 8(b). In this way, one can regard the set of magic N-cubes of arbitrary dimension as a category ,~ whose morphisms are generated by packings and unpackings. Since packing and unpacking are inverses of each other, every morphism of the category is an isomorphism. Thus the category At is a groupoid. Denote by @ the full subcategory of At consisting of the magic N-cubes constructed using Prouhet sequences. The groupoid @ is then unusual in that it possesses a large number of morphisms. This is a property not shared by the subcategories of ~ constructed by most other standard methods. However, the properties of these and similar groupoids will not concern us here. Instead we will discuss packing as a technique for constructing magic N-cubes and will begin with a simple example. In order to introduce this example, first note that the operations of packing and unpacking make sense for arrays and not just for magic N-cubes. We can view the 4 x 4 magic square of Figure 7 as being obtained by unpacking the 2 x 2 x 2 x 2 tessaract in Figure 10. The coordinates in this tessaract are interpreted in Figure 10 in the following way: the box with coordinates (i,j,k,l) is the (k,l)-th entry in the (i,j)-th 2 x 2 square, where i and k index rows and j and I index columns. Thus, the (1,0,0,1)-th entry is 10.
or
,,
J 1 4
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}
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!
The next condition is that A 9u = u, w h e r e u is the element of SKN all of w h o s e entries are equal to 1. This will be called the stochastic condition. It guarantees that if two points x,y of S KN a d d u p to (t - 1) 9 u, t h e n so do A 9x a n d A 9 y. As x r u n s o v e r a diagonal of the N-cube, so does (t - 1) 9u - x. T h e r e f o r e the stochastic condition implies that the s u m of the entries in a n y diagonal of the N-cube will be t K. (t KN - 1)
Figure 10. A 2 x 2 x 2 x 2 tessaract.
The reader m a y be quick to point out that according to one of the few general theorems on the subject of magic N-cubes, there cannot exist a 2 x 2 x 2 x 2 magic tessaract, a n d I quite agree. The tessaract in Figure 10 is not magic a n d so we are led to investigate the conditions u n d e r w h i c h a KN-dimensional cubic array of order t, which is n o t necessarily magic, will u n p a c k into a magic N-cube of order t K. Let u s a d o p t the convention that the origin is one of the corners of the KN-cube and that the opposite corner is the vector with all coordinates equal to t - 1. Again, one w o u l d like for simplicity to use affine transformations to define the entries of the KN-cube of order t. H o w e v e r , I'd like to make it e v e n simpler by considering a linear transformation 6(v) = A 9u w h e r e A is a KN x KN matrix with entries in S = Z/tZ. In order that the entries r u n over all the n u m b e r s f r o m 0 to t KN - 1, it is necessary a n d sufficient that the d e t e r m i n a n t of A be a unit of S. To guarantee that all the sums over orthogonals of the N-cube obtained b y u n p a c k i n g the KN-cube are the same, we can no longer use the trick of h a v i n g each of the numbers, 0, 1, 2 . . . . . t - I occur exactly once as, for example, the units digit of an entry, because t K will generally be larger t h a n t. H o w e v e r , w e can achieve the same result b y g u a r a n t e e i n g that for 1 ~ i ~ KN, the t n u m b e r s 0, 1 . . . . t - I occur equally often as the i-th digit. To accomplish this, we note that each orthogonal of the N-cube is obtained b y u n p a c k i n g the entries of a coset P' of a certain K-plane P of SKN. If ei d e n o t e s the projection of SKN onto its i-th factor, t h e n we are really asking that the composition e i o ~ of ei with ~b m a p the coset P' onto S. In terms of the matrix A, the condition can be expressed in the following way. Let A 1 d e n o t e the matrix m a d e u p of the first K columns of A, let A 2 d e n o t e the matrix m a d e u p of the next K columns a n d so on. T h e n we have N matrices A1, A 2. . . . . A N w h o s e concatenation is A. The primitivity condition t h e n says that for 1 ~ i ~ N, each of the rows of Ai is a primitive vector of S K, that is to say, that no divisor of t greater t h a n 1 divides all of the entries of a n y row of Ai. It is not difficult to s h o w that the primitivity condition is equivalent to the equidistribution for I ~ i ~ N of the i-th digits of the entries of an orthogonal of the N-cube.
But that is precisely the s u m of an orthogonal of the N-cube if the equidistribution condition holds. Therefore w e h a v e p r o v e d the following result. T H E O R E M 1. Let t, N and K be positive integers with t > I and N > 1. Let S = Z/tZ and let A be a N K x NK matrix with entries in S. Suppose that A is invertible and satisfies the stochastic condition and the primitivity condition. Then the function q~(v) = A 9v from S NK to itself defines a KNcube of order t that unpacks into a magic N-cube of order t K. This s u m m a r i z e s the principles u n d e r l y i n g the p r o o f of T h e o r e m 1 of [1] for the case of P r o u h e t sequences. For example, if, as in the p r e c e d i n g section, we take A to be the matrix w h o s e (i,j)-th e n t r y is I if i # j a n d 2 if i = j, t h e n the stochastic condition says that t divides KN. T h e o r e m 1 is more general t h a n the construction of [1] via P r o u h e t sequences since it leaves o p e n the choice of the matrix A. H o w e v e r , the construction via P r o u h e t sequences is still useful, since it shows that the t h e o r y is n o n e m p t y a n d e v e n contains remarkable examples. The emphasis o n u n p a c k i n g and on linear and affine m a p s as a principle of construction puts the construction via P r o u h e t sequences on the same footing as the construction of de la Loub~re. H o w e v e r , d e la Loub6re's construction is not a special case of Theo r e m 1 a n d more work is n e e d e d to find a c o m m o n generalization.
The 3-Adic
Zeta Function
Thanks to T h e o r e m 1, we are n o w faced with the task of finding invertible matrices A that satisfy the primitivity condition and the stochastic condition. While I d o n ' t k n o w of a general solution, I will indicate s o m e ways of looking for solutions. First note that if S' is a h o m o m o r p h i c image of the ring S, t h e n a matrix A with entries in S that satisfies the conditions of T h e o r e m 1 will d e t e r m i n e a matrix A' with entries in S' that also satisfies the conditions of T h e o r e m 1. Therefore, we might as well be efficient and find invertible matrices w i t h entries in the profinite completion ~ of Z that satisfy the primitivity condition a n d the stochastic condition, H o w e v e r , this THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
19
amounts to finding such matrices over the ring Zp of p-adic integers for all p, and I find it more convenient to work there instead. Next observe that if s~ is a Zp-algebra that is a free Zp-module of finite rank and if ~ is a unit of sg, then multiplication by ~ is an automorphism of the underlying Zp-module of sg. If we are given a basis for ~ , we obtain a matrix. We can then seek conditions on the algebra, the unit, and the basis which guarantee that we get a matrix satisfying the primitivity condition and the stochastic condition. For example, we can take sg to be the group algebra over Zp of a finite group G. In that case, the elements of G form a natural basis and we only have to find a suitable unit oL. Even the case of a cyclic group is interesting in this connection. Indeed, the matrix ot used to construct magic N-cubes from Prouhet sequences is a circulant! But n o w we observe that if a group H is a homomorphic image of the finite group G and if a is a unit of Zp[G], then oL will be mapped to a unit of Zp[H] under the induced homomorphism of group algebras. Hence, if F is a profinite group, we can try to work in the complete group algebra Zp[F] = lim Zp[F/U] where U runs over the open normal subgroups of F. More precisely, we can try to find units of Zp[F] whose images in Z/fZ[F/U] satisfy the primitivity condition and the stochastic condition for infinitely many (or perhaps all) pairs n, U. For the magic cubes constructed in the second section via Prouhet sequences, there is no such unit. Indeed, for magic N-cubes of order t K, the unit of Zp[Z/KNZ ] is ~0] +
E
[a~,
N-cubes constructed from Prouhet sequences arise in this way, so that the theory is not empty even if we require c, to be a unit, we can then look for other examples. Let us take F to be Zp. In that case, we know from Iwasawa's theory that Zp[F] is isomorphic to the ring Zp~x] of formal power series over Zp. It is therefore tempting to ask whether perhaps one of the formal power series associated to p-adic L-functions might give rise to suitable elements of Zp[F]. There are two naive reasons to explore this possibility. First, the p-adic L-functions are at hand for immediate use. Second, they are important objects of study in contemporary n u m b e r theory. By establishing a connection between p-adic L-functions and magic N-cubes it becomes possible to ask n e w questions about p-adic L-functions. I don't k n o w in general which p-adic L-functions are related to magic N-cubes and I don't know h o w difficult it is to work it out. However, I have looked at the case of p = 3 and have found that the numerator of the 3-adic zeta function gives rise to a suitable element. Furthermore,that element works without the technical modifications introduced above to handle the magic N-cubes associated to Prouhet sequences. This will now be stated more precisely and proved in Theorem 2 below. We follow the notation and conventions of w of Iwasawa's book [2], to which we refer the reader for details. To simplify the discussion, we will take p to be an odd prime and later specialize to the case p = 3. According to [2], p. 87, if X is a Dirichlet character of order v with • - 1) --- 1, we have the following formula for the p-adic L-function Lp(S,•
Lp(S,X) = 2f(~(1 + qo)3 - 1; 0) 2g(~(1 + q0)s - 1; 0) h(~(1 + qo)s - 1; 0)
(1)
a~Z/KNZ
where for a in Z/KNZ the symbol [a] denotes a viewed as an element of the group algebra Zp[Z/KNZ] and where we have [a] 9[b] = [a + b] ~a [a] + [b]. We will use the same notation [a] in connection with group algebras of arbitrary groups over arbitrary rings. As KN approaches infinity, the sum in (1) diverges. However, if we agree to throw that summation away, then the limit is [0]. Hence we can modify our original idea by looking in general for an element ot of Zp[F] such that for infinitely many n, U, the image ~ of cx in Z/fZ[F/U] becomes a unit satisfying the primitivity condition and the stochastic condition after a suitable multiple of
M aeF/U
is added to ~. With the reassurance that the magic 20
THE MATHEMATICAL INTELLIGENCER VOL, 14, NO. 3, 1992
Here, g(x; O) and h(x, O) are certain formal power series that depend on the character X or, more precisely, on the first factor 0 of X. The power series h(x; 0) is given by o~
h(x; O) = 1
11 ++ qx 0 _ 1 - (1 - q0) E
(-x) n
n=O
and really doesn't depend on 0. The series g(x; O) depends essentially on 0 and is defined to be that formal power series in (~[xB which corresponds to a certain element ~q= = ~qo of lira ~[F,,] under the canonical isomorphism of Lemma 2 on p. 71 of [2]. Here, ~ denotes the ring generated by Zp and the v-th roots of unity and F, is the group of units of Z/pn+lZ that are congruent to 1 modulo p. The element ~qoois prescribed by giving
its i m a g e ~qn = ,qO in ~[F,] for all n. This is d e f i n e d on p a g e 72 of [2] b y ~0 = (1 - (1 + qo)~n(1
+ q0)-l)~n,
where
w h e r e the coefficient [3(k) is to be d e t e r m i n e d . Each t e r m of this s u m m a t i o n is o b t a i n e d b y a d d i n g the t w o t e r m s of (2) c o r r e s p o n d i n g to v a l u e s of a for w h i c h ~&(a') = "y,(1 + 3k). But if a is o n e of the values, the other is e v i d e n t l y 3 "+1 - a. F r o m 4a = a' + 3 "+1 a", w e d e d u c e that
(a)
1 ~n =
~0 =
2
.-n+l ~' @
(a> 9 0 ( a ) 9 ~/n(a) - 1
P
~(k) = a" -~
a n d w h e r e the s u m m a t i o n r u n s over all integers a such that I ~< a ~< p" + 1 a n d s u c h that a is not divisible b y p. We h a v e u s e d the s y m b o l (a> to d e n o t e the unique p-adic unit that is c o n g r u e n t to 1 m o d u l o p a n d such that a = (a> 9 o~(a), w h e r e 00(a) is a (p - 1)-th root of unity in Zp. As for ~&(a), n o t e that (a> r e d u c e s m o d u l o p " + ~ t o an e l e m e n t b of F,. W e then define "y,(a) to be the e l e m e n t [b] of f3[F,]. Let us n o w take X = 0 = 1 a n d p = 3, so t h a t v = l a n d ~ = Z 3. T h e n w e h a v e 1 (a> ~" = - 2 9 3 ~+1 ~ ' ~.(a) "qn =
1
w h e r e the s u m m a t i o n r u n s o v e r all integers a such that I -< a -< 3" + 1 a n d s u c h that 3 d o s e s not divide a. H e r e we have
a w h e r e w e h a v e u s e d the L e g e n d r e s y m b o l . In Iwasaw a ' s p r o o f that "q, actually belongs to Z3[Fn], h e writes 4a in the f o r m a' + 3" + la" w h e r e 0 <-- a' < 3" + 1. H e t h e n arrives ([2], p. 74) at the expression
1 ~, a"Ol(a'___~) ~ln(a') " a=0
w h e r e in our case q,, = 3" + 1 a n d
We t h e n h a v e
~n = ~ ~ '
~.(a')
(2)
w h e r e ]~' has the s a m e m e a n i n g as before 9 We can collect the t e r m s that h a v e the s a m e d e n o m i n a t o r a n d obtain the e x p r e s s i o n
1 3~, - 1 "qn = ~
~(k) ~/n(1 + 3k)
k=0
(3n13 a)
In particular, [3(k) =- ---a" ( m o d 3).
(3)
We will take 4 to be the g e n e r a t o r of the cyclic g r o u p F , w h e n w e consider circulant matrices associated to t h e regular representation. T H E O R E M 2. The formal power series g(x; 1) which defines
~n,
"qn = ~
+ (3 - a")
the numerator of the 3-adic zeta function corresponds under the canonical isomorphism of Z3~x~ onto lim to the gi~-enZ3[F"] element "qoowhose image ~q, in Z3[F,] is by (2) for all n. Then "qo may be identified with a 3-adic unit and for every positive integer n we let H, denote the circulant matrix belonging to left multiplication by ~qJ'qo in Z3[F,]. For every pair n,r of positive integers let H,,r denote the matrix obtained by reducing the entries of H, modulo 3 r. If n is greater than 1, then H,, r satisfies the hypotheses of Theorem 1 with t = 3 r, N = 3 and K = 3"- 1. In particular, the matrix H,, r determines a magic cube of order 3 L, where L = r 9 3"- 1. PROOF: It is s h o w n on p a g e 81 of [2] that g(x;1) is a 3-adic unit. Therefore the e l e m e n t -q~ of lim Z3[Fn] a n d its i m a g e "qn in Z 3 [ F n ] is a unit for all n I> 0. It follows that the matrices H , a n d H,,r are all invertible. Furtherm o r e , if w e d e n o t e b y ~ the s u m of all of the e l e m e n t s of F,, t h e n w e h a v e -q, 9 or = "qo " ~r. Therefore, H , satisfies the stochastic condition of T h e o r e m 1 a n d so do all of the matrices H,,r. It r e m a i n s to the p r o v e the primitivity condition. It is e n o u g h to p r o v e it for H , . Since w e are taking N = 3, w e can write H , as the c o n c a t e n a t i o n of three m a t r i c e s A 1, A 2, A 3 of size 3" x 3 " - 1 . Since H , is a circulant, all three matrices h a v e the s a m e r o w s but in a different order. Since e a c h A i has r o w s of length 3"-1, it will therefore be e n o u g h to p r o v e that the top 3 " - 1 r o w s of A 1, A 2, a n d A 3 are primitive. Since the entries of H , are 3-adic integers, w e just h a v e to s h o w that the t o p 3 " - 1 r o w s of A 1, A 2, a n d A 3 each contain a 3-adic unit. The entries of the top r o w s of A 1, A 2, a n d A 3 are the coefficients of ~&(a)- 1 in ~ , as a r u n s o v e r the e l e m e n t s 4 i of F n w i t h 0 ~< i < 3"-1, 3 " - 1 ~ i < 2 93 " - 1 a n d 2 - 3 " - 1 ~ i < 3", r e s p e c t i v e l y . W e h a v e t h a t THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3, 1992
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Figure 11. A 3 x 3 • 3 magic cube.
immmmmmm immmmmmmmmEmmmmmmmH mgmmRmmm m/mEmmEmm/m mm /m mmm mEBmmmm mm mm /m mE/mmm
L Figure 12. One face of a 27 x 27 x 27 magic cube. 22
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
4 i = 1 + 3" i n F , i f i = 3 " - l a n d 4 i = 1 + 2 . 3 " i n F . if i = 2 9 3"-1. For these two cases, the coefficient of ,/,(4i+ 1)-1 is easily seen to be a 3-adic unit by a direct computation using (3). Since these entries lie in the first columns of the top rows of A 2 a n d A 3, they lie in all of the top 3 " - 1 rows of A 2 and A 3. This proves that the top 3"- l r o w s of A2 and A 3 are primitive. To prove that the top row of A1 is also primitive, we take a=
16 16+4.3
"-1
ifn = 2 i f n > 2.
One can verify that ~&(4a)-1 lies in the top row of A 1 a n d is a unit. Unfortunately it does not lie in the first column of A1 but in a column about one third of the w a y across. Therefore, it does not lie in all of the first 3 n-1 columns of A 1. To take care of the last 3 n-2 + 1 rows a m o n g the first 3 "-1 rows of A 1, we simply choose a suitable entry of the top row of A 3 which, as the elements are cyclically shifted to the right, enters A 1 just before ~&(4a)- 1 is shifted out of A 1. It is easy to verify that we can take that entry to be the last entry in the top row of A 3 if n = 2. If n > 2, t h e n examine the entry of the top row of A 3 which is in the same relative position in A 3 as ~n(4a) -1 is in AI: that position contains a unit if n = 3 a n d the position to the right of it contains a unit if n > 3. This proves T h e o r e m 2. It would be interesting to see w h e t h e r the magic cubes obtained in this w a y have any n o t e w o r t h y properties. Also, it w o u l d be desirable to k n o w w h e t h e r other p-adic L-functions can be used to m a k e magic N-cubes. It is natural to try to imitate the proof of Theorem 2 for p-adic zeta functions. The proof of the stochastic condition works in general. But the proof of the primitivity condition seems to be more difficult. The problem is that in order to determine the coefficient of ~&(a)-I for a given a, one needs to k n o w something about the values of 01(a') 9 a" not only for a itself but for all of the n u m b e r s of the form a 9~ w h e r e ~ is a (p - 1)-th root of unity in Zp. In general, these roots of unity are irrational a n d little can be said about their p-adic expansions. But for p = 3, they are rational and have obvious p-adic expansions, and that is the reason that the proof of T h e o r e m 2 works. It should also be noted that Theorem 2 is probably not the best result, n o t e v e n for the 3-adic zeta function. Rather one should interpret the t h e o r e m as saying that results of this type are possible. This is certainly n e w a n d the task of exploring this connection remains. If we violate the h y p o t h e s e s of Theorem 2 by taking n = 1, the unit ~4] of Z3[Z/3Z ] is obtained. This unit does not satisfy the primitivity condition of Theorem 1, but if we add to it the element ~1~ + [4] + ~7~, then we obtain [1] + 2 9 ~4] + [7], which does satisfy the primitivity condition. Therefore for n = 1 the unit constructed in Theorem 2 satisfies the more general primitivity condition described at the beginning of this section. The 3 x 3 x 3 magic cube one obtains is given in
Figure 11. The reader will note that it differs by a rotation from the 3 x 3 x 3 magic cube appearing in Figure 1 on page 621 of [1]. If one adheres strictly to the h y p o t h e s e s of Theorem 2, t h e n the smallest magic cube that can be obtained will be a 27 • 27 x 27 magic cube. We do not h a v e space to present the entire magic cube here, but one of the faces of the cube is s h o w n in Figure 12.
References 1. Adler, Allan, and S.-Y. Robert Li, "Magic N-Cubes and Prouhet Sequences," American Mathematical Monthly 84 (1977), 618-627. 2. Iwasawa, Kenkichi, Lectures on p-adic L-functions, Princeton, NJ: Princeton University Press and University of Tokyo Press, (1972). 3. Kraitchik, Maurice, Trait~ des Carrdes Magiques, Paris: Gauthier-Villars & Cie (1930). 4. Meyer, Jerome S., Fun With Mathematics, Cleveland: World Publishing Company (1952). 36 Rolens Drive #4C Kingston, RI 02881 USA
After 1992, see Combined Membership List of the AMS
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 2 3
9. . Deine A Reading
Sonia" from a Burned
Letter
Reinhard B011ing Translated by David E. Rowe
It was in January 1990. Finally, just two months after the Berlin Wall had fallen, I had the opportunity to spend a few days at the Mittag-Leffier Institute in Djursholm, a small town just northeast of Stockholm. The palatial villa that today houses the Institute is the former home of G6sta Mittag-Leffler (1846-1927), and on entering its doorway I felt as if I had taken a step back into the world in which he lived. For me, the Institute's single greatest attraction lay in its archival holdings, and particularly the extensive correspondence that linked Mittag-Leffler with many of the era's leading mathematicians. A former student of Karl Weierstrass (1815-1897), Mittag-Leffier sought to preserve everything he could get his hands on from the master's estate. Thanks to his efforts, m a n y letters, manuscripts, and other documents associated with Weierstrass have survived today. In Berlin, on the other hand, where Weierstrass passed his entire scientific career, there is no corresponding collection of materials. I had just completed my work on a new edition of the letters that Weierstrass wrote to Sonya Kovalevskaya (1850-1891), his trusted pupil and friend [cf. [10]; this is a reedition of [4], together with a detailed commentary on the biographical and mathematical contents of the letters, to be published in 1993 by Akademie Verlag (Berlin)]. Several questions remained open, and I hoped that during my visit I might find some new documents that could shed light on these matters, if not resolve them completely. With only three days available to look through the archival material, I had to make use of every second. From dawn to dusk and without a break, I scanned letters and leafed through large quantities of mathematical notes and sketches. Mittag-Leffier had already made a first attempt to put the papers of Weierstrass and Kova24
levskaya into some kind of order, but there still remained a good number of boxes and folders among which sheer chaos reigned (cf. also [2]). Nevertheless, there was compensation to be found in the adventure of the hunt, and as it turned out I was not to be disappointed. It was already late afternoon during the second day of my stay. I was just about to examine the contents of a box that contained material from Kovalevskaya's posthumous papers. At first glance, the documents appeared to be exclusively concerned with details surrounding the events of her death and the arrangements that had to be made afterward. I found various
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3 9 1992 Springer Veflag New York
bills and receipts in connection with the funeral, others from purchases she had made shortly before her death, and a number of telegrams from friends and acquaintances expressing condolences. Between these, I found some photographs and several pages filled with handwriting. Staring at these, illuminated by the glimmer of the table lamp, I discovered to m y surprise a text filled with crossed-out words and lines, and containing in places afterthoughts written in an almost illegible, scrawling hand that looked like Sonya's own. Then I noticed that the text, whatever it was, had been written in German. At first I thought that perhaps this was a sketch for a literary work, but then I noticed some comments about Mittag-Leffler and his lectures. Thunderstruck, I strained to decipher a few more passages; the words Dampfschiff (steamboat), and Stockholm flashed by, and then: Deine arme, kleine [ . . . ] Schfilerin. Unbelievable as it seemed, this appeared to be a page from the first draft of a letter that Kovalevskaya intended to send to Weierstrass. I knew very well--Mittag-Leffier had reported it--that Weierstrass had burned all the letters he had received from Kovalevskaya soon after her death in February 1891. Thus, these writings of Kovalevskaya are gone forever, but n o w the thought raced through my mind: could it be that what I held in my hands was a draft of such a letter from which we might gain an impression of what at least one of them actually looked like? With growing excitement and anticipation, I soon found another page which proved to be a continuation of the one I already had before me. After making a first transcription of the text, I could no longer doubt it: this was a fragment from the draft of a letter (shown on p. 28) composed by Kovalevskaya approximately two weeks after she first arrived in Stockholm in early December 1883. In all likelihood, it is the only remaining document that has survived from the many letters she addressed to Weierstrass. Whether it was accident or design that spared these pages from destruction is a mystery that will probably never be solved. Beyond the flavor of her idiosyncratic style, Kovalevskaya's letter is particularly significant because it was written at a critical turning point in her life. Since there are several good biographies of Kovalevskaya available (e.g., [1,3,5,7]), and, moreover, a previous article about her by Ann Hibner Koblitz in this journal (Mathematical Intelligencer, vol. 6 (1984), no. 1, 20-29), I shall restrict myself here to the events in her life directly relevant to the letter under consideration. Kovalevskaya had just come from St. Petersburg to Stockholm on November 18, 1883 in order to lecture at the newly founded H6gskola (which later became the University of Stockholm). Thus, she stood at the very beginning of her academic career. Only a little more than a half year earlier her husband, Vladimir, had taken his own life after his business ventures in connection with an oil company had brought him and his
The Mittag-Leffler Institute.
family to the brink of financial disaster from which he could see no way out. Even earlier, during the late 1870s, a series of building projects and real estate investments by the Kovalevskys had seriously depleted their financial resources, including the money Sonya had inherited following the death of her father. Between 1870 and 1874 Kovalevskaya had studied privately with Weierstrass in Berlin, and, on the latter's recommendation, G6ttingen University awarded her a doctorate (in absentia). She returned to Russia in August 1874. However, her efforts to find a suitable position in her native country proved futile, and so she gradually lost interest in mathematics. Eventually she gave it up altogether, and thereafter she fell out of contact with her teacher and friend, Karl Weierstrass. Three years went by during which he heard nothing more from her. In February 1876 Mittag-Leffier happened to be in St. Petersburg, and during his stay he paid Kovalevskaya a visit. This was their first meeting. Through his former Swedish pupil, Weierstrass again learned something about Sonya, but it was not until August 1878, two months before the birth of her daughter, that she wrote to him directly. Even so, it would take another two years before she began to correspond with him regularly. During this period, Vladimir was often away from home trying to manage the affairs of his oil company. Sonya sought to persuade her husband to break his ties with the company, which she rightly viewed as a dubious enterprise. She hoped in vain that he would eventually return to his scientific work. Vladimir, w h o had taken his doctorate from the University of Jena in 1872 and had since made a name for himself through his work in paleontology, received an appointment as a Dozent at Moscow University in 1880. But even this proved of little consequence. Her hopes crushed, Sonya's relationship with her husband grew more and THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
25
Sonya Kovalevskaya. more distant. Finally (evidently in the spring of 1882), she wrote him the following mournful words: "Our dispositions are so different; you are at times capable of making me really crazy and I can only come to my senses when I am left alone. If I look at matters soberly, then I find that you are completely right, and it is best for both of us if we live our lives separately" (translated from [5], p. 103). Weierstrass spoke of their " i r r e d e e m a b l y b r o k e n marriage" (letter to Kovalevskaya from 5 August, 1882 ([0], letter 111, p. 1 or [4], p. 102)). And indeed each went his separate way: Vladimir traveled to North America on business, and Sonya stayed in Paris where she joined her friend Maria Jankowska (1850-1909), a Polish socialist and journalist w h o m she met that year. According to a remark of Charles Hermite (1822-1901), a divorce procedure was planned, but it never came to pass. It was here in Paris that Sonya learned of her husband's suicide. For five days she refused to see anyone or even eat. After that she fell into a coma and the doctor, w h o m she refused to see earlier, could finally be brought in to treat her. Following her recovery, she traveled to Berlin in the summer of 1883 in order to meet with Weierstrass. There she discussed with him her work on the mathematical treatment of the refraction of light as well as her plans to begin lectures in Stockholm. Before her arrival, Weierstrass had no idea of the fateful blow that his former student had suffered, but her appearance told him right away that she was in very bad health. 26
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
Kovalevskaya's desire to take up an academic career did not result from Vladimir's death, rather it had been her intention for some time past. But now the realization of this early dream clearly took on considerable urgency for her. Back in early 1881 when he was associated with the University of Helsingfors, MittagLeffler already learned that she would be willing to accept a university appointment. His efforts there, however, proved unsuccessful, not so much because Kovalevskaya was a w o m a n but because she was a Russian. Just before he took up his new position at the Stockholm H6gskola in September 1881, Mittag-Leffler informed her that he thought the chances would be better at this new institution. His evident interest in the realization of this plan can be seen from a letter he wrote her on June 19, 1881: "I have no doubt that once you are here in Stockholm ours will be one of the first faculties in the world" ([6], p. 27). Weierstrass remained skeptical. He even feared that Mittag-Leffler's efforts on behalf of Kovalevskaya might compromise his own situation in Stockholm. For this reason, Kovalevskaya also thought that it w o u l d be best for all concerned if, for the time being, no further steps in this direction were taken, at least not until she found time to complete her ongoing work. MittagLeffier, on the other hand, did not wish to postpone plans for her appointment, partly because he was concerned that Stockholm might ultimately lose her to another institution. As he tried to assure her on February 25, 1882, "Don't you think that it is only right that the one w h o comes first s h o u l d also have the b e s t chances?" ([6], pp. 32-33). Kovalevskaya's health improved during the course of her stay in Berlin. It was during this time that the situation regarding her teaching position was finally resolved. In the meantime, she continued to work with success on her paper dealing with the refraction of light. Just before Weierstrass left Berlin for Switzerland (where he spent the winter of 1883-84), she gave him a small exposdsummarizing the results she had by then obtained. Already in August 1883 he communicated a detailed report on these results to Mittag-Leffier. That month, Kovalevskaya left for Odessa, where she participated in the Seventh Congress of Russian Natural Scientists and Physicians from August 30th to September 9th. On September 3rd, she lectured on the "Integration of the partial differential equations that determine the refraction of light in a transparent crystalline medium." Full of thanks, she wrote to Mittag-Leffier from Odessa, "I am very grateful to the Stockholm H6gskola, the first among all the European universities to open its doors to me . . . I hope to remain many years and to find there a second home" (Kovalevskaya to Mittag-Leffler, 9 September 1883 [6], p. 34). After the Odessa Congress, she spent some time in Moscow and Petersburg, and from there, as already mentioned, she traveled to Stockholm in mid-November. Origi-
nally she planned to spend two or three months once again in Berlin with Weierstrass in order to fill in some of the gaps in her work and to prepare her lectures. But this visit never came about due to Weierstrass's prolonged absence from Berlin. We now have reached the point w h e n Kovalevskaya composed her first letter to Weierstrass after arriving in Sweden. It should be noted that the English translation of the text that follows gives only a very rough idea of the highly idiosyncratic German of the original. In her biography of Kovalevskaya, Anne Charlotte Leffler (1849-1892), the sister of G6sta Mittag-Leffler, mentioned that although Sonya had studied for several years in Germany, she always spoke a rather broken German (cf. note 7). "She always spoke fluently, always succeeded in expressing what she wanted to say, and in giving an individual stamp to her utterances, however imperfectly she spoke the language she was using. When she had learned Swedish she had nearly forgotten all her German, and when she had been away from Sweden a few months, she spoke Swedish very badly on her return" ([7], p. 59). The original text will be presented as part of a more detailed article in a forthcoming issue of Historia Mathematica. The present text of Kovalevskaya's letter is followed by some brief explanations and commentary on its contents.
Sonya Kovalevskaya to Karl Weierstrass Stockholm, Early December 1883 9. . [the students] are all very talented and have a feel especially for function-theoretic views.1 Last semester Mittag-Leffler lectured on the theory of analytic functions, and now he has just finished a course on analytic functions of several variables based completely on your lithographed text. 2 Next semester, as I said, he will deal with ordinary differential equations, so that lectures on partial differential equations would appear to tie in naturally with his own. It is true that I regret somewhat that I did not choose from the first to lecture on the calculus of variations, as I o w n such a good copy [of your lectures] and you would have perhaps allowed me to base my lectures on these. 3 Unfortunately, this thought only occurred to me after it was too late. If, however, you declare yourself dissatisfied with my choice for my first lectures, perhaps I could still take refuge in this subject. But please, be so kind, my dear best friend, and help me by giving me your advice in my distress. 4 Also in another matter I very much need your support and your help. I wish to turn to the detailed work [Ausarbeitung] on my last study, 5 as it is most necessary that it appear this winter in Acta Mathematica, and without your help I cannot take a step forward. Would you not be so very kind and read through the small expos6 that you took with you and then write me as to how I
should begin my work. 6 If you have lost it, I will send you another right away. I have already gotten so used to flying to you in every emergency, so that I again with assurance turn to you. You, my supreme teacher, would not let your poor little bold 7 student drown without extending a hand to save her. The day after my arrival here in Stockholm all the papers announced this great world event. My arrival itself did not come off without a rather humorous adventure. The trip from Petersburg to Stockholm was quite arduous for me, since I still could not understand a word of Swedish. As bad luck would have it, the steamship that carried me to Stockholm had but one single man on board who spoke German or any language for that matter other than Swedish. He was quite friendly to me, and in my distress I was happy to take advantage of his company. But since I already k n e w that several n e w s p a p e r s , both Russian a n d Swedish, had written about me, I tried to behave as simply and modestly as possible so that he would not guess w h y I was travelling to Stockholm, and so as not to draw general attention. He appeared very interested to know w h y I was going to Stockholm so utterly alone and without knowing the language. 8 1 don't know exactly what I told him, but he apparently imagined that I was some kind of little governess, alone and unprotected in the world, on my w a y to Stockholm to take a position there with a family. As our steamship departed, I had telegraphed Mittag-Leffler [the telegram is pictured here], but the trip went so exceptionally well that we arrived in Stockholm two to three hours earlier than we had expected 9 Because of this, Mittag-Leffler was not at the harbor to meet me. Well, you can well imagine my forlorn state, especially since my letters to Mittag-Leffler had always been addressed to the University and I didn't even know his actual address 9
Telegram from Kovalevskayato Mittag-Leffler. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
27
Naturally I was relieved when the German-speaking ish gathering where Swedish alone is supposed to be man offered to accompany me to a hotel he evidently spoken! It will be interesting to see how I make out. knew of, and since there was nothing else I could do I implore you, my dearest friend, to write me very under the circumstances, I thankfully accepted his pro- soon. I await the next direct news from you with altoposal. But then, I suppose in accordance with his im- gether indescribable anticipation, all the more so, for pression of me, he took me to a really out-of-the-way even though I am [so honored?] 17 here, I feel really hotel where no one could understand a foreign word lonely and estranged. May the spring come quickly so and where the service was altogether [lacking?]. After that I can see you again, strong and healthy, in Berlin. I had passed enough time that this sad situation be- My best and warmest greetings to your dear sisters came clear to me, I wanted to find the man again and and please do not forget ask him to take me elsewhere, but as I approached the Your most devoted room he had taken, I noticed that there were already a Sonia number of other m e n with him. There were loud voices and laughter coming from his room, and I did The question arises naturally whether Weierstrass not dare to knock. So there was nothing left for me to ever received the letter corresponding to this fragment, do but return to my [room?] again and quietly wait and, if so, what kind of reply he gave. As a matter of several hungry hours in the saddest state in the world. fact, his answer to Kovalevskaya exists, for on the 27th Only around evening did Mittag-Leffler succeed in of December he wrote to her, " N o w I will answer your learning with [great trouble?] and inquiries what had letter from Stockholm (undated) in a suitable fashbecome of me, and then he came with his whole fam- ion . . . . You have thus crossed the Rubicon. I had ily, wife and sister, in the most elegant carriage to take me away. 9 . . . to find me in such a sad [state?] . . . . 10 I have seldom seen anything so peculiar as the astonishment of the hotel employees and my travelling companion when they learned that I am such a highly regarded lady. For you must know that Stockholm is the funniest little city in the world in which everything about everybody is known immediately and every little incident takes on the proportions of a world e v e n t ) 1 In order to give you an idea of what Stockholm is t . actually like, I will tell you about how a democratic newspaper announced my arrival the next day approximately as follows: This is not the visit of an insignificant prince or some other distinguished personality that we today have to report to our readers. No, it concerns something completely and incomparably different. The princess of science, Frau Sophie von Kowalevsky, has honored our city with her visit and plans to lecture at our university.112 How do you like that? Mittag-Leffier has already gathered a whole collection of newspaper announcements about me, but this one is by far the nicest. My letter has already become so long that I am compelled to bring it to a close. I would like to have told you about the grand soir6e that Mittag-Leffier gave in my honor, during which I was introduced to most of the notables of the University here. 13 1 liked Herr Nordenskj61d most of all. That is a wonderful man and so simple, and even a democrat as well. ~4 Herr, and especiaUy, Frau Gyld6n are also very nice people) 5 For the most part, I have been very busy learning the Swedish language and in the last two weeks I have ~.~'~V~.~.~ ~ > ~.Z.-- i ~ "~i-i~r made significant progress, although it is indeed difficult, much more so than I had believed. Still, I can already understand what I read and also a little of that which is spoken around me. Naturally I can't yet speak it myself. 16 Tomorrow I must go to an authentic Swed- The last page of the letter draft. 9
28
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
J
,
o
hardly thought that y o u w o u l d still decide to do so this winter" ([0], letter 124, pp. 1-2 or [4], p. 109). Weierstrass also gave her the advice she requested regarding her lectures (as m e n t i o n e d in note 4). He apologized for not going into her work on the refraction of light, saying that he f o u n d writing to be difficult, but promi s e d to do so l a t e r . R e g a r d i n g t h e c o m m o t i o n prompted by her arrival in Stockholm, he wrote, "That so m u c h has been a n d will continue to be spoken about you is not good, but that is Swedish nature and the good Mittag-Leffier himself is not free of this" ([0], loc. cit. or [4], p. 110). Weierstrass's a n s w e r also allows us to say something about the contents of the missing first part in the draft of Kovalevskaya's letter. This evidently began by inquiring about the health of her former teacher, because he describes in some detail his present rather unsatisfactory condition. Beyond w h a t is contained in the fragment, it is possible that Kovalevskaya also said something more in the beginning of her letter about the city of Stockholm and the provincial impression it made on her, as Weierstrass replied: " W h a t y o u write about the Kleinstfidterei in Stockholm is perfectly understandable to m e . " He also thanked her for telling him the " c h a r m i n g story a b o u t Mittag-Leffler's potato sacks over which we all got a good chuckle." This amusing episode can be reconstructed from the letters that Weierstrass's sisters exchanged w i t h Sonya. A shipload of 30 sacks of potatoes ordered by MittagLeffler had arrived in Stockholm. However, MittagLeffier had not received a permit to import them. Finally, after m a n y futile discussions, he h a d to personally consult with the King before he was granted the permit; but in the m e a n t i m e the ship carrying all his precious potatoes h a d departed for England. To close this mini-adventure into the remains of a burned letter, we cite the final words that Sonya's revered teacher wrote to her as she prepared herself for a fresh start in her n e w life and career ([0], letter 124, p. 6 or [4], p. 112): A n d n o w a w a r m e s t fare-thee-well, m y dear friend, and my heartfelt wishes that the N e w Year m a y bring you all the best, not in the form of wonderm e n t on the part of Stockholm journalists, but in the satisfaction that only serious striving a n d successful work can guarantee. To that, best of luck! Your true friend, K.W.
Acknowledgments I would like to t h a n k the Institut Mittag-Leffler for the kind permission of publication of the materials used in this paper that w e r e c o m p l e t e d together w i t h the translation during m y joint stay with D. E. Rowe in February 1991 at the Institut--just exactly 100 years after the death of Sonya Kovalevskaya.
Notes 1. Following Weierstrass's suggestion, Kovalevskaya's first lectures were to be held before a special group chosen by Mittag-Leffler as a kind of test to determine whether she possessed the requisite talent to be a successful teacher. It is conceivable that she is referring here to this special circle; another possibility, of course, would be that she is writing about those who were attending Mittag-Leffler's lectures. 2. As a young Dozent in Uppsala, Mittag-Leffler received a stipend to study abroad, which he used to study first in Paris and then under Weierstrass in Berlin during 18741875. Thereafter, he played a key role in disseminating Weierstrass's mathematics throughout Scandinavia. 3. In 1882, Weierstrass had a copy made of his lectures on the calculus of variations for his pupil. 4. Weierstrass approved of Kovalevskaya's choice for her lectures, particularly since in her dissertation she had already done some original work in this field (Theorem of Cauchy-Kovalevskaya). Beyond this, he gave her some hints about which methods and results she should present in detail. Kovalevskaya gave her first lecture on 30 January 1884 (she spoke in German). 5. During 1883, Kovalevskaya had worked intensively on the mathematical treatment of the refraction of light in a crystalline medium (integration of the Lain6 differential equations). 6. For his vacation on Lake Geneva, Weierstrass took the expos~ Sonya gave him. He also promised to study it and give her advice, but then he put this off over and again. In her work, Kovalevskaya applied an unpublished method of Weierstrass for the integration of linear partial differential equations with constant coefficients. During the summer of 1884, she was urging him to complete the final version of his work on this subject so that she could submit her own. In a letter of 13 September 1884, he wrote her that he was "horribly tired" and that this had "made him apathetic and filled him with antipathy for all thinking and writing," and he described himself as "mentally exhausted [gehirnmfide]" ([0], letter 131, p. 1 or [4], p. 115). He therefore left it to her to use his unpublished work as she wished. Her paper appeared the following year in volume 6 of Acta Mathematica. 7. The German reads wagkfihne, an invention of Kovalevskaya's that probably stands closest to the word wagemutig. Anne Charlotte Leffier mentions that Sonya's "German friends used to laugh at the ridiculous and often impossible words she coined. She never allowed herself to be stopped in the flow of her conversation by any such minor considerations as the correct choice of words" ([7], p. 61). 8. Kovalevskaya afterward made an addition to this sentence and then added about three more words that are impossible to decipher. 9. Possibly she meant to add two more words to complete the sentence. 10. Kovalevskaya added this sentence in which approximately four words appear to be missing. 11. The details given here about her arrival in Stockholm are entirely new. Anne Charlotte's biography merely states: "She arrived from Finland in the evening by boat, and came as a guest to my brother [Mittag-] Leffler's house" ([7], p. 55). According to the recollections of Anne Charlotte, she first greeted Sonya only on the morning of the following day. 12. On 19 November 1883, the Swedish newspaper Dagens Nyheter carried an article about Kovalevskaya entitled THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
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New, softcover edition available m H.-D. Ebbinghaus and H. Hermes, Universit~it Freiburg; F. Hirzebruch, Max-Planck-Institut fiir Mathematik, Bonn; M. Koecher (1924 - 1990) and R. Remmert, Universit~it MOnster; K. Mainzer, Universit~it Augsburg; J. Neukirch, Universit~it Regensburg; and A. Prestel, Universit~it Konstanz, all FRG With an Introduction by K. Lamolke English Edition edited by J. Ewing Translated by H.L.S. Orde
16. 17.
" A n Important Guest in Stockholm." The report began with the words: "This does not have to do with an insignificant king or prince from some friendly nation but rather a queen from the empire of science." Before giving a few details about the m o n a r c h herself, the article pointed out that an earlier published account in another newspaper indicating that the Russian widow Kovalevskaya had come to teach as a "Privatdozent at the Stockholm H6gskola" was incorrect, and her teaching activity had been arranged privately with Mittag-Leffler for a specially selected group. Anne Charlotte recalled this social gathering in these words: "My brother, as soon as she arrived, told her that he wanted to give a soiree in order to introduce her to his scientific friends. But she begged him to wait until she could speak Swedish. This seemed to us rather optimistic, but she kept her word. In a fortnight she could speak a little, and during the first winter she had mastered our l i t e r a t u r e . . . ' " ([7], p. 58). The reference is to Eric Nordenskj61d (1832-1901), Swedish geologist, geographer, and Arctic explorer. This couple was Hugo Gyld6n (1841-1896) and his wife Teresa. Gyld6n, a Swedish astronomer, was a wellk n o w n authority on celestial mechanics. Nordenskj61d and Gyld6n later played a vital role in securing Kovalevskaya's appointment to a five-year professorship beginning in June 1884. Following Kovalevskaya's death, Teresa Gyld6n cared for Sonya's daughter until she had completed primary school in Sweden, after which she returned to Moscow. Anne Charlotte reported that during the first weeks of her stay, Sonya did nothing but read Swedish from morning to night (cf. note 13). Meaning unclear.
Numbers This is a book about the system of numbers - - all kinds of numbers, from integers to p-adics, from rationals to octonions, from reals to infinitesimals. Who first used the standard notation for pi? Why was Hamilton obsessed with quaternions? What was the prospect for "quaternionic analysis" in the 19th century? This is the story about one of the major threads of mathematics over thousands of years. It is a story that will give the reader both a glimpse of the mystery surrounding imaginary numbers in the 17th century and also a view of some major developments in the 20th century. 1991/395 pp., 24 illus./Softcover/$39.00/ISBN 0-387-97497-0
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References 0. O. R. B611ing (ed.), Briefwechsel zwischen Karl Weierstrass und Sofja KowaIewskaja, Berlin: Akademie-Verlag (to appear in 1993). 1. R. Cooke, The Mathematics of Sonya Kovalevskaya, N e w York/Berlin: Springer-Verlag (1984). 2. I. Grattan-Guinness, Materials for the history of mathematics in the Institut Mittag-Leffler, Isis 62(3) (1971), no. 213, 363-374. 3. A. H. Koblitz, A Convergence of Lives: Sofia Kovalevskaia: Scientist, Writer, Revolutionary, Boston/Basel: Birkh/iuser (1983). 4. P. Ya. Kochina-Polubarinova (ed.), Pisma Karla Veiershtrassa k Sofye Kovalevskoy, Moscow: Nauka (1973). 5. P. Ya. Kochina, Sofya Vasilevna Kovalevskaya, Moscow: Nauka (1981). 6. P. Ya. Kochina and E. P. Ozhigova (eds.), Perepiska S. V.
Kovalevskoy i G. Mittag-Lefflera, Nauchnoye nasledstvo (A. P. Yushkevich (series ed.), Moscow: Nauka (1984), Vol. 7. 7. A. C. Leffler, Sonya Kovalevsky, London: Fisher Unwin (1895).
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Abraham Ezechiel Plessner (1900-1961): His Work and His Life Dieter Gaier
As an analyst w h o has worked at the University of Giessen for more than 30 years, I became interested in the mathematician Abraham Plessner for three reasons. 1. He is well k n o w n for his work on the summability of trigonometrical series, especially of conjugate Fourier series. 2. Every function-theorist knows Plessner's theorem relating to the b o u n d a r y behavior of functions meromolphic in the unit disc; it is often quoted and has been generalized in various directions. 3. Plessner received his doctoral degree from the University of Giessen in 1923. His dissertation, written under Ludwig Schlesinger (1864-1933), was published as Nr. 10 of our "Mitteilungen aus dem Mathematischen Seminar Giessen." This publication was founded in 1921 by Engel and Schlesinger, then directors of the Seminar, and has survived until now as a small but widely-distributed publication. Plessner stayed in Giessen until 1930 w h e n he tried to become Privat-Dozent at the university. These efforts were, however, without success, as we shall see below. After that, he seemed to have disappeared from the scene. What happened to him? Was he a victim of the Holocaust? Did he remain active as a mathematician?
coefficients of a function f E L1. The general problem studied in Plessner's dissertation and in his Habilitationsschrift is: Under what assumptions on (1) can we draw certain conclusions about (2)? In his dissertation (accepted 1922) Plessner assumes that (1) is a FS and asks for convergence or summability of (2). For example he shows: If (1) is a FS, then (1) and (2) are summable almost everywhere (a.e.) by the methods of Ces~ro (C,1), Abel, and Riemann. The sum for (1) is a.e. equal to f(x) and for (2) to f'(x), where 1 f_-~ x-t f(x): = ~ "rr fit)cot ~
dt
exists as a Cauchy principal value integral a.e. in (0,2~).
Mathematical Research Before 1930 On trigonometrical series.
We let
E (a,, cos nx + b n sin nx)
(1)
E (-b,, cos nx + a n sin nx)
(2)
and
be a trigonometrical series (TS) and its conjugate series (CTS). (1) is a Fourier series (FS) if the a n, bn are Fourier THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3 9 1992 Springer-Verlag New York
31
P l e s s n e r ' s Habilitationsschrift [6] was n e v e r p u b lished. Some of its results w e r e published in [5] a n d [7] w i t h o u t proofs. Marcinkiewicz writes in his review of [7] for the Jahrbuch, "This is an i m p o r t a n t t h e o r e m w h i c h has b e e n attacked b y mathematicians for a long time."
Abraham Ezechiel Plessner. Also, c o n v e r g e n c e factors are studied: If (1) is a FS, then
~P/z_,~anCOSnx + bnsin nx) 9
1 log
(1') n
and
s
1 nx + ansin nx) 9 log n
B o u n d a r y Behavior of H o l o m o r p h i c Functions. H e r e our starting point is Fatou's t h e o r e m (1906): If f is h o l o m o r p h i c and b o u n d e d in D, t h e n f has angular b o u n d a r y values a.e. on 3D. That is, for almost all ~COD the limit lim f(z) exists for z -+ ~, z E A, w h e r e A is a n y (symmetric) angle in D with vertex at ~, also called a "Stolz angle." W h a t can we say if the condition "f b o u n d e d in D" is r e m o v e d ? To formulate the result, we say that ~ E OD is a Fatou point for f if lim f(z) exists in C for z ~ ~, z E . A, for e v e r y Stolz angle A with vertex at ~. In this case f b e h a v e s "nicely" near ~. In contrast w e say that ~ @ 0D is a Plessner point for f if for each Stolz angle A with vertex at ~, and for each w 0 E C, there are points z n A with zn --~ ~ for which lim f(Zn) = w o. In this case f behaves " b a d l y " near ~. Plessner's theorem (1927) n o w asserts: If f is m e r o m o r phic in D, then those points o n aD which are neither Fatou points nor Plessner points, form a set N of Lebesgue m e a s u r e 0. In o t h e r words, 0D can be d e c o m p o s e d OD = F U P U N, w h e r e F and P contain the Fatou and Plessner points, respectively, and w h e r e N is of m e a s u r e 0. A n easy consequence is the following identity theorem, also due to Plessner, generalizing an earlier theo r e m of Lusin and Priwalow: If f is m e r o m o r p h i c in D, and if there is a set E C 3D of positive measure such that for each ~ E E there exists a Stolz angle A (how-
(2')
converge a.e. in (0,2~r). There is a remarkable b y p r o d u c t that b e c a m e useful in the s t u d y of H p spaces: If u is a h a r m o n i c function in the unit disc D with
f+j_ lu(r, rb)lddo <~ M
(0 < r < 1),
t h e n u = u 1 - u 2 w h e r e u I and 1/2 are positive harmonic functions in •. In his Habilitationsschrift (1929) it is no longer ass u m e d that (1) is a FS. Plessner s h o w s that if (1) is a TS that is summable by the m e t h o d (C,k) with k = 1,2 . . . . . on a set E of positive Lebesgue m e a s u r e , then the conjugate series (2) is almost e v e r y w h e r e summable (C,k) o n E.
Plessner's doctoral diploma. 32
THE MATHEMATICAL 1NTELLIGENCER VOL. 14, NO. 3, 1992
ever narrow) with vertex at ~ for which lim f(z) = 0 for z---*4, z E A , t h e n f = 0. Plessner's theorem became the starting point for many investigations regarding boundary behavior of holomorphic functions in D. Here we describe its generalization to functions of n complex variables; see Rudin [A7]. We consider n-tuples z = (Zl,Z2. . . . . z,) of complex numbers and let
Izl = (X Izj[2)'2, B = {z E C": IzI < 1}, s = {z E G": IzI = 1}. We assume that f maps B into C so that f becomes holomorphic in one variable if any n - 1 of the varizn are kept fixed. The analog of a ables zl,z 2. . . . . Stolz angle in I~ is n o w a Kordnyi approach region D~(O for any ~ E S and parameter c~ > 1. (Unexpectedly, these regions D~(4) permit tangential approach to the point 4 provided that n > 1.) We now say that 4 E S is a Fatou point for f if lim f(z) exists finitely for z ~ 4, z E D~(O, and for each oL > 1; whereas ~ ~ S is a Plessner point for f if for each o~ > 1, and for each w o E C, there are z E D~(0, z --* 4, for which lim f(z) = w o. In full analogy with Plessner's classical theorem we have the following theorem. THEOREM (Rudin [A7, p.83]). If f is holomorphic in B, then there is a (disjoint) decomposition S = F U P U N, where F and P contain the Fatou and Plessner points of f, respectively, and where N is a Lebesgue-null set. Even more can be said: Every (smooth) Jordan arc J c S can be decomposed into three disjoint sets F, P, N. On F limits exist, for restricted approach to 4 E F, while the cluster set is C for suitable approach to 4 E P, and N is again a null set. See Beatrous-Li [A2].
of his n e w student, and after three semesters in Giessen recommended he study in G6ttingen and Berlin. So during the summer of 1921 we see Plessner in G6ttingen. The records show that he took courses by Landau on Dirichlet series and on Galois theory, by Noether on the theory of algebraic number fields, and by Courant on calculus of variations. During the winter semester 1921/22 he went to Berlin to hear von Mises on differential and integral equations and Bieberbach on differential geometry. He also had a seminar with Schur. Coming back to Giessen, he presented his dissertation: "Zur Theorie der konjugierten trigonometrischen Reihen," and passed his doctoral examination in August 1922. During the winter semester 1921/22 Schlesinger had given a course on "Lebesguesche Integrale und Fouriersche Reihen." He now was preparing a book on that subject. With a stipend from the state of H e s s e n ("Sonderzuschuss" 150,000 Marks---but this was at the height of German inflation!) Plessner cooperated with Schlesinger, and the book appeared in 1926. From 1925 to 1928 Plessner was Privat-Assistent of the algebraist Hensel in Marburg; his job was to assist Hensel in editing the collected works of L. Kronecker. It was during this stay at Marburg that Plessner published his famous paper [3] on what is n o w known as Plessner's theorem, and in 1967 this was translated into Russian. In August 1928 Plessner returned to Giessen to become Privat-Assistent of Schlesinger with a small remuneration as Hilfs-Assistent. Incidentally, we should note that the Mathematisches Seminar had only one assistant at that time. The position was occupied by Harald Geppert from 1924 to 1930, and by Herbert Gr6tzsch from 1930 to 1935, after Geppert became Schlesinger's successor in 1930.
P l e s s n e r ' s Life U n t i l 1929 Plessner was born February 13, 1900 in L6d~., a large textile-manufacturing town belonging to Russia at that time. His father Ezechiel was a merchant and had a factory. In June 1918, after 9 years of secondary education at the " ' O b e r r e a l s c h u l e d e s V e r e i n s zur F6rderung mittlerer Handelsbildung," he got the Abitur. It is interesting to note that the language at that school was Russian up until 1916, then German (during the German administration) for one year, and then Polish after 1917. His diploma, written in Polish and signed by all of his teachers, is still in the archives of the University of Giessen. In June 1919 Plessner arrived in Giessen and began to study mathematics with Schlesinger and Engel. Schlesinger, the analyst, soon discovered the potential
Transcript of Plessner's courses. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
33
The Campaign to Make Plessner Privat-Dozent On February 12, 1929 Schlesinger and Engel submitted Plessner's Habilitationsschrift [6] to the faculty. Here they write: " . . . Never have we met a student who has acquired such extensive knowledge of mathematical works within a few semesters . . . . we became convinced that Plessner will become a brilliant mathematician . . . Considering his (financially) insecure position it is astonishing that he had the strength to make deep speculations and we e x p e c t . . , that he will make further most important contributions." Fifteen days later Plessner presents a lecture "Uber neuere U n t e r s u c h u n g e n zu den G r u n d l a g e n der Mathematik'" before the faculty to prove his teaching abilities, and the faculty is satisfied ("ausreichend") although six out of fourteen professors abstained. Thus, having presented his thesis and having passed his lecture, Plessner qualified himself to become PrivatDozent. (This would entitle him to give courses at the university.) However, the Senate of the University of Giessen still had to give its approval, and here the drama begins: Plessner was a Russian citizen! 1 And although the University laws (Habilitationsordnung of 1911) did not formally require German citizenship to get the venia legendi, the Senate voted 27 to 18 that Plessner should first acquire German citizenship. When Plessner applied with the city administration, he got the answer that his application would have a chance only after 20 years of continuous stay in Germany. Thereupon Schlesinger and Engel appealed to the Rector of the university that it was unfair to postpone Plessner's appointment that long, and the Rector asked the city to speed up the procedure. In the meantime, Plessner moved to Berlin (June 1929), because--according to Schlesinger he found better possibilities there for earning a living. And sure enough, his application came back without any action from the city administration because it was no longer their responsibility to deal with the case. On January 24, 1930 Engel and Schlesinger launched another attempt with the Rector: The Senate should give Plessner the venia legendi without the stipulation of German citizenship. The Rector asked two members of the Senate to report on the matter. One of them favored Plessner's appointment warmly, the other recommended that the Minister of the Interior of the state of Hessen be consulted about the question of citizenship. Finally, on February 26 the Senate voted to postpone the whole matter until Schlesinger's successor (Geppert, see above) could be heard. This was the end of 1 The records of t h e city of Giessen dearly s h o w that Plessner had Russian citizenship as early as 1921, most probably even w h e n he came to Giessen in 1919. In [A4] it is reported that Plessner became a Soviet citizen in 1922.
34 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3, 1992
the story. Plessner had completed all requirements but never obtained the venia legendi in Giessen. All prospects for a scientific career in Germany vanished.
Plessner in M o s c o w We do not know how long Plessner stayed in Berlin since the city records were destroyed in the war. However, according to [A9], p. 544, Plessner worked after 1932 in the Mathematics department of the University of Moscow as well as in the Mathematical Institute of the Academy. In [A4] one can read that for some years he belonged to the editorial staff of a scientific publisher. In 1935 he got the degree of Dr. fiz.-mat, nauk, and in 1938 the title of professor. Why did Plessner move to Moscow? There was a practical reason--he was a Soviet citizen, as we noted earlier. More important for him, we assume, were scientific reasons. Under the leadership of Egorov and Lusin, Moscow University had developed in the 1920s into a very lively center of research in analysis. In particular, Lusin had created a school of many outstanding mathematicians ("Lusitania") including P. S. A l e k s a n d r o v , N. K. Bari, L. V. Keldych, M. A. Lavrentiev, A. A. Lyapunov, L. A. Lyusternik, D. E. Menshov, L. G. Schnirelman, M. Y. Suslin, and P. S. Urysohn. As Phillips [A5] writes: "There was a veritable explosion of mathematical activity at Moscow University in the 1920s." Lusin's mathematical interests were to a great extent the same as those of Plessner. Thus, the title of Lusin's dissertation (1915) "The integral and trigonometrical series" is almost identical with that of the book which Schlesinger and Plessner published in 1926, and the identity theorems of Lusin and Priwalow treat the same questions as Plessner did later. So it seems quite natural that Plessner joined this active group of researchers. On the other hand, it is somewhat striking that he changed his field of interest radically after coming to Moscow. For the rest of his life he worked almost exclusively in functional analysis and here, in particular, in spectral theory. Fortunately, I am in possession of two letters 2 from contemporaries of Abraham Plessner who had k n o w n him personally during his time in Moscow. They are: Boris Yakovlevich Levin and Boris Moisejevich Levitan. For example, Levin writes that Plessner is considered by Soviet mathematicians as one of the founders of the Moscow school of functional analysis, and that Plessner had an "immense reputation." 3 Levin continues: 2 Both letters were sent to S. L. Kruschkal in 1989 a n d translated b y R. K~ihnau for use in this article. 3 In [A1], p. 41, it is m e n t i o n e d that Plessner had a big influence in forming the Moscow school of functional analysis, and that m a n y y o u n g mathematicians including Gelfand attended his lectures a n d seminars.
It was in 1934, shortly before the second Congress of Mathematicians. I was sitting in the mathematical cabinet of the University when Gelfand (who was an aspirant like myself) approached me, showed me Banach's book "Throrie des oprrations linraires," and said: "Plessner says one must study this." Possibly we see here the trigger to Plessner's surprising change of interests: He was fascinated by Banach's book of 1932. After 1939 several Doklady notes and three long and fundamental survey articles on spectral theory appeared that were based on lectures Plessner had given in 1938 and 1939 at Moscow University. They deal with maximal operators, Hermitian operators, the conjugate graph of an operator and the like. Eduard Tsekanovskii told me during my recent visit to Donetsk that the papers [13] and [14] were "the Bible for the Russian students who wanted to learn spectral theory," and indeed [14] and [15] were translated into English 20 years after the original appeared. Incidentally, Plessner was also invited to other universities to lecture on spectral theory, for example by Achieser to Kharkov, as Levitan remembers. Levin mentions in his letter that Plessner was very critical of himself and of the work of others. He loved the phrases "this is trivial" and "this is false." The aspirants of Moscow University had unlimited respect for him, but proposed the following definition: "Paradise in the sense of Plessner is an abstract space in which all theorems are both false and trivial." Plessner began to write his book Spectral Theory of Linear Operators around 1948, but had to interrupt his work frequently because of illness during the next ten years. It was completed by L. M. Abramov and B. M. Makarov after Plessner's death in 1961 and it appeared in 1965. Virtually u n k n o w n is Plessner's contribution to the founding of the theory of Banach algebras. According to Levitan, Plessner was possibly the first person who expressed the need for a theory of normed rings. In the early 1930s he realized the unique connections between functional analysis and abstract algebra. Levitan continues, "The first paper (at least in the USSR) on the theory of Banach algebras was written by V. A. Ditkin [A3], an aspirant of Plessner's, and only afterwards the remarkable works of Gelfand and ~ilov appeared." [We note that the latter papers appeared at about the same time as Ditkin's, and they do not mention the aspirant Ditkin.] During the war years, Plessner and the Mathematical Institute of the Academy were evacuated to Kazan, and after the war there is only one more Doklady note [16] and of course work on the book we mentioned above. In 1949 Plessner was dismissed from both of his posts, at the Academy and at the University. To understand this, one must know (Levin) that during the period from 1949 to 1952 I. M. Vinogradov was direc-
tor of the Steklov Institute as well as Vice-Rector for "Kader" at the University. The next ten years were extremely difficult for Plessner. Having been in service for only about 17 years, his pension was very small. In addition, his health was failing. In 1922 he had to undergo a complicated knee operation and was in the University hospital for over 7 months. Professor M.-P. Geppert (now at Tiibingen) recalls that already in 1925, w h e n she was a first-year student at Giessen, Plessner had to use a cane. Now he had increasing difficulty standing at the blackboard. Levin adds, "His lectures were always a little confused and unclear, and hence he could not lecture at a Polytechnic School." Some of his pupils were V. A. Ditkin, V. A. Rohlin, G. A. Suhomlinov, and A. A. Gurevich. Plessner was married to Nina Andreevna (t 1982) but they had no children. Plessner died on April 18, 1961. Once more Levin: "Abram Ezekiilovich knew so much that it seemed he knew everything. He understood works in any field and every young mathematician tried to tell him of his new results." And Levitan: " O n l y if we look back do we realize the important role played by A. E. Plessner in the development of mathematics in the Soviet Union."
Acknowledgment The author would like to thank his friend Bob Burckel for his assistance in the preparation of this article.
Bibliography 1. Plessner, A. "Zur Theorie der konjugierten trigonometrischen Reihen," (Dissertation). Mitt. Math. Sem. Giessen 10 (1923), 1-36, JB 49, 204. 2. - "Uber Konvergenz von trigonometrischen Reihen," J. Reine Angew. Math. 155 (1926), 15-25. JB 51, 220. 3. - "l]ber das Verhalten analytischer Funktionen am Rande ihres Definitionsbereichs," J. Reine Angew. Math. 158 (1927), 219-227. JB 53, 284. Russian Translation: Uspehi Mat. Nauk 22, no. 1 (1967), 125-136. Zb1177, 106; MR 34, 824. 4. - "Eine Kennzeichnung der totalstetigen Funktionen," J. Reine Angew. Math. 160 (1929), 26-32. JB 55, 143. 5. - "Trigonometrische Reihen," In: Repertorium der h6heren Analysis. E. Salkowski, ed. Leipzig-Berlin: B. G. Teubner, 1929. Kap. XXV, pp. 1325-1396. JB 55, 119. 6. - "Ober Summierbarkeit der trigonometrischen Reihen durch arithmetische Mittel," Unpublished manuscript (Habilitationsschrift), 1929. 54 pages. 7. - 'q[lber konjugierte trigonometrische Reihen," Dokl. Acad. Sci. URSS (N.S.) 9 (1935), 251-253, JB 61, 1123; Zbl 13, 350. THE MATHEMATICAL [NTELLIGENCER VOL. 14, NO. 3, 1992
35
8. - "Ober das Gesetz der grot~en Zahlen," Mat. Sbornik I (1936), 165-168. JB 62, 1335; Zbl 14, 168. 9. " Z u r Spektraltheorie maximaler Operatoren," Dokl. Acad. Sci. URSS (N.S.) 22 (1939), 227-230. JB 65, 1319; Zbl 20, 369. 10. - "Uber Funktionen eines maximalen Operators," Dokl. Acad. Sci. URSS (N.S.) 23 (1939), 327-330. JB 65, 1319; Zbl 21, 234. 11. - "Uber halbunit~ire Operatoren." Dokl. Acad. Sci. URSS (N.S.) 25 (1939), 710-712. JB 65, 1320; Zbl 22, 357; MR 1, 338. 12. - "[Voer die Einordnung des Heavisideschen Operationskalk~ils in die S p e k t r a l t h e o r i e m a x i m a l e r Operatoren," Dokl. Acad. Sci. URSS (N.S.) 26 (1940), 1012. JB 66, 548; Zbl 22, 356; MR 2, 311. 13. "Spectral theory of linear operators. I," (Russian). Uspehi Mat. Nauk 9 (1941), 3-125. MR 3, 210. 14. Plessner, A. E. with V. A. Rohlin, "Spectral theory of linear operators. II," (Russian). Uspehi Mat. Nauk I (11), no. 1 (1946), 71-191. MR 9, 43. English translation: AMS Translations, Ser. 2, 62 (1967), 29-175. 15. Plessner, A. E., "Fundamental ideas of the spectral theory of Hermitian operators," (Russian). Uspehi Mat. Nauk 1 (11), no. 1 (1946), 192-216. MR 9, 43. English translation: AMS Translations, Ser. 2, 62 (1967), 1-28. 16. - "The structure of the conjugate graph of a selfadjoint operator," (Russian). Dokl. Akad. Nauk SSSR (N.S.) 66 (1949), 557-560. Zbl 40, 61; MR 10, 718. 17. Plessner, A. I., Spectral theory of linear operators (Russian), Moscow: Izdat. "Nauka", 1965. Zbl 147, 343; MR 33 # 3106. English translation: New York: F. Ungar Publishing Co., 1969. Zbl 188, 444; MR 39 # 6106. 18. Schlesinger, L. with A. Plessner, Lebesguesche Integrale und Fouriersche Reihen, Berlin-Leipzig: deGruyter, 1926 JB 52, 237.
Additional References A1. Aleksandrov, P. S., "Mathematics in the University of Moscow'during the first half of the 20th century" (Russian). Istor.-Mat. Issled. 8 (1955), 9-54. MR 17, 1. A2. Beatrous, F.; Li, S., "A Plessner decomposition along transverse curves," Can. J. Math. 40 (1988), 1243-1255. MR 89i # 32009. A3. Ditkin, V., "On the structure of ideals in certain normed rings" (Russian). U~en. Zapiski Moskov. Gos. Univ. Matematika 30 (1939), 83-130. MR 1, 336. A4. Ljusternik, L. A., Men~ov, D. E., Naimark, M. A., Uljanov, P. L., "Abram Iezekiilovi~ Plesner, On his sixtieth birthday" (Russian). Uspehi Mat. Nauk 16 (97), no. 1 (1961), 213-218. MR 23A # 1508. A5. Phillips, E. R., "Nicolai Nicolaevich Luzin and the Moscow school of the theory of functions," Historia Math. 5 (1978), 275-305. MR 81c # 01027. A6. Pommerenke, Ch., Univalent functions, G6ttingen: Vandenhoeck & Ruprecht, 1975. MR 58 # 22526. A7. Rudin, W., Function theory in the unit ball of C', New York: Springer, 1980. MR 82i # 32002. A8. Shields, A. "Luzin and Egorov," Math. Intelligencer 9, no. 4 (1987), 24--27. MR 89b # 01060. Part 2, ibid. 11, no. 2 (1989), 5-8. MR 90h # 01015. A9. Mat 40: Matematika v SSSR za 40 let: 1917-1957 (Russian). Gos. Izdat. Fiz.-mat. Lit. Moscow 1959. Mathematisches Institut der Universitdt D-6300 Giessen Federal Republic of Germany 36
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
Minimal Surfaces, Crystals, Shortest Networks, and Undergraduate Research Frank Morgan
S e c t i o n 1. M i n i m a l
Surfaces
What is the least-area surface on a tetrahedral frame? (See Fig. I.) Because soap film tends to minimize surface energy or area, a good candidate may be obtained by dipping a tetrahedral wire frame in soap solution. Six triangular pieces of film stretch from the six edges to the singular point at the center. They meet in threes along four seams running from the four vertices of the tetrahedron to the center. This surface is called the cone
over the tetrahedral frame. Only fairly recently have mathematicians vindicated the soap film's behavior: T H E O R E M 1.1. (Taylor [19, W.6], Lawlor, Morgan
[10, 1.1]). The surface of the cone over the regular tetrahedron in R 3 (R ") is area-minimizing. Admissible surfaces are assumed to have the given boundary and separate the four faces of the tetrahedron. Taylor's original proof was only by the process of elimination. More recent work of Lawlor and me has produced the following simple direct proof, which generalizes to R n (the hypercone over the regular simplex in R" is area-minimizing). So let us start right off with a proof and enjoy greater leisure thereafter.
Figure 1. This soap film on a tetrahedral frame is areaminimizing. Admissible surfaces are assumed to have the given boundary and separate the four faces of the tetrahedron. Photograph by F. Goro. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3 9 1992 Springer-Verlag New York
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Figure 2. The proof uses the four equidistant points Pi at the centers of the faces of the tetrahedron. Note that Pl - Pj is normal to the triangular piece of cone surface it pierces. The proof considers the flux of the constant vector fields Pi and Pi - Pj.
Figure 3. It is an open question whether there is a smaller surface of higher topological type which does not separate space into four regions. (Computer graphics by Jean Taylor, Rutgers University and The Geometry Center, after an idea by R. Hardt.)
Proof: Let Pi be the point at the center of face F i. See Figure 2. By scaling, we may assume the Pi are unit distance apart. The vector Pi - P / f r o m Pi to pj is the unit normal to the triangular piece of the cone surface which it pierces. N o w consider any competing surface S dividing space into regions labeled R i according to which face F i they contain. Denote by Sq the part of S separating R~ from Rj. N o w we will consider the constant vector fields Pi and argue that
Note how the proof makes essential use of the four (pairwise) equidistant points Pi at the vertices of a regular tetrahedron dual to the given tetrahedron. The similar proof that the cone hypersurface over the regular simplex in R n is area-minimizing uses n + 1 points Pi at the vertices of a dual regular simplex. This write-up deliberately suppresses concern about -+ signs; rest assured, they work out OK.
area S = ~
area Sij
~ (flux (pi - pj) through Sij) ij = ~
(flux pi through Fi). i
The inequality follows from the fact that Pi - Pj is a unit vector field. Equality holds only if the vector field is normal to the surface, as it is for the triangular pieces of the cone. To obtain the last equality, focus on the occurrences of Pi in the second-last expression, which contribute the flux of the constant vector field Pi out of R i through all the bounding surfaces Sq. Because it is a constant vector field, this contribution equals the flux of Pi into R i through the face F i, yielding the last equality. Note that the final expression does not mention the surface S; it is just a constant. Because the area of any admissible surface S is greater than this constant, with equality for the cone surface, the cone surface is areaminimizing. 38
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
Remarks.
O p e n Q u e s t i o n . Theorem 1.1 admits only surfaces that separate the four faces of the tetrahedron. It is an open question whether there might be a smaller surface of higher topological type that does not separate the faces. Figure 3 imagines such a surface with tunnels connecting the faces in pairs, front to back and right to bottom. Actually, for general theoretical reasons any other soap film (without bubbles) on the whole tetrahedral frame would have to have less area than the cone surface. Therefore, this open question is one that any child might solve--by dipping a tetrahedral frame into soap solution and discovering another surface.
S e c t i o n 2. Crystals a n d N o r m s o n R"
Crystals. Crystals try to minimize a surface energy which depends on the direction of the unit normal n with respect to the underlying crystal lattice. For many crystals, certain directions are very cheap, and these are the only directions occurring in the crystal surface: The crystal is a polyhedron. For example, if you look closely, you can tell that a salt crystal is a tiny cube. See
It follows immediately that x.y
~ ~(x) cI)*(y).
(1)
If equality holds, one says that x and y are dual. If 9 is just Euclidean length, then cI)* is also just E u c l i d e a n l e n g t h . The i n e q u a l i t y is the f a m o u s Schwarz inequality that the dot product of two vectors is less than or equal to the product of their lengths, with equality if the two vectors are parallel. Thus, dual is just a generalization to other norms of parallel. Theorem 1 on area-minimizing surfaces generalizes to cI)-minimizing surfaces, w h e r e (I) is a norm on, say, R 3 and the energy ~(S) of a surface S is given by r
=
fs r
with n the unit normal to the surface.
Figure 4. Many crystals are polyhedralmhave finitely many faces---because the surface energy for those orientations happens to be very cheap. (Photographs of S. Smale [20] and E. Breiskorn, from Mathematics and Optimal Form by S. Hildebrandt and A. Tromba [9].)
Figure 4. For other materials, such as solid helium, the energy is nearly isotropic and the crystal is smooth and almost round. In the borderline case, as for pivalic acid, the crystal is s m o o t h but distinctly n o n r o u n d (see Fig. 5). Compare also [18]. Norms. Consider a general surface energy given by a norm cI)(n) on vectors n in R 3 or R n. A norm c~: R ~ -~ R is h o m o g e n e o u s [cI)()~x) = )~cI)(x)] and convex [(I)(x + y) ~< ~(x) + (I)(y)]. A n o r m is characterized b y its unit ball, the set of all vectors of norm at m o s t 1. For the unit norm ball of Figure 6, the vertical direction is cheap because y o u can go far with unit cost. The horizontal direction is expensive. The convexity of a norm is equivalent to the convexity of its unit ball. The s m o o t h n e s s of a n o r m (always ignoring the nonsmooth point 0) is equivalent to the s m o o t h n e s s of its unit ball. Norms come in pairs. Associated to a n y n o r m 9 is a dual norm ~* defined by (I)*(y) = sup{ x 9y : cI)(x) ~< 1 }.
Figure 5. A pivalic acid crystal is smooth but not round, a kind of interpolation between round and diamond shapes. (Photograph from Glicksman and Singh [7].)
Figure 6. A norm ~ is characterized by its unit ball, the set of all vectors of norm at most 1. For this unit ball, the vertical direction is cheap because you can go far with unit cost. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
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THEOREM 2.1. Sample theorem on ~-Minimizing Surfaces. Given a norm 9 on R 3, if there exist four ~-equidistant points ("an equilateral tetrahedron"), then the cone over the dual tetrahedron is ~*-minimizing. Remark. As before we admit only surfaces which separate the regions containing the four faces of the tetrahedron. The proof is a minor variation on the proof of theorem 1.1 (see Fig. 7). Proof: We may assume the Pi are ~-unit distance apart. Consider a "dual" tetrahedron such that a triangular piece of the cone surface has normal dual to Pi - Pj. N o w consider any competing surface S dividing space into regions labeled Ri according to which face Fi they contain. Denote by Sly the part of S separating R i from Rj. N o w we will consider the constant vector fields Pi and argue that
9 *(s) =
0
=
=
fs
X fs,,(Pi-Pl)'n = ~ (flUX (pi -- pj) through Sij) /j = ~
(flux pi through Fi). i
The first two equalities are immediate from the definitions. The third equality follows from the fact that Pi pj is a D-unit vector field. The inequality is the basic norm inequality (1) earlier in this section. Equality holds only if the vector field is dual to the normal to the surface, as it is for the triangular pieces of the cone. The second-last equality is the definition of flux. To obtain the last equality, focus on the occurrences of Pi in the second-last expression, which contribute the flux of the constant vector field Pi out of Ri through all the bounding surfaces Sq. Because it is a constant vector field, this contribution equals the flux of Pi into R; through the face F i, yielding the last equality. Note that the final expression does not mention the surface S; it is just a constant. Because the D-energy of any admissible surface S is greater than this constant, with equality for the cone surface, the cone surface is ~-minimizing. A natural question about Theorem 2.1 asks h o w many norms 9 satisfy the hypothesis that four cI)-equidistant points exist. The following theorem due to C. M. Petty says that all norms do. THEOREM 2.2. [16, Theorem 4]. For any norm 9 on R 3, there are four C~-equidistant points. Interestingly enough, the result breaks down in higher dimensions: Open Question. Are there n + 1 equidistant points for any smooth norm on R"? The question is open even for R 4. There are some more surprises: Proposition [10, 3.4]. For some smooth norm c~ on R 3, there are five equidistant points. Consequently, the cone over a dual triangular prism is ~*-minimizing. The five points and the unit D-ball can be as in Figure 7. At the vertex of the minimizing cone, nine triangular surfaces and six curves meet at a point. This is a n e w kind of singularity that does not occur in soap films. It seems to have been recognized as a minimizer in mathematics before being discovered in nature! Open Questions. Can there be six equidistant points for a smooth norm on R3? H o w many in Rn? For the nonsmooth norm on R 3 with cubical unit ball, the eight vertices of the cube are equidistant. In general, for arbitrary norms, 2 n gives a sharp b o u n d on the number of equidistant points in R" [16, Theorem 4]. S e c t i o n 3. M i n i m a l N e t w o r k s
Figure 7. For this unit @-ball, these five points are equidistant. Consequently, the cone over the dual triangular prism is ~*-minimizing---a new kind of singularity. 40
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
Warm-up Question. What is the shortest network connecting the four corners of a square?
Solution to Old Puzzle: How Short a Shortcut? By GINA KOLATA
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design of networks thai has enormous practical importance but has baffled some of the
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sharpest minds in the business. Dr. Frank Hwang of A.T.&T. Bell Laboratories in Murray q i l l N.J., and Dr. Ding Xhu Du, a pc._,doctoral student at Princeton University, announced at a meeting of theoretical computer scientists .last week that they had found a precise limit to the design of paths connecting three or more points. Designers of things like computer circuits, long-distance telephone lines, or mail reutings place g r e a t
The Shorter Route Thelength of string needed to join the three comers of a triangle is shorter if a point is added in the middle, as in the figure at right.
stake in finding the shortest path to connect a set of points, like the cities an airline will travel to or the switching stations for long-distance tele-
Figure 8. The shortest network connecting the four corners of a unit square is the double Y. In general, lengthminimizing networks meet in threes.
One possibility is the U, of length 3 (see Fig. 8). A very good solution has four segments meeting at the center in an X, of length 2V2 ~ 2.82. The best solution of all is a double Y, of length V~ + 1 ~ 2.73. Indeed, it is a classical fact that at interior junctions
length-minimizing networks meet only in threes (cf. [6, pp. 354-361, 3921). The interstate highway system probably should look more like Figure 9. A recent article in The New York Times described a new theorem on the value of using such triple junctions (see Fig. 10). This principle can be well illustrated by soap films. Etch a map of the United States on parallel plexiglass plates, joined by plastic pins at the locations of cities. Dip it in soap solution. The soap film, in its attempt to
phone lines. Dr. H w a n g and Dr. Du proved, without using a n y calculations, that an old trick of adding extra points to a network to make the path shorter can reduce the length of the path by no more than 13.4 percent. " T h i s problem has been open for22 years," said Dr. Ronald L Graham, a research director at A.T.&T. Bell Laboratories who spent years trying in vain to solve iL ':The problem is of tremendous interest at Bell Laboratories, for obvious reasons."
In 1968. two other mathematicians guessed the proper answer, but until now no one had proved or disproved Illeir conjecture. In the tradition of Dr. Paul Erdos, an eccentric /-Iunganan mathematician who offers prizes for solutions to hard problems that interest him, Dr. Graham offered $500 for the solution to th|s one. He said he is about to mail his check.
Figure 10. This article from The New York Times, October 30, 1990, described a new theorem of Du and Hwang on the value of triple junctions. Copyright 9 1990 by The N e w York Times Company. Reprinted by permission.
"I
Figure 9. The shortest highway system connecting these cities meets only in threes (From The Shortest-Network Problem, Marshall W. Bern and Ronald L. Graham. Copyright 9 1990 by Scientific American, Inc. All rights reserved. [3]).
Figure 11. A soap film between plexiglass plates exhibits a shortest network, with roads meeting in threes. Photograph from Schecter [17]. (Courtesy of Gordon Graham/Prism.) THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
41
minimize length or energy, connects the cities by a network of roads meeting in threes (see Fig. 11). Now replace length by a norm ~(x) and try to minimize the ~-length of a curve C 9 (C) = f c $ ( T ) , where T is the unit tangent vector. We will assume that is uniformly convex, that is, that the boundary of the unit ~-ball has positive inward curvature, bounded away from 0. This assumption guarantees that a straight line is the unique cheapest path between two points, so that ~-minimizing networks consist of straight line segments, in general, meeting at auxiliary nodes. The following theorem says they still meet in threes, at least for smooth norms in the plane. THEOREM 3.1.
(Adam Levy, Williams '88 undergraduate thesis [11], [2]). If ~ is smooth and uniformly convex, ~-minimizing networks in R2 meet only in threes (at interior nodes). His proof showed that junctions of four or more segments are unstable. Only recently has Levy's theorem been generalized to higher dimensions. For length, the bound of 3 holds in all dimensions, but for the class of differentiable norms cI), n + 1 is the sharp bound on the number of segments meeting at a node in a cI)-minimizing network in R" [10, Theorem 4.4]. The summer after his thesis, Levy led a group in the Williams SMALL undergraduate research project to examine the smoothness hypothesis in his theorem. They came up with the following surprise. THEOREM 3.2. (SMALL Geometry Group 1988: Alfaro, Conger, Hodges, Kochar, Kuklinski, Levy, Mahmood, and yon Haam [1,2].) For some piecewise smooth, uniformly convex norms 9 in R2, rb-minimizing networks sometimes meet in fours (but never in fives or more). Indeed, they showed that an X is minimizing for the unit norm ball of Figure 12. Note that the ball is uni-
8
t-t.
Figure 12. The X is ~-minimizing for this unit ball, which is uniformly convex, and smooth except at the four comers, which stick out to make the diagonal directions, which occur in the X, cheap. Thus, for some uniformly convex norms, minimizing planar networks can meet in fours. 42
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
formly convex, and smooth except at the four comers, which stick out to make the diagonal directions, which occur in the X, cheap. The proof uses some elegant symmetry arguments to reduce the analysis to a calculus problem. These results have just appeared in an article [2] in the Pacific Journal of Mathematics. The authors are distinguished both by being undergraduates and by numbering eight! Another interesting feature of the article is that of the 15 references, 6 are earlier work by undergraduates. E. J. Cockayne [4] had earlier considered uniformly convex norms (I), but did not discuss the dependence on the smoothness of ~. M. Hanan [8] had observed that for the nonuniformly convex "rectilinear" norm (with a square unit ball), minimizing networks in R 2 can meet in fours. (With a cubical unit ball in R", 2" segments can meet at a point. It is an open question whether there ever could be more.) The following year, one member of the group, Mark Conger '89, went on in his thesis [5] to generalize Theorem 3.2 to six segments meeting along the axes in R 3. His conjecture that the sharp bound for uniformly convex norms in R" is 2n remains an open question. I must say that working with these students has helped me in my own work on minimizing surfaces in general dimension and codimension. They do read m y manuscripts, check details, and edit, but their main contributions have gone beyond that. The principles and heuristics they develop working on minimizing networks have given me new ideas and corrected m y prejudices on the higher-dimensional problems. A survey of some of these results appears in [13, Chapter 10].
S e c t i o n 4. U n d e r g r a d u a t e R e s e a r c h
In ever-increasing numbers, undergraduates are doing mathematics, proving theorems, writing papers for publication, and giving talks at mathematics meetings. An active special session at the annual mathematics meetings in San Francisco, January 1991, was dedicated to undergraduate research. In the first two years of our SMALL undergraduate summer research project at Williams, the number of majors per year rose from 11 to 28. We think this increase is partly due to SMALL. The SMALL project involves about 20 students, including a few from outside Williams, awarded about $2500 for 10 summer weeks' work in mathematics and computer science. Each student belongs to two of seven research groups, each with a student leader and a faculty advisor. The student groups work largely independently, although the faculty advisor usually provides the problem and some guidance. The students generally are
quite busy and do not like to be interrupted by the faculty! The less-experienced students learn from the others, and the most advanced like to bounce ideas off the rest. The students range from seniors to freshman, including men and women, blacks and Hispanics, and sometimes students not even intending to major in mathematics or computer science. One woman, who had no intentions of attending graduate school, changed her mind during the summer project and is now a graduate student in computer science at Wisconsin. A nonmajor, who used to consider mathematics just an avocation, is starting graduate school in mathematics at UC Berkeley this fall. A Hispanic student, who never seemed to settle down in his coursework, s u c c e e d e d o u t s t a n d i n g l y in the SMALL project his freshman year and has written an honors thesis in mathematics. Although focused on 10 summer weeks, SMALL activities continue throughout the school year. At one of the first colloquia of the fall, SMALL students describe the project to faculty, friends, and potential n e w SMALL participants. SMALL alumni often participate in colloquia, pursue independent work, or write theses. This year, student colloquium audiences num-
bered 50. Likewise, 50 Williams students applied for SMALL, in addition to the 40 applicants from outside Williams. The demands of the summer project on the faculty advisors are alleviated by the large number of faculty involved (six or seven) and the major role played by the student group leaders. These student leaders, sometimes seniors who have already written honors theses, are the ones who really run the groups. As a result, faculty already immersed in research need only spend about an additional 5 hours a week on the project and can be away for up to 2 weeks at a time. But faculty participation does assume a summer-long commitment to research. Faculty have received modest stipends of about $2500 for the 10 weeks, occasionally supplemented for major administrative tasks. The SMALL project is now an official NSF Research Experiences for Undergraduates (REU) site, with additional funding from the Ford Foundation, NECUSE (New England Consortium for Undergraduate Science Education), NSF grants, G.T.E., Shell, and Williams College. The name SMALL, in case you are wondering, is an acronym for the faculty on the original proposal.
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Figure 13. Cindy Cero began her mathematical correspondence at age 12. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 4 3
In addition to REU sites, the NSF is now supporting summer geometry institutes involving researchers, graduate students, undergraduates, and high school teachers and students. Write to the Mathematical Sciences Division at the NSF for information. It probably is good to start research early on. I got one letter that began "I'm Cindy Cero and I'm 12 years old." Cindy described her experiments with soap films that led her to ask the fundamental questions about structure and singularities: "Can you help me understand why they formed what they did in the middle" (see Fig. 13). By the end, she sounded exactly like any other mathematician: "I'm having a hard time finding information on bubbles so any other resources you could give would be very helpful." Soon she surpassed us all: "P.S. Enclosed is a s e l f - a d d r e s s e d envel o p e . . . ;" but then relapsed into the common closing: "Need information before March." I got feeling sorry that it was not until graduate school that my advisor, Fred Almgren, got me interested in minimal surfaces . . . until an old document reminded me that I, myself, started research at an early age, with an exceptional teacher. See Figure 14.
References
1. Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, Segments can meet in fours in energyminimizing networks, J. Undergrad. Math. 22 (1990), 9-20. 2. Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, The structure of singularities in cI)-minimizingnetworks in R 2, Pacific J. Math. 149 (1991), 201-210. 3. Marshall W. Bern and Ronald L. Graham, The shortestnetwork problem, Scientific American (January 1989), 8489. 4. E. J. Cockayne, On the Steiner problem, Can. Math. Bull. 10 (1967), 431-450. 5. Mark A. Conger, Energy-minimizing networks in R", Honors thesis, Williams College, 1989, expanded 1989. 6. R. Courant and H. Robbins, What is Mathematics?, Oxford: Oxford Univ. Press (1941). 7. Martin E. Glicksman and Narsingh B. Singh, Microstructural scaling laws for dentritically solidified aluminum alloys, Special Technical Pub. 890, Philadelphia: American Society for Testing and Materials (1986), 4~ 61. 8. M. Hanan, On Steiner's problem with rectilinear distance, J. SIAM Appl. Math. 14 (1966), 255-265. 9. Stefan Hildebrandt and Anthony Tromba, Mathematics Acknowledgments and Optimal Form, New York: Scientific American Books, This article is based on an AMS-MAA address [15]. Inc. (1985). The work was partially supported by the National Sci- 10. Gary Lawlor and Frank Morgan, Paired calibrations applied to soapfilms, immiscible fluids, and surfaces or netence F o u n d a t i o n a n d the Institute for Advanced works minimizing other norms, preprint (1991). Study. 11. Adam Levy, Energy-minimizing networks meet only in threes, J. Undergrad. Math. 22 (1990), 53-59. 12. Frank Morgan, Geometric Measure Theory: a Beginner's Guide, Boston: Academic Press (1988). 13. Frank Morgan, Riemannian Geometry: a Beginner's Guide, Boston: Jones and Bartlett (1992). 14. Frank Morgan, Soap bubbles and soap films, in Mathematical Vistas: New and Recent Publications in Mathematics from the New York Academy of Sciences (Joseph Malkevitch and Donald McCarthy, eds.), New York: New York Academy of Sciences (1990), Vol. 607. 15. Frank Morgan, Compound soap bubbles, shortest networks, and minimal surfaces, video of AMS-MAA address, Winter Mathematics Meetings, San Francisco, 1991. 16. C. M. Petty, Equilateral sets in Minkowski spaces, Proc. AMS 29 (1971), 369-374. 17. Bruce Schecter, Bubbles that bend the mind, Science 84 (March 1984). 18. Jean E. Taylor, Crystalline variational problems, Bull. AMS 84 (1987), 568-588. 19. Jean E. Taylor, The structure of singularities in soapbubble-like and soap-film-like minimal surfaces, Ann. Math. 103 (1976), 489-539. 20. Steve Smale, Beautiful crystals calendar, 69 Highgale Road, Kensington, CA 94707.
Figure 14. Here, at age 2, I watched my morn blow soap bubbles [12, p. ii].
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THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
Department of Mathematics Williams College Williamstown, MA 01267 USA E-mail:
[email protected]
Karen V. H. Parshall*
Bolzano's Analytic Programme Steve Russ
Our modern definitions of the convergence of an infinite series, and of the continuity of a real-valued function, appeared in print for the first time 175 years ago in a short work written by an obscure Bohemian priest [1]. The work in question used these definitions in a general and careful style to construct the first "purely analytic proof" of the intermediate value theorem, which states that if a continuous function takes values of opposite sign at the endpoints of some interval, then it has a zero within that interval. The priest in question was Bernard Bolzano (1781-1848), and the theorem is occasionally referred to as "Bolzano's T h e o r e m . " Apart from that, and recognition in the "BolzanoWeie~strass Theorem," the mathematical obscurity which frustrated the priest for much of his lifetime has hardly diminished, until perhaps very recently. Yet his publications (from 1804) and manuscripts (still being published) have revealed a wide range of remarkable insights [2]. He gave the first topological definitions of line, surface, and solid, and he stated the Jordan curve theorem [3]. Banishing the usual illdefined infinitesimals of his time and skillfully employing an arithmetic limit concept, he defined convergence, continuity, and derivatives in a modern sense, including one-sided continuity and derivatives. He also constructed, in the early 1830s, the first example of a function everywhere continuous and nowhere differentiable. In the same period, he began an elaborate but successful construction of the real numbers--his socalled "measurable numbers"---and a comprehensive theory of real functions. He developed an elementary theory of infinite sets, and clearly understood that such sets were characterised by the fact that they always have a 1-1 correspondence with some proper subset of themselves. In his work on logic, published
in 1837 [4], he showed he understood the notion of a variable as a place-holder capable of arbitrary substitutions from a given class of values, and subsequently gave the first formal account of satisfiability, truth, and semantic entailment, in very much the style that is usually attributed to Tarski in the 1920s. Indeed, Bolzano had most of the insights just listed at least two or three decades (in some cases a century) before the
* Column Editor's address: Departments of Mathematics and History, University of Virginia, Charlottesville, VA 22903 USA. THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3 9 1992Springer-VerlagNew York 45
Copper engraving of Marienplatz, Prague (about 1740). Bolzano was born in the house on the extreme right (No.101); in the background is the Klementinum where he worked as Professor of Theology [P1. III.2, p.248"].
results were published by the mathematicians to w h o m they are now usually attributed (such as Dedekind and Weierstrass). Nor were these just lucky accidents. While he was never a professional mathematidan and while he worked largely in isolation, Bolzano formulated a carefully considered methodology, which appeared in outline in his first publication [5] and which acted as a guiding programme for his research work throughout the rest of his life. Bernard Bolzano was born in Prague on 5 October 1781. Thus, the state of almost uninterrupted war which existed in most of Europe between 1789 and 1815 formed the backdrop for his childhood and early career. At this time, Bohemia, with its main town of Prague, comprised part of the Habsburg Empire ruled from Vienna by the Emperor Franz I. Bohemia's dominant intellectual movement was the so-called "Catholic Enlightenment," which emphasised the themes of rationality and usefulness in all things and did much to promote education at all levels. Bolzano's father, an Italian art dealer, had emigrated to Prague in the 1760s and there married a German woman, Cecilia Maurer. Of their 12 children only two survived to adulthood. Bolzano himself was not a strong child, but, in spite of headaches and a weak heart, he wrote, "I was a very lively boy who never rested for a moment'' [6, p. 56]. 46
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This disposition to incessant activity in the face of frequent illness did not abate as he grew older. There are, for instance, over 8000 sheets of manuscript in his mathematical diaries in addition to similar diaries in logic, philosophy, and ethics. (The whole of his writings, much of it previously unpublished, is now being made available through the Bolzano-Gesamtausgabeedition, of which the mathematical volumes come under the general editorship of Professors Jan Berg and Bob van Rootselaar [7].) In 1796, Bolzano entered the Philosophy Faculty in the University of Prague, and for four years he followed courses mainly in philosophy and mathematics. Although he found both subjects initially rather difficult, he soon discovered in pure mathematics ample scope for the foundational and conceptual investigations which appealed to him so strongly. He referred specifically to K/istner's Anfangsgffinde der Mathematik, where "he proved what is generally passed over because everyone already knows it, i.e., he sought to make the reader clearly aware of the basis [Grund] on which his judgements rest. That was what I liked most of all. My special pleasure in mathematics rested therefore particularly on its purely speculative parts, in other words I prized only that part of mathematics which was at the same time philosophy" [6, p. 64].
In the autumn of 1800, Bolzano began three years of He was primarily concerned in this work to find (for theological study. Although he was basically an ortho- the first time, in his view) the correct proofs of elemendox Catholic, he found that his rationalist inclinations tary geometry. He held that such proofs must necesdid not fit as comfortably as he had hoped with his sarily employ only concepts appropriate to the theotheological studies. He came to realise that teaching rems concerned: and not ministering defined his true vocation. EducaI could not be satisfied with a completely strict proof if it tional value no doubt influenced his constant concern were not derived from concepts which the thesis to be for the clarity and correct ordering of concepts in any proved contained, but rather made use of some fortuitous, alien, intermediate concept, which is always an erroneous exposition. (His major work on logic [4] has a long transition to another kind. [5, Preface] section on h o w textbooks should be organised and written!) While p u r s u i n g his theological studies, According to Bolzano, then, logical or formal correctBolzano also prepared his doctoral thesis on geometry, ness is not the sole criterion of an adequate or correct which was published in 1804 [5]. Unable to obtain a proof: The concepts involved in the deduction must be mathematics post and torn over his choice of career, appropriate, in some sense, to the conclusion. For exBolzano seemed initially to face an unsure future. ample, with respect to the elementary theory of the However, after he decided in favour of a theological triangle and parallel lines, the concepts of straight line post, events moved swiftly. On 5 April 1804, Bolzano and direction are appropriate, whereas those of motion was awarded his doctorate; on 7 April, he was or- and the plane are not. Motion, he believed, belongs to dained; and on 19 April, he was appointed to the a different subject from geometry, and its use in a geonewly formed professorship in theology at the Univer- metrical proof exemplifies a "transition to another sity of Prague. Such a post had been created at all kind" (a type of error to which Aristotle first drew universities with a view to curtailing the then current attention in the context of using a numerical fact to wave of liberalism and free-thinking. In addition to courses of lectures, Bolzano was required to give twice-weekly sermons to the students and citizens of Prague. He performed these duties with seriousness and enthusiasm and soon became highly respected and popular in Prague, with over a thousand people regularly attending his sermons. Despite his successes in the pulpit, Bolzano was never politically suited to such a post, and his appointment was viewed from the start with suspicion by the authorities in Vienna. He would only use the authorised textbook in order to criticise it, and he held distinctly pacifist and socialist views. After a long process (which he resisted strongly), Bolzano was dismissed in 1819 for heresy, put under police supervision, and forbidden to publish. This enforced early retirement probably greatly lengthened his life---he suffered from tuberculosis--for he was subsequently able to spend much of his time recuperating, and writing, as a guest on the estate of his friends, Joseph and Anna Hoffmann, at T~chobuz in southern Bohemia. After this crucial turning point in his career, Bolzano began to work on his two major projects: the Wissenschaftslehre [4] on logic and the Gr6ssenlehre [8] on mathematics. Although the restrictions on him were gradually lifted after Franz I died in 1835, Bolzano took no further active part in politics, or in the revolution of 1848, the year of his death. With this brief introduction to Bolzano's life, we now turn to an examination of the principles which guided his choice of research. Beginning in his thesis [5] of 1804, Bolzano formu- Oil painting of Anna Hoffmann [Pl. II.24, p. 228*]. lated a series of general views about mathematics (a "philosophy of mathematics," in a modern sense, would be too grand a description), which included his * R e f e r e n c e s to P l a t e s a n d p a g e s a r e to t h e v o l u m e of t h e B o l z a n o notions of correctness for both concepts and proofs. Gesamtausgabe ( S t u t t g a r t ) Vol. 4 1/1 Bildnisse Bolzanos (1986). THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3, 1992
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prove a geometrical result). In many cases, it is also logically circular because it is the geometrical theorem which proves the possibility of the motion concerned. Bolzano also seemed to conceive of a kind of hierarchy of concepts, which proceeds from the simpler to the more complex and which takes its structure from the definitions. Thus, for the proof of a result involving concepts at a certain level in the hierarchy, no concepts from a higher level should be presupposed. For example, the use of Euclid's parallel postulate presupposes the plane and so does the treatment of angle as a quantity. In this light, Bolzano's elementary theory of lines and triangles, in which angles cannot be added or subtracted as quantities, represents an unusual development. Proofs which ignore this hierarchical principle will, of course, often succeed in the sense that they are logically correct and convincing. Adherence to the principle, however, should result in proofs which follow and reflect the "objective dependence" [9, w between truths. Indeed, this goal defined Bolzano's central motive in his early mathematical works. To modern eyes, it is remarkable that such a dogmatic and essentialist view yielded a methodology of such extraordinary fruitfulness in his hands.
The methodology underlying his early geometrical work is consolidated in his work on the foundations of mathematics published in 1810 [9], and it ultimately i n f o r m e d the mathematical research p r o g r a m m e Bolzano embarked upon for the remainder of his life [10]. He had intended [9] to be the first of a series containing reorganisation of all mathematical theories (including the simplest ones of arithmetic and geometry) according to his hierarchical methodology, but it did not progress further. Although parts of the second installment were written, they were not published until recently [7, Vol. 2A5]. The failure of Bolzano's programme did not result from lack of enthusiasm on his part, however. He simply n e e d e d some reassurance from other mathematicians that his ideas were being given some critical attention. Because this was forthc o m i n g in neither reviews nor c o r r e s p o n d e n c e , Bolzano decided to postpone the major work on foun-" dations and, as he candidly acknowledged, "make myself better known to the learned world by publishing some papers which, by their titles, would be more suited to arouse attention" [1, Preface]. He explained in the same passage that this strategy motivated the publication of his papers [11,12] in 1816 and 1817, re-
The Hoffmanns' house at T~chobuz where Bolzano wrote his most important scientific works [P1. III.10a, p.256"].
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spectively. The topics of these works, and [1], were therefore chosen chiefly for publicity purposes. There are few n e w results proved in these papers; each aims primarily to give new (and, in Bolzano's view, the first rigorous) proofs of several well-known theorems in analysis. Once again, this attention to the proofs of basic theorems stemmed from one of Bolzano's fundamental conceptual requirements: the removal from analysis not only of ideas of infinity and infinitesimals but also of the remnants of geometrical intuitions. His papers [11,12] thus merely indicate the kind of reorganisation Bolzano envisioned for analysis. He put it bluntly in the Preface of [11], when he described the paper as "a sample of a new way of developing analysis." The aim of attracting attention to his ideas through these works was eventually achieved, but only long after Bolzano's death. The first important recognition of Bolzano's work came in connection with [11] in Hankel's article, "Grenze [Limit]," in Ersch and Gr(iber's Allgemeine Encyklopddie of 1871. In order to get the flavour of Bolzano's approach to analysis, consider the 1817 paper [1] in which he gave his proof of the intermediate value theorem. Bolzano opened this work with a clear statement of his methodological position relative to the theorem under consideration: Precisely because the theorem can be stated in purely analytic terms, no proof using notions of time, motion, or geometry could be correct: If we consider that scientific proof should not merely be confirmations [Gewissmachungen], but rather justifications [Begriindungen], i.e., presentations of the objective reason for the truth concerned, then it is self-evident that the genuinely scientific proof or the objective reason, of a truth which holds equally for all quantities, whether in space or not, cannot possibly lie in a truth which holds merely for quantities which are in space. [1, Preface] For Bolzano, there simply had to be a purely analytic proof, and he proceeded in the remainder of the paper to produce it. The underlying structure of the work is simple. In the Preface, which is as long as the rest of the paper, Bolzano criticised earlier attempted proofs both for technical reasons (for instance, the inadequate definition of a continuous function as a function taking any value which is between any two of its values) and for the methodological reasons which, as we have seen, motivated him so highly. He then repeated his own definition of the continuity of a function (which had appeared first in [11]) before moving into the main part of the paper. There, he opened with a short discussion of infinite series prefatory to his statement of the convergence criterion and to his two main theorems: Theorem 1, an early form of the Bolzano--Weierstrass theorem, which uses the convergence criterion; and Theorem 2, the intermediate value theorem, which hinges on both the continuity definition and Theorem 1.
In this work and, indeed, in his broader programme, Bolzano did not seek merely to "arithmetise analysis,'" that is, he did not wish only to purge the concepts of limit, convergence, and derivative of geometrical components and replace them by purely arithmetic concepts. He was aware of a deeper problem: the need to refine and enrich the concept of number itself. It is within this broader framework that the importance and effectiveness of Bolzano's definitions of convergence and continuity can best be understood. Bolzano also brought to the task important insights into the use of symbols and variables in mathematics. In making use of an arithmetic limit concept, for instance, he understood clearly the need for two distinct variables and for their quantification. He also appreciated the fact that one of the variables must have a given or chosen value. In Bolzano's time, it was still common, in the context of a convergent series, to speak as though one simply set the series of partial sums off, rather like a clockwork engine, and watched them gradually approach some boundary value. Bolzano instead spoke of first choosing a value for any desired degree of approximation, or tolerance, and then explicitly finding sufficiently large partial sums which guarantee sums always differing by less than that tolerance. Postponing, for a moment, the issue of the existence of limits in such cases, we note that this emphasis on choice brings out clearly the "transactional" nature of the modern concept of a limit. It is striking that in the early analysis works Bolzano nowhere referred to the limit concept, although he used it extensively, in a thoroughly modern manner, and more fluently than many of his contemporaries. In doing so, he underscored an important aspect of his methodology of conceptual correctness, namely, to reject emphatically the use of infinitesimals in name and concept. He always spoke instead of "quantities which can be as small as desired," and he made it clear that these are ordinary arithmetic quantities conceived as small, chosen values which remain constant throughout a given context. Bolzano also introduced a conventional notation of o~ and fl with various superscripts to distinguish different "arbitrarily small quantities.'" They had in common the universal quantification implied in the idea of choice. Thus, in the Preface to [1], after criticism of several unsatisfactory definitions of continuity that were in current use, we read: According to a correct definition, the expression, that a function fx varies according to the law of continuity for all values of x inside certain limits means just that, if x is some such value, the difference fix + ~o) - fx can be made smaller than any given quantity provided ~0can be taken as small as we please . i.e., f(x + o~) = fx + f~. .
.
In w of [1] he used this definition to prove that any polynomial is continuous, and in [12] he proved several elementary theorems on continuous functions. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
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The definition is clear and formulated for easy arithmetical manipulation. It is also original in the sense that this combination of the concept (which was already u n d e r s t o o d by m a n y mathematicians informally) and its precise arithmetic formulation and use appeared in print here for the first time. For his proof of the intermediate value theorem, however, Bolzano crucially required some form like a bounded set of real numbers has a limit point---of the Bolzano-Weierstrass theorem. (Of course, at this time there was a formal definition neither of the real numbers nor even of the rationals or the integers. The historical development of such definitions or constructions followed roughly the reverse of a "logical" order.) To obtain this, he needed a convergence criterion for infinite series, and he considered in the first few sections of [1] a power series in x with the partial sums forming a new series which he represented in [13] as
(ii), Bolzano said that "the assumption of a constant quantity (X) with this property of proximity to the terms of our series contains nothing impossible because on this assumption it is possible to determine this quantity as accurately as desired" [1, w As for (iii), he asserted that "the true value of X therefore differs from the value of Fnx by at most d, and can therefore be determined as accurately as desired since d can be taken arbitrarily small. There is therefore a real quantity [reelle Gr6sse] to which the terms of the series being discussed approach as much as desired if the series is continued far enough" [1, w In order to understand Bolzano's reasoning here, we first need to explain the phrase "contains no impossibility." From the early work, it seems Bolzano believed that only the possibility of a concept must be proved before it may be used legitimately by the mathematician. Several times in [5] he emphasised, or at least exhibited, the possibility of an object before using it. Fix, F2x, F3x. . . . . Fnx . . . . F, +rx . . . . . Although mathematics at this time was usually defined The value of x here is constant, and there is no concept as a "science of quantity," in the Preface of [9] Bolzano at this stage of uniform convergence. He then stated defined it quite abstractly as "a science which deals with the general laws according to which things are the crucial result on convergence in w regulated in their existence." He made clear that it is If a series of quantifies not for mathematics to prove the actual existence of anything (that is a matter for metaphysics). Rather, Flx, F2x, F 3 x . . . . . Fnx ..... F.+,x . . . . . mathematics is concerned only with the "conditions has the property that the difference between its nth term, for the possibility" [9, w of a thing. To Bolzano's w a y Fnx, and every later one, F,+,.x, however far from the of thinking, then, to show the possibility of a quantity former this is, remains smaller than any given quantity if X sufficed for its introduction and use [which is pern has been taken large enough, then there is always a haps all that he intended by (iii)]. As Kitcher suggests, certain constant quantity and indeed only one, which the terms of this series approach and to which they can come Bolzano's thinking behind the clairn of (ii) may have been that if any contradiction could be derived from as near as desired if the series is continued far enough. the possibility of X, the same contradiction could be A sequence with the property described here usually derived from a sufficiently close approximation to X. bears Cauchy's name. Four years after Bolzano pubThat is, he might have seen (i) as a kind of relative lished his paper in 1817 in Prague, Cauchy described consistency proof of (ii). the same property in his widely read Cours d'Analyse. N o w many commentators (e.g., Steele in [16]) say (Grattan-Guinness in [14] actually brought forward that, of course, there is a logical error in the whole some quite elaborate evidence that Cauchy might inproof due to the lack of a proper existence proof, and deed have taken the idea of this p r o p e r t y from that Bolzano was doing the best he could in the abBolzano, but it is not at all convincing.) Following sence of a definition of real numbers. We could argue Kitcher in [15], we shall call a sequence with this propalmost the opposite, however: Rather than committing erty a "B-sequence" (for "Bolzano" or the "bunching" a logical error, Bolzano both argued for, and then apof the partial sums). pealed to, a profound insight into the nature of numBolzano gave an interesting proof of this B-sequence ber, namely, that "a correct concept" of number is one result, which fell into the following four parts: in which every B-sequence has a limit. Using such a (i) if such a constant quantity (X) exists, it can be perception as an essential part of a proof bears comdetermined as accurately as desired; parison both with adopting the existence of such a (ii) therefore, the assumption of such a quantity X limit as an axiom and with introducing a definition "contains no impossibility"; through certain infinite sets (cuts or equivalence (iii) therefore, there is such a real quantity X; and classes). Because we cannot hope to give formal proof (iv) this quantity X is unique. of the consistency of such axioms or definitions, the He provided clear, rigorous proofs of steps (i) and (iv), consistency of their introduction is a belief grounded in although his proof of (i) is curiously long-winded. In, the common intuition, or insight, which they capture. or between, steps (ii) and (iii), however, the proof This is not to deny, of course, the necessity of a defiseems suspect to the modern reader. In justification of nitional, or axiomatic, approach to real numbers, but to 50
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point out that these are only the formal counterparts of crucial intuitions into number which must also have been experienced by Dedekind and Cantor, among others, much later in the century. Bolzano's "proof" is useful precisely because it does succeed at an informal level and thereby points convincingly to the properties required of a formal concept of number. He was attempting to give some grounds [namely (i) and (ii)] for holding that numbers, rightly understood, are such that the property of being a B-sequence is sufficient to ensure convergence. Bolzano may not have been the first to have such an insight into number, but he was certainly the first to express it so usefully in terms of the B-sequence property. With this necessary foundation, Bolzano proceeded to state and prove the first main theorem in his 1817 paper [1]: Theorem 1. If a property M does not apply to all values of a variable quantity x, but does apply to all values smaller than a certain u, then there is always a quantity U which is the greatest of those of which it can be asserted that all smaller x possess the property M. [1, w If A is a nonempty sequence of real numbers and the property M signifies "not being a member of A," then the theorem asserts that if A has a lower bound, it must have a greatest lower bound. It is, therefore, equivalent to any of the family of theorems that often go under the name of the Bolzano-Weierstrass theorem. The proof Bolzano gave is well-organised, clear, and rigorous. Starting from the interval [L,R] (where L = u and R = u + D for some positive quantity D) with the property that M applies to all x < L but not to all x < R, it proceeds by repeated bisection to produce a sequence of nested intervals for which the endpoints maintain this property and, therefore, enclose the value U, as claimed. Then either U coincides with one of the points of bisection or there are infinitely many intervals. After comparison with a geometrical progression of common ratio 1/2, the convergence criterion is used to s h o w their endpoints converge to the value U. Whereas such an argument must have been well known for a finite number of bisections in the context of solving equations numerically, and whereas it resembles certain geometrical arguments in Euclid Books X and XII which hinge on an indefinite number of bisections, Bolzano's application of the bisection argument in the context of a process which may never end and in which the endpoints may never specify (by becoming arbitrarily close to) a limit value seems original. Bolzano next proved a general case of the intermediate value theorem: Theorem 2. If two functions of x, fx and qbx,vary according to the law of continuity either for all values of x, or for those lying between (~ and 6, and iff~t < ~cx andf~ > ~b~, then there is always a certain value of x between ~t and for which fx = ~x. [1, w
Portrait of Bernard Bolzano by Joseph Kriehuber 1849 with motto "To be happy and to make happy, that is our mission." [P1. 1.21b, p.176"]
The proof Bolzano gave of this result is an elegant application of his continuity definition and the completeness result of Theorem 1. He surnmarised it towards the end of the Preface in an outstanding example of h o w to convey the "bones" of a proof, before moving on to give the formal details. It could almost be given verbatim as an instructional model to undergraduates in an analysis course today. Due to Bolzano's dismissal and the censorship then imposed u p o n him, the three short installments on analysis, [11] (published in 1816) and [1,12] (published in 1817), were his last mathematical publications until the 1840s. Nevertheless, from the late 1820s until his death in 1848, he laboured on a major series, Gr6ssenlehre, which was to be a systematic development of the whole of mathematics from its proper foundations to the latest areas of research. In one of the most interesting parts of this manuscript, Bolzano developed a theory of real numbers (or measurable numbers as he calls them) by means of applying an equivalence relation to certain kinds of number expressions [17]. This THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
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then enabled him to prove that his measurable numbers form an ordered field which is complete in that every bounded nonempty set has a limit point. Thus, he vindicated his earlier insight into number and reformulated, and proved, his convergence criterion in terms of B-sequences of measurable numbers having a measurable limit value [18]. (For details of some of this work see the papers of Laugwitz [19] and Spalt [20].) On realising in the 1840s that he could not complete the Gr6ssenlehre, Bolzano began again to publish installments. He h o p e d his pupils might continue his work to some completion and see it into publication as a whole. Although this was not to happen, his friend P~ihonsk~ did edit and publish posthumously the volume, Paradoxien des Unendlichen [16], by which Bolzano became better k n o w n to mathematicians than through any work published in his lifetime. However brief this survey of Bolzano's early contributions to analysis, we must make mention of the function Bolzano discovered which is nowhere differentiable and yet everywhere continuous in a given interval. In [11] of 1816, Bolzano used the notion of arbitrarily small quantities to devise beautifully fluent proofs about derivatives (and to give the basic limit definition of derivative which, while standard now, was rare then). These arguments contrasted markedly with the vague language of ill-defined infinitesimals, which professional mathematicians of the day would long continue to employ. For example, in [11] Bolzano proved in a straightforward fashion that the differentiability of a function at a point implied its continuity there. It was an impressive insight into the nature of the family of concepts--function, continuity, differentiability--that, within a dozen years or so, Bolzano was also able to give an example showing that the converse was not true. It is not possible to do justice here to the details of what he actually does, but here is the broad outline. In w of the Functionenlehre of the early 1830s (see [18]), Bolzano constructed a function F with the property that the value of Fx is the limit of the values of y , x as n increases indefinitely [21], where y~ is defined recursively as follows: y~ is on the straight line joining (a,A) to (b,B) so B - A
yffx) = A
+ (x-a)
b-a
then Y2 has a graph consisting of the four segments formed by joining consecutive pairs in the sequence (a,A), (a + 3/8 (b - a), A + 5/8 (B - A)), (1/2 (a + b), 1/2 (A + B)), (a + 7/8 (b - a), A + 9/a (B - A)), (b,B). Then for all n, the function Yn+ 1 is obtained by decomposing each segment of yn into four new segments in the same way that Y2 was obtained from Yl. Taking the initial segment joining the origin to (1,1), for y~, the fourth iteration Y4 of this function is illustrated in Figure 1. At 52
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Figure 1.4th iteration of Bolzano's function with initial segment join of (0,0) to (1,1).
its first introduction, Bolzano simply assumed, without proof, that the limit of a sequence of continuous functions is itself continuous: a result which, while false in general, does hold in this case. In fact, he only showed that the function F is not monotonic in any arbitrarily small neighbourhood of a point in the initial interval. Later on in w he proved that it fails to have a finite derivative at a dense subset of points in the interval. It has subsequently been shown not to have a derivative at any point [22]. The limit function F also represents what is, presumably, the first analytically defined fractal set. In the discussion which followed, Bolzano clearly revealed his awareness of the need for a much wider concept of function than was usual at the time. From an historical point of view, Bolzano's mathematical work raises an interesting issue: For all its originality and fruitfulness, it seemingly had not the slightest significant influence on the work or thought of any subsequent mathematician. It was either misunderstood, unread, or unknown until much too late to have any effect. What value, then, can there be for those trying to understand the maelstrom of history, in understanding the mixture of insights and mistakes of this isolated amateur? Was his work not just a curious dead-end in the historical maze? The history of ideas involves charting, and making sense of, changes in ideas and theories. It has more interest in sources, motives, and values, and in determining the "light of the times," than in seeking to measure how much "'influence" A had on B. Something interesting was happening in the mind of a man who, with only moderate technical ability and with very limited time devoted to
research, gained such a range of fruitful a n d subtle insights into w h a t h a v e become central concepts of mathematics. It is, therefore, w o r t h w h i l e to examine a n d assess not only w h a t h e did, b u t also the prog r a m m e of m e t h o d s a n d principles w h i c h g u i d e d him so successfully. Such a n analysis could well point to n e w approaches, a n d n e w solutions, to current and future problems. I am pleased to a c k n o w l e d g e m a n y helpful comm e n t s o n this article from Karen Parshall a n d David Fowler, and from Meurig B e y n o n w h o also p r o v i d e d Figure 1. References 1. B. Bolzano, Rein analytischer Beweis des Lehrsatzes, dass
2.
3.
4.
5. 6.
7.
8.
zwischen je zwey Werthen, die ein entgegengesetztes Resultat gew~hren, wenigstens eine reelle Wurzel der Gleichung liege. Prague: Haase (1817). A new edition, with P. E. B. Jourdain as editor, appeared in Ostwald's Klassiker der exakten Wissenschaflen, No. 153, Leipzig (1905). An English translation also appears by the present author in Historia Mathematica 7 (1980), 156-185. For what is still the best survey in English of Bolzano's life and mathematical work, see the article "Bolzano, Bernard" by Professor Bob van Rootselaar in the Dictionary of Scientific Biography New York: Scribners (1973). Although now out of date in some respects (e.g., the bibliographic details and the remarks on the real number work), this article also gives a useful summary of Bolzano's contributions to logic in [4]. For an excellent survey of Bolzano's topological contributions see the article, Dale M. Johnson, Prelude to dimension theory: The geometrical investigations of Bernard Bolzano, Arch. History Exact Sci. 17 (1976), 275-296. B. Bolzano, Wissenschaftslehre. Versuch einer ausfiihrlichen und gr6sstentheils neuen Darstellung der Logik mit steter Riicksicht auf deren bisherigen Bearbeiter, 1-4. Sulzbach: Seidel (1837). Substantial parts of this work are translated in R. George (ed.), Theory of Science, Oxford: Basil BlackweU (1972), and in J. Berg (ed.), Theory of Science, Dordrecht: Reidel (1973). B. Bolzano, Betrachtungen fiber einige Gegenstande der Elementargeometrie, Prague: Barth (1804). M. J. Fesl (ed.), Lebensbeschreibung des Dr Bernard Bolzano mit einigen seiner ungedruckten Aufsfitze und dem Bildnisse des Verfassers. Sulzbach: Seidel (1836). The page numbers quoted refer, however, to the more easily available extract from this work which appears in E. Winter (ed.), Bernard Bolzano: Ausgewfihlte Schriften. Berlin: Union Verlag (1976). E. Winter, J. Berg, et al. (eds.), B. Botzano-Gesamtausgabe. Stuttgart: Frommann Holzboog (1969). Well over half of the proposed 56 volumes of this magnificent and scholarly edition have already been published. The bibliographic series of this edition, starting with Vol. E2/1 (1972), is now the most authoritative source for the extensive literature on Bolzano and the numerous editions of his works. A useful survey in English of the contents of the Gr6ssenlehre is given in Jan Berg, Bolzano's Logic, Stockholm: Almquist and Wiksell (1962). Some volumes of the original manuscripts, enjoying the meticulous and helpful editorship of Jan Berg and Bob van Rootselaar, have already appeared in [7]. All the early works [1,5,9,11,12] and important parts of the work on real numbers and the
theory of functions also have English translations forthcoming in S. B. Russ, The Mathematical Works of Bernard BoIzano. Oxford: Oxford University Press (to appear). 9. B. Bolzano, Beytrdge zu einer begriindeteren Darstellung der Mathematik. Erste Lieferung. Prague: Widtmann (1810). 10. In the late 1960s Lakatos analysed scientific theories in terms of the notion of a "research programme", a theoretical structure which provides guidance for future research. See I. Lakatos and A. Musgrave (eds.), Criticism and the Growth of Knowledge, Cambridge: Cambridge University Press (1970), 91-195. The paper "Cauchy and the continuum" in J. Worrall and G. Currie (eds.), Imre Lakatos, Philosophical Papers. Vol. 2, Mathematics, Science and Epistemology, Cambridge: Cambridge University Press (1978), 43-60, applies such principles in a particular case. The recent work, T. Koetsier, Lakatos's Philosophy of Mathematics: A Historical Approach, Amsterdam: NorthHolland (1991), is a study of these principles generally in mathematics. Because of the explicit accounts of his methodology, and its fruitfulness, Bolzano's work would make a particularly interesting case study to analyse within the Lakatos' framework. 11. B. Bolzano, Der binomische Lehrsatz und als Folgerung aus
12.
13. 14. 15. 16.
17. 18.
19. 20. 21. 22.
ihm der polynomische und die Reihen, die zur Berechnung der Logarithmen und Exponentialgr6ssen dienen, genauer als bisher erwiesen. Prague: Enders (1816). B. Bolzano, Die drey Probleme der Rectification, der Complanation und der Cubirung, ohne Betrachtung des Unendlich Kleinen, ohne die Annahmen des Archimedes, und ohne irgend eine nicht streng erweisliche Voraussetzung gelfst; zugleich als Probe einer gfinzlichen Umstaltung der Raumwissenschaft, allen Mathematikern zur Priifung vorgelegt, Leipzig: Kummer (1817). Bolzano actually uses superscripts centred over the F for the partial sums, but they are rendered here as subscripts for convenience. See Ivor Grattan-Guinness, Bolzano, Cauchy and the 'New Analysis' of the nineteenth century, Arch. History Exact Sci. 6 (1970), 372-400. Philip Kitcher, Bolzano's ideal of algebraic analysis, Stud. History Philos. Sci. 6 (1975), 229-269. D. A. Steele, Paradoxes of the Infinite by Dr Bernard Bolzano, London: Routledge and Kegan Paul (1950). This is a translation of the work of Bolzano, F. Pf'ihonsk~" (ed.), Paradoxien des Unendlichen, Leipzig: Reclam (1851). This appears in a section headed Unendliche Gr6ssenbegriffe and is in Vol. 2A8 of [7]. This is found in w of the Functionenlehre which is a part of the Gr6ssenlehre that appears in Vol. 2A10 of [7]. Berg dates this part of the manuscript as being completed in 1834. It was also published as Vol. 1 of K. Rychlik, Bernard Bolzanos Schriflen. Prague (1930). Deflef Laugwitz, Bolzano's infinitesimal numbers, Czechoslovak Math. J. 32(107) (1982), 667-670. Detlef Spalt, Bolzanos Lehre von den messbaren Zahlen 1830-1989, Arch. History Exact Sci. 42(1) (1991), 15-70. Bolzano uses superscript notation here y", yl, etc. We have used corresponding subscripts to avoid confusion with powers. Vojt~ch Jarnik, Bolzano and the Foundations of Mathematical Analysis. Prague: Society of Czechoslovak Mathematicians and Physicists (1981). See the section, "On Bolzano's Function," 67-81.
Department of Computer Science University of Warwick Coventry, CV4 7 AL England THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 5 3
The Ubiquity of Free Groups A. M. W. Glass
Consider the functions g(z) = z + 2 and h(z) = 1/z on the extended complex plane C U {~}. Note that g-1 and h -1 exist and are given by g - l ( z ) = z - 2 and h - l ( z ) = 1/z = h(z). We will write gm for g composed with itself m times and g-m for g-1 composed with itself m times (m a positive integer). If @ is the unit disc with centre 0, then, clearly, gn(@) is properly contained in @', the outside of @, for any nonzero integer n, whereas h interchanges @ with @'. This "ping-pong" effect has as a consequence: if n 1. . . . . n k are nonzero integers, then hg nk 9 9 9 hg ~1 ( ~ ) is a proper subset of @; so hg ~ 9 9 9 hg"l(Zo) ~ z o for some z 0 E @. Thus, hg "k 9 9 9 hg ~ is not the identity function; that is, there is no interrelation between g and h. In group theory parlance, the cyclic subgroups (g) and (h) (generated by g and h, respectively) generate their free product. However, h 2 is the identity. N o w let f = h - l g h , s o f ( z ) = z / ( 2 z + 1). T h e n f ~ is not the identity if n # 0, and the same holds for g. Moreover, by the above, f m ~ g , ~ . . , f m~g,i is not the identity if m 1. . . . . ink, n1 . . . . . nk are all nonzero. Therefore, the group generated by f and g is free; that is, there is no collapsing of "words" in f and g. Under the identification of the M6bius transformation z ~
az + b CZ + d
The finitely generated subgroups are, in fact predominately free, as shown by Epstein [5]. If "almost all" means "except for a meagre set" (with respect to the usual metric in the special case of G L ( n , C ) ) , then: If G is a finite-dimensional, connected,, nonsolvable Lie group and n is a positive integer, then almost all n-tuples from G freely generate a free group.
with [ca b]
(a,b,c,d ~ C, ad ~ bc), we obtain that [~ 0] and [5 2]
generate a free group inside SL(2,Z) C SL(2,C) C GL(2,C), where G L ( 2 , R ) is the general linear group of 2 x 2 matrices over a ring R and SL(2,R) is the subgroup of all such matrices with determinant 1. This result that GL(2,C) contains a free group is but a pale shadow of a very much deeper result, the Tits D i c h o t o m y [15], a special case of which is: Every finitely generated subgroup of GL(2,C) either includes a free non-Abelian group or is an extension of a solvable group by a finite group. 54
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General Permutation Group Settings for Free Groups
m-order-transitive for all positive integers m. More generally, if fl is a set with a set of binary relations defined on it, we say that a subgroup G of the autoNow the free group on a finite or countable number of morphism group of f~ (respecting all the binary relagenerators is countable and so, by Cayley's theorem, tions on f~) is m-homogeneous if whenever (a 1. . . . . am) "is" a subgroup of Sym(N), the group of all one-to-one and (61. . . . . [3,,) are m-tuples of distinct elements of onto functions from the set of natural numbers, ~, to 1~ and for every relation R on fL R(oLjyg) if and only if itself. We define the distance p(f,g) between two disR(Bj, f3k) (j,k E {1. . . . . m}), then for some g E G, g(ai) tinct functions f and g in Sym(~) as follows: p(f,g) = = Bi (i -- 1. . . . . m). Highly homogeneous is defined 1/2m if f and g agree on {0. . . . . m - 1} and so do f - 1 analogously. Now, by methods of FraissG it is possible and g - l , but either tim) ~ g(m) or f-l(m) ~ g-l(m). to show the existence of a unique (to within isomorWith respect to this distance function, Sym(~) is a phism) countable partially o r d e r e d set (poset) ll complete metric space and both composition and in(whose automorphism group is highly transitive) converse are continuous operations [11]. As usual, a taining all finite posets as posubsets, such that if A~ countable union of nowhere dense sets is called meaand A2 are any finite posets and 0:A1 --~ A2 is a poset gre, Bake's theorem applies, and the complement of a embedding, then for any poset embedding q~: h 1 --~ fl, meagre set is of 2nd category. With the product topolthere is a poset embedding 0:A2 ~ fl such that 00 = ogy on Sym(~) x Sym(N), one can show the lovely 6. The same is true for graphs. The resulting poset result of Dixon [3] suggested by Epstein's splendid the(graph) is called the countable universal poser (graph). orem above: Indeed, together with the ideas of Macpherson [12] Almost every two-generator subgroup of Sym(~) is and Droste [4], Glass, McCleary, and Rubin [7] have free shown that where "almost every" means except for a meagre set. There is nothing sacred about the number 2; the same result holds for "finitely generated'" with the product topology on the product of the appropriate number of copies of Sym(N). A group G of permutations of a set f~ is said to be m-transitive if for any pair (oh . . . . . O~m), ( 6 1 , " 9 9 , ~ m ) of m-tuples of 1~, each with m distinct elements, there is g ~ G such that g(og) = [3j (j = 1. . . . . m). If G is m-transitive for every positive integer m, we say that G is highly transitive on ft. These are very "rich" groups. Dixon [3] proved the somewhat more surprising result: Almost every finitely generated highly transitive group in Sym(~) is free. Because ~ and Q, the set of rational numbers, are in one-to-one correspondence, the groups Sym(~) and Sym(Q) are isomorphic; so Dixon's results hold for Sym(Q) too. Inside Sym(Q) are two natural subgroups, Homeo(Q) and Aut(Q,~), the groups of homeomorphisms (respectively, order-preserving permutations) of the rational line. Both are meagre in Sym(Q) (in the metric obtained from Sym(~) using any one-to-one map of N onto Q), but Glass, McCleary, and Rubin [7] have shown: Almost every finitely generated subgroup of Homeo(Q) and of Aut(Q,~) is free, and almost all finitely generated highly (order) transitive subgroups are free. Of course, we can only map (oh . . . . . olin) to (f~l, . . . . [3m)by an element of Aut(Q,~) if [3j < Bk precisely w h e n o~j < o~k (j,k E {1. . . . . m}). We say that a subgroup G of Aut(Q,~) is m-order-transitive if G is m-transitive on such m-tuples and highly order-transitive if it is
If (fl,~) is any countable, highly homogeneous poset, then almost all finitely generated subgroups of Aut(fL~) are free and, with the possible exception of the case where (fl,~) is the countable universal poset, 1 almost all finitely generated, highly homogeneous subgroups are free. The reader is encouraged to take his or her favourite countable structure and see if free groups are ubiquitous in its automorphism group.
Concrete Realisations of Free Groups Dixon's result is rather akin to Cantor's theorem that almost all real numbers are transcendental. Neither gives explicitly free groups or transcendental numbers, merely the assurance that you cannot really miss getting one. To show, for example, that ~r is transcendental was a totally different undertaking. Just as we gave two complex maps (matrices) that generate a free group, it would be nice to give such elements of Sym(~). This was done--semiconstructively--by McDonough [13] and Glass and McCleary [6], the maps given generating free groups acting highly transitively on •. [The idea of the proof is to divorce the high transitivity from the embedding and then sew t h e two I The automorphism group of the countable universal graph has exactly the same properties vis-a-vis free subgroups as that of the countable universal poset; see [16]. Both of these groups are simple; in the automorphism group of the countable universal graph, every element can be written as the product of at most 5 conjugates of any given nonidentity element and its inverse [16], whereas for the countable universal poset the same holds but with 16 instead of 5; see Glass, McCleary, and Rubin [7]. THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3, 1992 55
together in an appropriate way; the highly transitive (not necessarily faithful) action is achieved inductively on the even natural numbers, the embedding into the symmetric group on the odd natural numbers by Cayley's theorem, and the sewing by appropriate modification of the latter.] However, more natural maps in Sym(R) are tl: x ~ x + I and e3: x ~ x 3, where ~ is the set of real numbers. (Take the orbit of a particular real number to obtain elements of the symmetric group on a countable set.) Do these generate a free group? Several false or incomplete "proofs" were given before White [17] proved that tl and e3 generate a free group. Indeed, he showed that this holds if 1 is replaced by any nonzero real number a (t d x ~ x + a) and 3 is replaced by any prime p > 2 (ep: x ~ xP). The proof uses Galois theory and is most ingenious and tricky. The ideas can be extended to show ta and ep generate a free group for any nonzero real number a and any odd positive integer p > 1.2 See [1] for the proof, which shows that if T = {ta: a E R}, P = {ep/q: p,q odd positive coprime integers}, and M = {x ~-* rex: m E ~ and m # 0}, then the group generated by the Abelian groups T and P is just their free product, and the group generated by M, P, and T, the group of arithmetic permutations, is just the free product of M T and M P with the subgroup M amalgamated. The proof proceeds as follows: Let w be any permutation of the real line obtained by alternately taking nonidentity elements of P and T, and let K D Q be an algebraically closed field containing all elements a ~ such that ta occurs in the expression of w. We further require ~ ~ K. Let ~ E R be transcendental over K. Then, by induction on the number of "constituents" of w, it is shown that the field K(~,w(O) is an extension of the field K(0 of degree equal to the product of the q's of the various ep/q s that occur in w. This is achieved by simultaneously proving that for each divisor m of [K(Gw(~)); K(O], there is at most one subfield of K(~,w(0 ) of degree m over K(0. Indeed, the unique subfield of degree m (when it exists) is completely described. Consequently, w(0 # ~ and the associated Riemann surface has no degeneracy. Therefore, {~ E C: w(~) = ~} has size at most the product of the p's times the product of the q's which occur in w as constituent epl q' S,9in particular, this set is finite (cf. number of roots of polynomials). Unfortunately, when the restriction 2 In contrast, Sunil Koswatta has recently proved that if f,g E Aut(R,~ <) with supp(f) = {ct ~ R: a f # ~t} = (a,b) and supp(g) = (c,d), thenfand g do not generate a free group except, possibly, when a = c or b = d; that is, two strictly increasing continuous real-valued functions that are the identity except on an interval, do not generate a free group unless the lower endpoints (or the upper endpoints) of these intervals are the same. 56
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
that p and q are odd is removed, the complex permutations (with appropriate branching) b e h a v e less kindly. The "uniqueness" of subfields can disappear and the degree can fail to realise the number described above. Nonetheless, by using the earlier work of Ritt [14], Cohen [2] was able to prove that the full complex arithmetic permutation group (with arbitrary rational powers) is the free product with amalgamation (as described above). The proof, though similar in spirit to that of [1] in its use of Galois theory and "'simultaneous induction," is far more difficult and is a great tribute to Cohen's considerable ingenuity. This leads to the following natural question first asked by Graham Higman: Let T = {t,: a E R} and P = {er: r E ~+}. Do T a n d P generate their free product? By any of these results, we obtain 2 ~~ two-generator free groups explicitly in Sym(~), the maximum possible (in terms of cardinality). These results on free groups in Sym(~) lead naturally back to free groups in certain groups of complex maps. Let G(77) = {z + a2z2 + a3z3 + 9 9 9: aj E 71, j = 2,3 . . . . .
lira l a, I TM # m}; rt-~oo
so G(Z) is the set of all complex functions analytic in a neighbourhood of 0, having a simple zero at 0 and integer coefficients in their p o w e r series expansions with leading coefficient 1. Then G(77) is a group under composition [10]. We now use White's theorem to give explicitly two elements of G(77) that generate a free group; the clever trick employed is J. S. Wilson's (see
[10]). Let tl and e3 be as above and let h(z) = 1/z as before. Because tl and e3 have no interrelation, it follows that % - 1 t i e 3 and t 1 do not either (as before), and the same is true for b = h - I (e31tle3)h and c = h - l t l h . These maps are given by b(z) = z(z3 + 1) -1/3
and
c(z) = z/(z + 1).
Finally, if d(z) = 3z, then d - l b d and d - l c d generate a free group because b and c do. But (d-lbd)(z) = z((3z) 3 + 1) -1/3 and (d-lcd)(z) = z/(3z + 1) and both of these maps belong to G(Z). Hence, G(77) contains a free group on two generators. Are almost all finitely generated subgroups of G(Z) free? If we replace Z by the Galois field Fp with p
elements, does G(Fp) contain a free group and, if so, are such groups ubiquitous?
A Generalisation to Free Products Finally, we mention a generalisation of some of the results of this talk. In their paper [6], Glass and McCleary proved: If G and H are countable or finite nontrivial groups, one of which has an element of infinite order, then their free product is isomorphic to a highly transitive subgroup of Sym(N). However, if G and H are both cyclic of order 2, their free product has no homomorphic image in Sym(~) that is highly transitive (even 3-transitive). It turns out that this is the only exception. If G and H are countable or finite nontrivial groups not both cyclic of order 2, then their free product is isomorphic to a h i g h l y transitive s u b g r o u p of Sym(N). This theorem was proved using a geometric approach by my Ph.D. student Steven V. Gunhouse [8]. It was also obtained independently by Hickin [9] using spelling (Nielsen-Schreier) and building on his very deep work on algebraically closed groups. Indeed, Hickin's results also include free products with an amalgamated subgroup when the amalgamated subgroup satisfies certain appropriate hypotheses. Included are, for example, (xl, x2, x3, x4; [Xl,X2] = [x3,x4]). (This latter example answers a question of Neumann.) Hickin's proof has recently been extended by Gunhouse to obtain a theorem which has as a consequence If G and H are countable or finite nontrivial groups with a common proper finite subgroup A, then the free product of G and H with A amalgamated is isomorphic to a highly transitive subgroup of Sym(N) if and only if the only subgroup of A normal in both G and H is {1}. The "only if" part of this corollary is false if the stipulation that "A is finite" is removed; we believe the "if" part to be true, however, even when A is infinite. By taking appropriate concrete functions, I have attempted to circumvent the usual abstract definitions of free groups and free products and have shown that they occur naturally, explicitly, and certainly more ubiquitously than might have been expected. I hope that in doing so, I have made them a little more "real"----or complex!
Acknowledgments This is a talk given at The Queen's College, Oxford, Queen Mary & Westfield College, London, Michigan State University, and Bowling Green State University. I am most grateful to audiences at all, and especially to Peter M. Neumann, for many helpful comments which have been incorporated. Some of the research mentioned was funded by the NSF US-UK program and some by the Bowling Green State University Faculty Research Committee. The author also thanks the members of the Department of Pure Mathematics and Mathematical Statistics, Cambridge and the Mathematical Institute, Oxford for their hospitality.
References 1. S. A. Adeleke, A. M. W. Glass, and L. Morley, Arithmetic permutations, J. London Math. Soc. (to appear). 2. S. D. Cohen, The group of translations and rational powers is free, (submitted). 3. J. D. Dixon, Most finitely generated permutation groups are free, Bull. London Math. Soc. 22 (1990), 222-226. 4. M. Droste, Structure of partially ordered sets with transitive automorphism groups, Mem. Amer. Math. Soc. 334 (1985). 5. D. B. A. Epstein, Almost all subgroups of a Lie group are free, ]. Algebra 19 (1971), 261-262. 6. A. M. W. Glass and S. H. McCleary, Highly transitive representations of free groups and free products, Bull. Austral. Math. Soc. 43 (1991), 19-36. 7. A. M. W. Glass, S. H. McCleary, and M. Rubin, The automorphism group of the countable universal poset (submitted). 8. S. V. Gunhouse, Highly transitive representations of free products on the natural numbers, Arch. Math. (to appear). 9. K. K. Hickin, Highly transitive Jordan representations of free products, J. London Math. Soc. (to appear). 10. D. L. Johnson, The group of formal power series under substitution, ]. Austral. Math. Soc. (Series A) 45 (1988), 296-302. 11. A. Karrass and D. Solitar, Some remarks on the infinite symmetric group, Math. Z. 66 (1956), 64--69. 12. H. D. Macpherson, Groups of automorphisms of Rocategorical structures, Quart. I. Math. Oxford (2) 37 (1986), 449-465. 13. T. P. McDonough, A permutation representation of a free group, Quart. I. Math. Oxford (2) 28 (1977), 353-356. 14. J. F. Ritt, Prime and compositive polynomials, Trans. Amer. Math. Soc. 23 (1922), 51-66. 15. J. Tits, Free subgroups in linear groups. J. Algebra 20 (1972), 250-270. 16. J. K. Truss, The group of the countable universal graph, Math. Proc. Cambridge Phil. Soc. 98 (1985), 213-245. 17. S. White, The group generated by x ~ x + 1 and x ~ xp is free, J. Algebra 118 (1988), 408-422. Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403-0221 USA THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 5 7
Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the caf~ where the famous conjecture was made, the desk where the
famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.
Quasicrystal Sculpture in Bad Ragaz Istv in Hargittai My wife and I spent a day in October 1990, in the small, quiet, and exclusively elegant Swiss resort Bad Ragaz, visiting some friends. We stayed overnight in the Quellenhof. Early in the morning we walked around the hotel and noticed a beautiful geometrical * Column Editor's address: Mathematics Institute, University of Warwick, CoventryCV4 7AL England.
piece of sculpture in the rising sunshine. Bad Ragaz is near Liechtenstein on the train line from Zfirich just before it splits for Chur and Davos. The photographs show two views of this sculpture. It consists of two incomplete star pentagon polyhedra. These are nice shapes indeed, worthy of a look by the mathematical tourist. However, in our age of the quasicrystaI [1] these star pentagons appear to us also as parts of rhombic
T w o v i e w s of the sculpture in Bad Ragaz. 58
THE MATHEMATICALINTELL1GENCERVOL. 14, NO. 3 9 1992 Springer Veflag New York
triacontahedra which are building blocks of the threedimensional Penrose pattern. The top figure shows a rhombic triacontahedron [2] whose construction followed Mackay [3], and the photograph at the bottom shows an actual quasicrystal [4]. This particular quasicrystal strikes us as if it served as model for the Bad Ragaz sculpture.
Budapest Technical University Szt. Gell~rt 4. H-1521 Budapest, XL Hungary References
Rhombic triacontahedron (a) composed of 10 oblate (b), and 10 prolate (c) rhombohedra, and second-generation cluster of rhombic triacontahedra (d) forming a small segment of a three-dimensional Penrose tiling [2]. Reproduced by permission of Professor L. A. Bursill.
1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range order and no translational symmetry." Phys. Rev. Lett. 53 (1984), 1951-3. 2. Peng Ju Lin and L. A. Bursill, "Computer-Simulated Images of Icosahedral, Pentagonal, and Decagonal Clusters of Atoms." In: I. Hargittai, ed., Quasicrystals, Networks, and Molecules of Fivefold Symmetry, Chapter 4, pp. 49--67. VCH, New York, 1990. 3. A. L. Mackay, "Crystallography and the Pentrose pattern." Physica 11A (1982), 609-13. 4. F. D6noyer, G. Heger, and M. Lambert, "X-Ray Diffraction Study of Slowly Solidified Icosahedral Alloys." In: I. Hargittai, ed., Quasicrystals, Networks, and Molecules of Fivefold Symmetry, Chapter 5, pp. 69-82. VCH, New York, 1990.
AI6LiaCu quasicrystal [4]. Reproduced by permission of Professor F. D~noyer. THE MATHEMATICALINTELLIGENCER VOL. 14, NO. 3, 1992 ~ 9
David Gale* For the general philosophy of this section see Vol. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.
More on Protocols: Playing Games over the Telephone For almost all of what follows I am indebted to Imre Barany.
1. Coin-Tossing A mathematician in Moscow and another in N e w York are talking long distance about a joint paper they have written which they have been invited to present at a meeting in Paris. Because both of them would like to go and only one can, they decide the only fair thing is to leave it to chance. "All right," says the Russian, "I just tossed a coin. Is it heads or tails? . . . . Wait a minute," says the American, "that's not fair." "What's the matter?" says the Russian. "Don't you trust me? All right then, I give you the number 3235765057108466121397469644230097884888 530062931908450094100370625655448971039595527235 426169795539, and if you tell me correctly whether its largest prime factor is congruent to 1 or - 1 modulo 4 you can give the paper." After the American makes his guess the Russian reveals the prime fact o r i z a t i o n , w h i c h in this case h a p p e n s to be 56123699566021020558766279166381074847903158831451 x 57654165390541998801236990031588314500658098016489, and the American can verify the multiplication and the primality of the factors. So until and unless some one finds a fast algorithm for factoring, this is a pretty good cheat-proof method to replace coin-tossing.
2. Scissors, Paper, Rock--and Two-Person Games in General The above trick extends easily to playing any two-person game which does not involve chance moves. Suppose, for example, we want to play scissors, paper, rock. I will then send you a big number * Column editor's address: D e p a r t m e n t of Mathematics, University of California, Berkeley, CA 94720 USA
whose largest prime factor is congruent to 1, 2, or 3 modulo 5, based on whether I want to play scissors, paper, or rock. You then announce your choice directly, no encoding being necessary, after which I reveal the prime factorization of the transmitted number. The same thing will obviously work for games with n strategies by encoding them in numbers whose largest prime factors are congruent to 1,2 . . . . . n for a suitable modulus. Thanks to Dirichlet, we know we will always be able to find the needed primes. 3. Games with Three or More Players Interestingly, the protocols become much simpler and more secure when there are more than two players. In an n-person game, each player has a given finite set of strategies, and a play of the game consists in each of the players picking simultaneously elements from these sets. It is the simultaneity feature which presents the problem for implementing the game by long distance. The solution lies in the mechanism of strategy "sharing." Namely, if player i has m possible strategies, these are numbered 0 to m - I and this numbering is known to all of the players. If the player n o w decides to play strategy k, he chooses n - I numbers modulo m whose sum is k and distributes one of these numbers to each of the other players. After every one has done this, the strategies are revealed, and, clearly, any player w h o lies about which strategy he has chosen will be caught by the others. Note that, unlike the two-person situation, this protocol is "unconditionally" secure, meaning it does not depend on the assumption that certain calculations are computationally infeasible. 4. Bridge Bridge and, indeed, all card games do not fall under example 3 because they involve not only choices by the players but also chance moves of Nature, namely, the deal of the cards. Nevertheless, there are nice randomizing protocols. This one for bridge is
60 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3 9 1992 SpringerVerIagNew York
due to Dmitrij Grigoryev: East, West, and South agree on an encoding of the cards of the deck, say, with the numbers from 1 to 52. North, not knowing the encoding, randomly assigns 13 numbers to each of the others w h o will then know their hands. The unassigned numbers make up North's hand but, of course, he does not as yet know what these numbers represent. No problem. To find out whether he has the ace of spades, he tosses a coin and, say, it falls heads. He tells the result to West. If she does not have the ace, she tells heads to South, and if she does have the ace, she tells him tails, and, in general, players either pass on or "reverse" the message they receive depending on whether or not they have the ace. Thus, North has the ace if and only if East tells him heads. Proceeding through the deck in this manner gives North the needed information without revealing anything to the other players. It is amusing to observe that it is possible to implement the procedure above while imposing an additional condition that players shall not be allowed to communicate directly with their partners. This requirement presents a problem because North must deal a hand to South. The problem is solved as follows: South invents a new encoding of the cards. Thus, 1 is zig, 2 is zag, etc. This new encoding is passed along to North using East as the intermediary. North then deals 13 of the zigs and zags to South using West as the intermediary, and again no one but South gains any new information. This example raises the question as to what sort of communication networks can be used for the carddealing operation. The answer is that the necessary and sufficient condition is that the network be doubly connected, that is, there must be at least two disjoint communication routes between every pair of players. One sees how this property was used in the example above in which two different routes had to be used from North to South. The necessity of the condition means that if the graph contains a vertex whose deletion separates the graph into two components, then a secret deal of the cards will not be possible. We will come back to this point later. 5. Poker Note that the bridge-dealing protocol depends crucially on the fact that the entire deck is being dealt out. For games like poker this is, of course, no longer true, so a different protocol must be devised. Here is Barany's description of the method of Barany and Furedi in the case of three players, an American, a Russian, and a Hungarian. The American has a Russian-Hungarian dictionary (giving the names of the cards), the Russian has an American-Hungarian dictionary, and the Hungarian knows all three languages. The American now deals the Russian 5 cards by sending her their Russian and Hungarian names which, of course, are meaningless to the American. The Russian recognizes her 5 cards and now also knows their Hun-
garian names, so she then scratches out the words with these names in her American-Hungarian dictionary, after which she deals 5 of the remaining cards back to the American. Finally either of the two may deal cards to the Hungarian. Of course, this will work for only one deal of the cards because in the course of the deal the American and the Russian are starting to acquire a Hungarian vocabulary. For each deal, therefore, the Hungarian must make u p a new pair of dictionaries. In this column last October, I described how protocols could be used not for playing games but for gaining information without giving away secrets. A typical example was the case of people with secret numbers, their salaries or ages, for example, w h o were interested in learning only the sum of these numbers. As in example 3, the solution lay in sharing. Each player chooses n - 1 numbers whose sum is his salary and distributes them to the others. Then everyone announces the sum of the numbers they have received, and the sum of these numbers is, of course, the sum of the original secret numbers. This simple procedure has the further feature that it is "n-private," meaning that even if any subset of the people get together and pool their information, they will still not be able to find out anything about the secret numbers of the others (aside from w h a t they w o u l d k n o w a n y w a y from their knowledge of the sum). It was pointed out, however, that the sum function was special in that it is, in a suitable sense, the only function which can be computed n-privately. Thus, for example, if you wanted to know the maximum rather than the sum of the salaries, the best you can achieve is "'minority privacy,'" meaning that there is a protocol such that no set of fewer than half of the participants can by pooling information learn anything about the complementary set. Indeed, there is what might be called the fundamental theorem of protocols which asserts roughly that any function can be computed minority-privately, but only a few like the sum can be computed n-privately. Thus, for most functions there is no protocol which is guaranteed to preserve privacy if some group of half or more of the participants is interested in gaining additional information. We will illustrate this impossibility result for the special case of the unanimity problem. The argument, due to Barany, admittedly requires somewhat more concentration than is normally expected in this column, but I think readers who are willing to make the mental effort will find it rewarding. A motion brought before committee C will pass if and only if it is not vetoed by any of the members. The committee wishes to know whether or not the motion passes and nothing else. The assertion is that there can be no protocol for this problem such that for any partition of C into sets A and B neither the members of A nor those of B will get additional information. It then THE MATHEMATICAL 1NTELLIGENCER VOL. 14, NO. 3, 1992
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follows as a corollary that no protocol for this problem can be majority-private. The argument runs as follows: Consider the case where the motion fails and there are negative votes in both A and B. N o w imagine an outsider who knows the protocol and the fact that the motion has failed and is given a transcript of all the messages MAB sent between members of A and members of B. Then it must be the case that the outsider cannot from this information eliminate the possibility that all members of A approved the motion; for if he could then so could the members of B because they have access to the same information, so they would be able to conclude that at least one member of A cast a veto, which they are not allowed to know. This means that there exists a sequence of messages MAA between members of A which is consistent with the protocol in the case where all members of A favor the motion. Symmetrically, there exist messages MBB which together with MAB would be consistent with the protocol if all members of B favored the motion. But then MRB is consistent with MAB and MAA because the messages MAA cannot affect the legitimacy of the messages MBB. This gives a contradiction because it shows that a set of messages which w o u l d occur when all voters favor the motion would, nevertheless, yield the conclusion that the motion has failed. The reader w h o has persevered through the above reasoning should note that we have nowhere defined what is meant by a protocol or even a message, and have used only the property that the set of messages, whatever one may mean by this, is able to give exactly the desired information and nothing else. Recall finally that in the problem of secretly distributing a deck of cards, we required that the communication network be doubly connected. This condition fits in with the impossibility result above. Namely, suppose removing some player P splits the remaining players into sets A and B whose only means of communication is t h r o u g h P. N o w the messages exchanged between A and B must determine uniquely how the cards not held by P are divided between A and B, but P has access to all of these messages and therefore, knowing the protocol, would know this distribution, which is not allowed. A corollary of this is that in games with only two players, there is no protocol for dealing secretly. (On the other hand, one of the earliest results in this area, by Adleman, Rivest, and Shamir, shows that if one assumes the existence of computationally hard problems, for example, factoring, then the two-player case can also be handled.)
Popular Mathematics
H o w do you convince the man on the street about the Marvels of Mathematics? Two of the better-known attention-getters are the story of the emperor and the wheat and the one about the monkeys and the typewriters. The first story, you will recall, involves the grateful emperor who promised to reward his adviser by giving him grains of wheat on the squares of a checker board, one grain the first day on the first square and 2 n- ~ on the nth day on the nth square. The point was that this was more than the annual wheat p r o d u c t i o n of the e n t i r e e m p i r e (to be exact, 18,446,744,073,709,551,615 grains, though this number was probably not given in the original tale). The mathematical moral of the story is, I suppose, that 264 - 1 is a bigger number than most emperors or men on the street realize. As for the monkeys, there are usually six of them and they sit all day at their typewriters pecking away "at random," and the assertion is that in time they will "according to the laws of probability" type out all of Shakespeare's plays or, if you prefer, all the books in the British Museum in alphabetical order. Upon closer scrutiny this turns out to be a rather empty assertion because the laws of probability, quote, unquote, turn out to mean exactly that the monkeys will do what they are supposed to do. Nevertheless, the claim is supposed to elicit ooh's and ah's from the intended audience. In this note I will suggest an illustration from the Wonderful World of Mathematics which includes both the powers of two and the British Museum and has the advantage that it is a precise, true statement (or, as we mathematicians say, a theorem), which nevertheless may seem to the uninitiated rather improbable. In fact, even the initiated may need a bit of convincing. The statement is that if the emperor had continued doubling the amounts of grain there would eventually come a day when the leading (left-most) zillion decimal digits of the number of grains would be exactly a digital encoding of the books in the British Museum. Furthermore, unlike the situation with the monkeys, one can pin down with certainty a definite date before which the desired number will have appeared. If we are a bit more modest in our demands and ask, say, for the day on which the number of grains added starts out with the digits 1992 we can exhibit the number explicitly. It happens to be 4077, because 24o77 = 199 201370508795262659407179015469433322899675187565 570904585549027243405908881656045689928423881065 052202245085609295432660593688618811341717383592 Bibliography 195771270360175509747259139698097532297970527327 I. Adleman, R. Rivest, and A. Shamir, Mental poker, The 4284496329986469700577390364451814166397733086289 Mathematical Gardner, London: Wadsworth International 571032518020455752952011244361970405804842213569 (1981). Imre Barany, and Zoltan Furedi, "Mental poker with three or 598738857324612974050477756095477956591854458352 more players," Informationand Control 59 (1983), 84-93. 929305228144517561853986791031728857088119454192
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THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
862919264872920673456064061857571371353807387048 4888433959864892774707585169773262349351407883993 794451733264390721694007301018073452223453811908 661381477128826783309006143922669341433058743019 071586844344804050363054126636829830293768181717 125429760654069664165918404279140428508802665119 508323796752843572233612755191236217024322204874 175245961379676637905113440547833845268843347369 847734151454764458652171217413606969170060521343 134040597782175502078927760751322716533966205843 842981716595660622451346489263942349072378214190 233740012032631697281797408521686200442571129300 122590233653983306139144796219210553134483041738 024837350879350484938262831338033830178875321688 031458529831869663147033206367958245585806059674 908652956376992134662982433124093647599427395439 396364873350091299670503904558487495571467978179 1859138116280251318272. For later reference, we note that the distribution of digits in the number above is quite uniform. There are only 118 zeros and there are 137 threes where the mean is 124, but these deviations from uniformity are well within the usual allowable limits. (Something which I have not been able to account for mathematically, however, is the fact that 4077 also turns out to be the last four digits of my social security number.) The theorem in more formal terms states that for any sequence of decimal digits N, there exists a natural number n such that the leading digits of 2 n are the sequence N. This result may even have commercial possibilities. Send us the sequence made up of the day, month, and year of your birth (along with d dollars) and you will receive by return mail your own personal n, guaranteed to last a lifetime. Another nice feature of this theorem is that its proof depends only on knowing how to do long division, plus a mathematical fact that even an emperor should be able to understand, namely, the famous Schubfachprinzip associated, once again, with Dirichlet, which says that if you put a zillion and one objects in a zillion boxes, then one of the boxes must contain more than one object. (Emperor: You mean you have to go to graduate school in mathematics to learn that?) How does this apply in the present case? Well, start with the first n such that 2n has at least d + 1 digits. Then somewhere among the next 10a + 1 + I powers of 2 you will get two different numbers whose first d + 1 digits are the same. Now divide the larger by the smaller and recall the rules of long division. The number you get will either (A) start with a one followed by at least d + 1 zeros (and then other stuff) or (B) start with d + 1 nines. (Whether you get A or B depends on the relative sizes of the first digit where the two numbers differ.) Let us denote the quotient in case (A) by Z. The claim is that if one takes successive powers of Z, the first d digits of the numbers obtained will run through all d-digit numbers in their natural order. This is because
if any number N is multiplied by Z, the first d digits of the product will be either the same as those of N or else the dth digit will increase by 1. To see this, consider the number Z' with the same number of digits as Z whose leading digit is a I followed by d zeros, then another I and then the rest zeros. Now Z' is greater than Z and it clearly has the italicized property: For if N = a b . . . xy . . . . where x is the dth digit, then the calculation of N x Z' is given by
ab...xy...O...0 + ab ab... and the dth digit of the product will be x or x + 1 depending on whether a + y is or is not greater than 9. The argument for case (B) is similar except that it involves borrowing rather than carrying. Notice that this proof is constructive and gives an algorithm for finding the n from the N, but the b o u n d given by the proof is astronomical. For a d-digit number it is roughly 21~ In fact, in theory and practice if we look for the first n that works, it turns out that n is on average roughly of the same size as N. For those who are up on such things, we present a Mathematica program, written by Stephan Heilmayr, which finds this smallest n. fract[x_l: = If[x = = Floor[x],l,x - Floor[x]] power2[x__]: = Block[{1 = fract[Log[10,x]],n = 1, u = fract[Log[10,x + 1]],pow = Log[10,2],t = pow}, Whfle[N[t < 1111N[t> = ul, pow = pow + Log[10,2]; t = fract[pow]; n = n + 1];n] The fact that n is small is to be expected because one is dealing with a uniform distribution p h e n o m e n o n . More precisely, the multiples of log210 are uniformly distributed mod 1. Using this fact, one can deduce further properties of the powers of 2. For example, the powers of 2 Which contain a million consecutive 7's (or any other given sequence) in their decimal expansions have density I in the set of all powers of 2. I thank Jack Feldman for pointing this out to me. What about the stronger statement that almost all, that is, all but a finite number of powers of 2 contain a million consecutive 7's, or even at least one 7? That is, are there infinitely many powers of 2 which have no sevens at all in their decimal expansions? If you think it unlikely, consider the fact that "practically all" integers contain a million consecutive 7's meaning that the set of integers with this property has density 1. For note that among all million-digit numbers the fraction that do not consist entirely of 7's is (1 - 10-6) n and among all n-million-digit numbers the fraction such that neither their first million, second million. . . . . (n - 1)st million consist entirely of 7's is (1 - 106)n-----SOyOU see? On the other hand, of course, there are infinitely m a n y THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
63
integers with no 7's at all. In other words, statistical information is not really relevant for questions of this sort. As another illustration, recall that in the number 24o77each digit occurs with density approximately V10th and presumably this behavior is typical, from which one might be tempted to conclude that perhaps all powers of 2 from some point on contain at least one 7. But again, if we think of the sequence of all integers in their natural order, then by the law of large numbers, those in which the percentage of 7's is less than, say, 9.99 have density zero. Nevertheless, again there are plenty of sevenless integers. Perhaps the question of whether or not there exist infinitely many sevenless powers of 2 is "unanswerable" in the sense that there is no proof either way from our usual axioms. Would it then nevertheless be either true or false? That, of course, depends on one's mathematical philosophy. Or, to fantasize wildly, suppose someone were to show that this question was equivalent to the existence of some unfathomable cardinal? At that point the problem would become theological rather than mathematical--which is probably a good place to stop.
if and only if q is a quadratic residue of p (p is a quadratic residue of q).
Problems "Jeu de Taquin" (92-1) by G. Rousseau (Leicester, England). The "pq - 1 Puzzle" is like the famous "15 Puzzle" or "jeu de taquin" of Sam Loyd, except that the frame is p x q instead of 4 x 4, p and q being distinct odd primes. The movable counters are numbered 1,2 . . . . , pq - 1, with the blank square (0) at the top left. What are the conditions on p and q for it to be possible to move the counters from row order to column order? from row order to diagonal order? from column order to diagonal order? (In diagonal order, counter k is in row k mod p and column k rood q, where the rows are numbered 0,1 . . . . . p - 1 and the columns 0,1 . . . . , q - 1.) THE 14 PUZZLE (p = 5, q = 3) (0) 3 6 9 12
1 4 7 10 13
2 5 8 11 14
Row Order
(0) 1 2 3 4
5 6 7 8 9
10 11 12 13 14
Column Order
(0) 6 12 3 9
10 1 7 13 4
5 11 2 8 14
Diagonal Order
Prove: it is possible to pass from row to column order except when p and q are congruent to 3 (rood 4), and it is possible to pass from row (column) to diagonal order 64
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
A Nonmathematical Problem 1. Name the only midwestern state of the United States that does not have an Indian name (midwestern means between the Appalachian and the Rocky Mountains and excludes the southern states on the Gulf of Mexico). 2. Why was this problem included in a column on mathematics?
Duplicity Across Down 1. W.W. II female org. 1. U.S. Pres. Wine country Old Asian capital 5. Dickens character 2. Change Dickens expression Distant 8. Berkeley school 3. Hinge or wire Garden thing Excuse 11. breve 4. Pod tree Mideast bigwig (var.) Pianist 12. Wreath 5. Unclear Latin I verb Spicy food 13. Marx follower 6, Pictures Danish cent Breather 14. Nice evening 7. Pious Indian Bring up (the) Lava Bed Indian 15. McNamara doctrine 8. Algebraic adjective Keats household item Famous mathematician 16. Automotive curse 9. Left bed Half of a swindle Reed 17. Transient 10. Constructive toy Karenina Acting award 18. Past 16. Eye part Cylinder Cozy up to 19. Follow 21. Poet lord facto Coin animal 20. Topological object (2 wds.)22, Glycerine preceder Topological object (2 wds.) Acting award 23. Affirmation 25. Famous mathematician Affirmation Topological object 24. Project follower 26. Major error Sun. speech Ridley Scott film 25. Tabloid 29. Kind of crown Byelorussia? Ways for words 27, Animator Avery 30. Get exposure Lynx, e.g High point 28. Sick 31. Water buffalo 46 across in Madrid Bathroom sign 29. Salt 32. Cowboy's rope (var.) Debtor nation Belief 31. Famous mathematician 33. Half of a tree (var.) Topological object (2 wds.) Out of bed 36. Publisher -Davis 34. Very dry country Kingdom of the dead? Black scratchpad 37. Era anagram 35. Temptress Pooh character Boxing lake 38. Perjurer 36. Gate output Pickens Abnormal sleep 40. Gulf 39. Cellist times two Jacob's brother Chimed 41. Physics unit Era 42. Mideast bigwig O'Hara's place 43. Bathroom sign Sinbad's bird 44. Chemical suffix Garden or party 45. Thing Element 46. Gate output Abby's twin 47. Question Color 48. Medit. wind Rockfish genus
Kuperberg and Zieve, 8/28/91
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Jet Wimp*
C u r v e s and Surfaces for Computer Aided
Geometric Design (2nd edition) b y Gerald Farin Boston: Academic Press, 1990. xvii + 444 pp. US $39.95 (ISBN 0-12-249051-7).
Reviewed by Len Bos From a mathematician's point of view, ComputerAided Geometric Design is simply constructive differential geometry. After all, the central problem is to construct a curve or surface of some smoothness and a desired shape. Some might say that it represents a rebirth of classical geometry, others that it is merely archaeology--the retrieval and reinvention of the elementary. It is true that much of the theory of the subject, in its constituent pieces, is classical, and probably at one time, generations ago, already well understood. But the point of view is rather different, and the computer has dramatically changed our notions of both what problems should be solved and what problems can be solved. The whole is greater than the sum of its parts. The result is an approach, due essentially to de Casteljau [1], to (piecewise) polynomial curves and surfaces that is simple, elegant, and applicable. Let me briefly describe what this approach is. Suppose that p : ~'~ ~ R s with s > m is a polynomial function. The image of p is some parametric polynomial surface. Of course, we may write p explicitly as
,(x)-
(n) x~ I~1<-,
* C o l u m n Editor's address: D e p a r t m e n t of Mathematics, Drexel University, Philadelphia, PA 19104 USA.
where n is the degree of p, a~ E R s, and we make use of standard multinomial notation. N o w I claim that to each polynomial p(x) there is associated another polynomial P(x 1, x 2. . . . . xn) : (~m), __~ Rs with the properties: (1) P(xi . . . . . x,,) is symmetric in the xi; (2) P(xi . . . . . x,,) is degree 1 in each variable, xi, separately; (3) P(x . . . . . x) = p(x); i.e., P reproduces p on the diagonal. P is known as the polar form of p. It is a classical notion, but it seems to have, over the years, lost any significance it might once have had--although a simple example of a polar form, well known to us all, is to take p(x) = []x[[2, the Euclidean norm squared, in which case the polar form is P(Xl,X2) = ( x l , x 2 ), the Euclidean inner product. In fact, it is precisely the advantages of inner products over norms that are gained by the use of the polar form. It is easy to show the existence of the polar form. In fact, in one variable, m = 1, we may take
P(xl. . . . .
x,) = ~ a~k(x~. . . . .
xn),
k=0
where Crk(X1. . . . . X,) is the degree k elementary symmetric function in n variables. In several variables, such a construction requires somewhat more bookkeeping but is not essentially any more difficult. I should remark that the polar form was rediscovered by Ramshaw [3], who gave it the perhaps more useful, and certainly more expressive, name of the blossom of p. How, precisely, is the polar form useful? Suppose that Vl,V 2. . . . . ~m+l are the vertices of a non-de-
66 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3 9 1992 SpringerVeflagNew York
generate simplex in R m. We may write x E R m in barycentric form m+l
X = E
tivi
i=1
for some tl, 9 9 . , t i n + 1 E R such that E m+1i=1 t i = 1. Then, since P is degree 1 in each variable separately, m+l
p(x) = P ( x . . . . .
x) = E
tiP(vi,x . . . . .
x).
i=1
Continuing this expansion in each of the variables, making use of the symmetry of P, and noticing that a factor of ti appears for each evaluation of the vertex vi, we see that
p(x) = P ( x . . . . .
x) = E
the curve or surface by giving its poles and to interactively modify its shape by manoevering the location of the poles on a computer graphics screen. This simple but profound idea is the basis of much of the emerging field of Computer-Aided Geometric Design. It is sometimes referred to as the B6zier approach to curves and surfaces (see for example Farouki [2]). What else does the polar form tell us? It should first be noted that the polar form gives the coefficients of a polynomial written in the Bernstein basis; i.e., the polar form gives the dual functionals to the Bemstein basis. The value of this h a p p y circumstance is easily appreciated. Further, if for 0 ~< r ~< n and 161 = n - r we set b~(x)
=
P ( x , . . . , x, Vl, 9 9 9 Vl,, . . . r
cf~tf~ x
, Vm+l,
61
9
9
9
,
Vm+l)
6m+1
I~1= , + 1 tivi" then by expanding in and again write x = •mi=1 one of the x's, we see that P(Vl,
9 9 9
, V l , V2, 9 9 9 ,
V2, . . . , V m + l ,
9 9 9,
Vm+l) m+l
v
[~1
62
br~(x) = E
~m+l
,-1 (x), tib~+ei
i=1
for some constants %, where t: = (t I . . . . . tm + 1), and [3 E N m+ 1 is a multi-index. In fact, it is not difficult to see that c~ is the multinomial coefficient
c0( )
Given the control points b~ E R s, 161 = n and Em + 1 RS i=1 tivi ~ (1) for 16[ = n set b~ = b~ (2) for r = 1. . . . . n and [61 = n - r compute br~ = ~p=+lI t bi rf~+ei -1
X
The polynomials
with tin+ 1 = 1 -- "~im=1 ti are known classically as the Bernstein polynomials. The Rs points b~: =
P(VD
9 9 9,
where e i is the standard basis vector in ~1m+l. This provides a simple recursive algorithm for the evaluation of p(x) known as the de Casteljau algorithm:
V l , V2, 9 9 9 ,
V2 . . . .
, Vm+l,
~1
99 9,
Vm+l)
~m+l
are called the poles or B~zier points of p(x). With this notation we may write p(x) = E
B~(t)b~,
I~1=, and we see that p is completely determined by its poles. Again, this fact may not be particularly new, but what is new is to use the poles as control points; to specify
=
At the end of this recursion, b~ = p(x). If we restrict x to lie within the simplex with vertices v i, we obtain a surface patch which may be glued to other such patches to form a (spline) surface. The smoothness of the resulting surface is equivalent to a corresponding discrete "smoothness" of the control nets. Indeed, in general, the control net behaves as a qualitative, discrete model of the surface. After this preachment on the virtues of the polar form, the reader may be surprised by its absence from Farin's book. This is probably due to the fact that the book was written before the relation between so-called B~zier techniques and the polar form was generally understood (and certainly before I myself learned of it). Farin begins his development with the de Casteljau algorithm for curves. The basic properties of the resulting curve follow reasonably easily from the algorithm and/or the Bernstein basis form. No facts are lost by not using the polar form, but the reader should be THE
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aware that often the mathematical content could be ex- The Mathematics of Great Amateurs (2nd plained more succinctly through its use. This is espe- edition) cially true about the chapter on B-splines where Farin, by Julian Lowell Coolidge with mixed success, introduces B-spline curves via Oxford: Clarendon Press, 1990. 240 pp.; US $29.95, a procedure k n o w n as knot insertion. This should paper (ISBN 0-19-853939-8). be compared with a polar form development as in Reviewed by David M. Burton Seidel [4]. This is a good book. It is possible to learn much from The book under review is a textually unaltered reissue it. In fact, if you are interested in the practice of curve of the original 1949 Oxford University Press edition and surface construction then this is the book to read. (reprinted by Dover in 1963). Jeremy Grey has brought However, I do have some criticisms to make. The first, it up to date with the addition of an introductory essay, less serious, is that occasionally, the author offers an a biographical note on Coolidge, and a bibliography. argument that doesn't meet the usual mathematical Grey's seven-page introduction serves to describe substandard of proof. For instance, on page 58, a sequence stantial pieces of information not available w h e n the of Riemann sums of approximations to a function is as- book was first published and indicates where recent serted to converge to the integral of the function, scholars have differed significantly from Coolidge. seemingly as if this were in general true. Again, on Great Amateurs is a tour through important portions page 158, the formula for the dimension of a spline of the mathematical work of 16 men " w h o were prinspace is obtained in a somewhat non-rigorous manner. cipally known for some other activity, yet whose sucThis doesn't really detract from the value of the book. cess in the field of mathematics enabled them to make Perhaps it is justifiable to sacrifice rigour for pedagog- contributions of permanent value." Half of the indiical clarity, but it should be made clear exactly what is viduals accorded amateur standing are artists or phia heuristic argument and what is a proof. losophers, with the majority living between the 16th I have a more serious criticism. Within the CAGD and 18th centuries. They are, in order of appearance, community, there is a historical confusion between the Plato, Omar Khayyam, Pietro dei Franceschi, Leonconcepts of a curve or surface being a smooth function ardo da Vinci, Albrecht D~irer, John Napier, Blaise Pasof its parameters and being smooth in the usual sense cal, Antoine Arnauld, Jan de Witt, Johann Hudde, Visof a smooth manifold. As a result they use the notation count William Brouncker, the Marquis de L'Hospital, Ca-surface to refer to a parametric surface that is k times the Comte de Buffon, Denis Diderot, William Horner, continuously differentiable as a function of its param- and Bernhard Bolzano. eters and G~(G for geometric) for what is in fact a CkConsidering the measurable progress of w o m e n in manifold. Unfortunately, this is also the notation that the mathematical community, it is somewhat startling Farin uses in this book. It is perfectly understandable to see a list of 16 amateurs which fails to include, say, that an idea irrelevant to its time and long since for- Maria Agnesi or Sophie Germain. Another surprising gotten could be rediscovered and renamed, but the omission in a book of this sort is the "Prince of Amaidea of a smooth manifold is neither irrelevant nor for- teurs," Pierre de Fermat. Coolidge denies him amateur gotten. I see no value in having this n e w notation. In status on the grounds that Fermat "was really so great some sense it seems remarkable that those in the busi- that he could count as a professional." Of course, the ness of constructing smooth surfaces do not know same reasoning might be applied to Pascal, w h o did what a manifold is, but is this their fault or is it just a join the select field. (For that matter, w h y was the consequence of "pure" mathematics having become so philosopher Descartes omitted?) But these are minor distanced from constructive aspects that practitioners quibbles; it is impossible to have everyone's favorite find it easier to reinvent than to read? a m a t e u r in a small v o l u m e of s o m e 200 pages. Coolidge is to be commended instead for drawing attention to some figures w h o seldom command more References 1. P. de Casteljau, "Courbes et surfaces a poles," Technical than a few lines in general histories of mathematics. Coolidge admits that he included Plato among the Report, A. Citroen, Paris, 1963. 2. R. Farouki, "Computing with barycentric polynomials," select few because he did not want to omit the Greeks The Mathematical Intelligencer 13, no. 4 (1991), 61-69. altogether, and confesses that "a productive mathe3. L. Ramshaw, "Blossoming: A connect-the-dots approach matical scholar he certainly was not." Such a candid to splines," Digital Systems Research Center, Palo Alto evaluation of the mathematical contributions of the in(1987). dividual in question is typical of the author's approach. 4. H.-P. Seidel, "A new multiaffine approach to B-splines," Referring to Napier's major accomplishment, De Arte Computer Aided Geometric Design 6 (1989), 23-32. Logistica, Coolidge says, "I cannot on the whole bestow Department of Mathematics and Statistics high praise on the Logistica, either as a contribution to University of Calgary pure mathematics or as an aid to calculation." ArCalgary, Alberta T2N 1N4 Canada nauld's Les Nouveaux l~l~ments de G~om~trie is dismissed 68
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as "not satisfactory either mathematically or pedagogicaUy," whereas Buffon "cannot be looked upon as a great mathematician, but a bright man interested in mathematics." The present volume naturally invites comparison with other works having a similar flavor, such as Men of Mathematics by E. T. Bell, or the more recent Makers of Mathematics by S. Hollingdale. In contrast to these popularizations, Coolidge concentrates on giving a mathematically sound account (in modern symbolism) of the contents, at the expense of biographical details. The lack of personal information is to be regretted, for it would be interesting to read a little more of the eventful lives of some of the lesser-known practitioners. A more annoying shortcoming of Great Amateurs is the lack of an index of names or subjects; this is offset to a degree by an Index of Authors Quoted. Still, the reader will find a good deal here. The Mathematics of Great Amateurs has stood the test of time to become an essential reference on the shelf of every working historian of mathematics. It remains a delightful "read" for the general mathematician with even a passing interest in the development of the subject.
Department of Mathematics University of New Hampshire Durham, NH 03824 USA
Mathematics and Its History by John Stillwen Springer-Verlag, New York, 1989, x + 350. US $49.80 (ISBN 0-387-96981-0). Reviewed by John Fauvet and Abe Shenitzer There are many books on the history of mathematics in
matter. The great merit of Sfillwell's book is that it makes this task easier, that it builds bridges from pre1800 mathematics to the tangled concepts and issues of the mathematics of the last two centuries. All topics are covered in considerable depth. Moreover, Stillwell manages to invest the material with a sense of newness that makes the reader sit up and take notice and decide that familiar or not it is worth reflecting upon. This is achieved by the avoidance of padding, by the blending of familiar material with less well-known facts, by frequent refreshingly unorthodox comments, and by linking the origins of an issue to its subsequent stages of evolution. A quick overview of the contents of the book. The first four chapters deal with the Greek heritage and some later developments rooted in it. Chapters 5-9 deal with polynomial equations, analytic geometry, projective geometry, calculus, and infinite series, respectively. The key part of Chapter 10 is devoted to rational points on quadratic and cubic curves. Chapter 11 deals with elliptic functions, including addition theorems for elliptic integrals and the connection between these theorems and the question of rational points on elliptic curves. These themes are amplified and intertwined in Chapters 14 and 15. Chapter 13 discusses the tangled story of complex numbers, and presents an excellent comparison of the proofs of the fundamental theorem of algebra by Gauss and d'Alembert. Chapter 12, titled Mechanics, links a number of very different concerns. Differential geometry, non-euclidean geometry, group theory, and topology are insightfully discussed in Chapters 16, 17, 18, and 19, respectively. Chapter 20, titled Sets, logic, and computation, deals with quintessentially modern concerns. Thumbnail sketches of some of the 20 chapters of the book follow. These sketches are often derived by splicing key sentences lifted from the book.
which mathematics is subordinated to history. This is a book in which history is definitely subordinated to mathematics. It can be described as a collection of critical historical essays dealing with a large variety of mathematical disciplines and issues, and intended for a broad audience. Roughly half the book is devoted to the mathematics of the last two centuries. This emphasis on newer mathematics is one of the book's special virtues. Thus, a student with a modest mathematical background and an interest in its history will have no difficulty acquiring a substantial amount of knowledge of Greek mathematics, of algebra in the 16th century, of the discovery and triumphs of the calculus, of the discovery and role of non-euclidean geometry, and, if she tries hard enough, of the discovery and role of Lebesgue measure and integration. Acquring a more comprehensive view of the history and content of mathematics, and especially of the history and content of the mathematics of the last two centuries, is not a simple
Chapter 1: Pythagoras's theorem. What m a k e s Pythagoras's theorem important is that it straddles the divide between the discrete and the continuous, between arithmetic and geometry. On the arithmetical side, the theorem is linked to the ancient problem of Pythagorean triples and to its near equivalent, the problem of rational points on a circle. The latter is a simple instance of the problem of finding rational points on algebraic curves, dealt with systematically first by Diophantus (the method of secants and tangents), and then by Fermat, Euler, Jacobi, PoincarG and, in our own time, Faltings. See Chapers 1, 3, 10, 14, and 15. On the geometric side, Pythagoras's theorem was involved in the shattering discovery that "not all is number.'" There exist magnitudes, such as the side and diagonal of the unit square, whose ratio is not the ratio of whole numbers. This led to a split between the theories of number and of space. Nevertheless, the Greek THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992
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geometers developed ingenious techniques for the precise handling of ratios of arbitrary (like) magnitudes in terms of rationals. In the 19th century, Dedekind reconsidered these techniques and realized that they provided an arithmetical interpretation of irrational quantities after all (see Chapter 4). It was then possible, as Hilbert showed, to reconcile the apparent conflict between arithmetic and geometry. The key role of Pythagoras's theorem in Hilbert's "number model" of (plane euclidean) geometry is obvious from the definition of distance between points in that model. See 1.6 (Section 6 of Chapter 1).
Chapter 2: Greek geometry. Here we restrict ourselves to two comments: (a) The axiomatic method was one of the greatest intellectual achievements of the Greek mind, and Euclid's Elements was the embodiment of the axiomatic method. The critical difference between Euclid's axiomatics and modern axiomatics is that axioms are no longer regarded as self-evident. (b) The discovery of analytic geometry moved synthetic euclidean geometry to the sidelines. But algebra caught on in elementary geometry only at the end of the 18th century. Euclid's system was overthrown in the 19th century as a result of the discovery of noneuclidean geometries and the axiomatization of the real number system.
Chapter 3: Greek number theory. Diophantus was interested in finding rational solutions of algebraic equations, or, to use modern geometric terminology, rational points on algebraic curves. To do this he (presumably) made systematic use of two related methods, namely the method of chords and the method of tangents. This calls for an explanation. The Riemann surface of a (complex) quadratic curve or of a singular cubic curve is topologically a sphere, and so has genus (number of holes) 0. Such curves admit parametrization in terms of a pair of rational functions of a single parameter. On the other hand, the Riemann surface of a nonsingular cubic is topologically a torus, and so has genus 1. Such curves admit parametrization in terms of a pair of elliptic functions. The same applies to other algebraic curves of genus 1. (Poincar6 and Koebe showed that all algebraic curves admit parametrization in terms of a pair of automorphic functions of a single parameter.) Consider rational curves of genus 0. The rational points on such a curve, if any, are obtainable by assigning rational values to the parameter in the two rational functions that parametrize it. In the case of a rational curve of genus 1 we must rely on the addition of two rational points to find a third. The geometric 70
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sense of this operation is that of Diophantus's (tangent or) secant method. The latter consists in intersecting the curve with the secant determined by two given rational points on the curve to obtain a third one. A remarkable theorem, conjectured by Poincar6 and proved by Mordell, states that all rational points on a cubic curve can be generated by tangent and chord constructions applied to finitely many points.
Chapter 5: Polynomial equations. Here we restrict ourselves to a comment on the nonalgebraic solution of polynomial equations. While he helped free algebra from the domination of geometry in one sense, Vi~te strengthened its ties with geometry at a higher level by relating algebra to trigonometry. He reduced x3 + ax + b = 0 to the oneparameter equation 4y3 - 3y = c, and, by putting y = cos 0, reduced the latter to the equation cos 30 = c. In 1786 Bring reduced the general quintic to the oneparameter form x5 - x - A = 0. This opened the way for its solution by elliptic modular functions (Hermite, 1858) analogous to Vi6te's solution of the cubic by circular functions. Hermite's astonishing discovery was later placed in a geometric setting by Klein (see Chapter 18). However, this approach to the solution of algebraic equations was abandoned once Galois's methods became generally known. Galois answered the age-old question of solvability of algebraic equations by radicals. What was far more important, however, was that his novel approach initiated the slow transformation of algebra from a discipline concerned with the study of algebraic equations to a discipline concerned with the study of fields, rings, groups, and so on.
Chapter 6: Analytic geometry. For the Greeks, an equation of a curve was a by-product of its geometric construction. Following Descartes's coordinatization of the plane, algebraic methods became dominant in the study of curves. An equation defined a curve, and (polynomial) equations became the key to the theory of (algebraic) curves. For example, Wallis derived the equations of the conic sections from their geometric definitions, but then deduced their properties from their equations. Again, in his classification of cubics, the first great achievement of curve theory beyond the Greeks, Newton used the tools of analytic geometry. He also adopted the projective viewpoint (see next Chapter). The final step to the arithmetization of geometry was taken by Hilbert, who introduced the algebraic model of plane euclidean geometry mentioned in Chapter 1: It is worth noting that Hilbert was motivated in part by the desire to provide a secure logical foundation for geometry!
Chapter 7: Projective geometry. Here we restrict ourselves to two comments. (a) One of the great insights involving projective geometry is due to Cayley and Klein. They showed that euclidean and hyperbolic geometry can be viewed as subgeometries of projective geometry. This was a major step in the transition from one geometry to many. (b) The very validity of certain mathematical statements depends on the use of particular settings. Thus the familiar assertion that a polynomial equation of degree n has n roots is only true provided that we work with complex numbers. Again, two algebraic curves of degrees m and n, respectively, have mn points of intersection (the theorem of B6zout) provided that we describe them in terms of complex projective coordinates. This makes clear the importance of the complex projective setting in algebraic geometry.
Chapter 8: Calculus. What is calculus? Archimedes's computation of the area A of a parabolic segment involved finding an exhausting sequence A 1, A 2, A 3. . . . of A, guessing the sum B of the series A~ + A 2 + a 3 + . . . . and proving (by double contradiction) that A = B. The key to the calculus computation of A is the evaluation of a definite integral of the form f~ x2dx = a3/3. Thus invention and proof are replaced by a calculating routine. At first the calculating routines involved integration and differentiation of polynomials and implicit differentiation of polynomials in x and y. Allied to algebra and analytic geometry, this was sufficient to find tangents as well as extreme values for all algebraic curves. This was soon extended to integration and differentiation of "infinite polynomials," that is, all functions representable by power series (Newton). The subsequent development of calculus centered on the question of its logical justification and did little to make it a more useful computational tool. To Newton the evaluation of f flx)dx was the problem of expanding f(x) in a series to be integrated termwise. He was not overly concerned about apt notation. Leibniz is sometimes said to have invented a remarkable system of notation (rather than calculus). He preferred "closed form" expressions to infinite series. Thus, for him, the evaluation of f flx)dx was the question of finding a combination of elementary functions whose derivative was fix). This is a much more difficult problem than term-byterm integration, and nothing like a "calculus" was found for it in Leibniz's time. In fact, the problem of deciding which algebraic functions may be integrated in closed form was solved only over the last 20 years (by Risch, Davenport, and others). While the search for closed-form expressions for f f(x)dx was a wild goose chase, it led to worthwhile results in other directions. Attempts to integrate ratio-
nal functions raised the problem of factorization of polynomials and led to the fundamental theorem of algebra. Attempts to integrate 1/(1 - x4)1/2 led to the theory of elliptic functions.
Chapter 9: Infinite series. Newton was well aware of the limitations of power-series representations of functions. In 1671 he discovered that any algebraic function y can be expressed as a fractional power series in x: y = ao + a l x r l + a2 xr2 + 9 9 9
where r 1, r2, r 3. . . . are rational numbers. Furthermore, this can be rewritten as ao + blx ~1Pl(x) + 999+ bnx~~ Pn(x), where Pl(x) . . . . . Pn(x) are power series and sl . . . . . sn are rational numbers. A more rigorous derivation of Newton's series, with x, y complex, was given in 1850 by Puiseux (Puiseux expansions).
Comment on Chapters 10, 11, 14, and 15. These chapters deal very effectively with the complex of issues associated with elliptic integrals and elliptic curves. The clarification of the issues involved requires a veritable catalogue of tools from analysis and topology. At the same time one can see how these tools can be used to solve a riddle that challenged the ingenuity of Euler, Jacobi, and Poincar6. The early part of Chapter 10 discusses some results in number theory, which is not only the origin of elliptic curves, but also the scene of many of their current applications.
Chapter 16: Differential geometry. In contrast to algebraic curves, which could be studied in some depth by purely algebraic methods, transcendental curves (Leibniz's term; Descartes called them mechanical curves) were inseparable from the methods of calculus.
An early by-product of the investigation of transcendental curves by means of calculus was the solution of the ancient problem of arc length, thought by Descartes to be unsolvable. Hence the "infinitesimal" ideas of differential geometry first emerged from the investigation of transcendental curves. An early by-product of the investigation of transcendental curves by means of calculus was the solution of the ancient problem of arc length, thought by Descartes to be unsolvable. The geometric and physical significance of curvature cannot be overestimated. This significance grows as THE MATHEMATICALINTELLIGENCERVOL, 14, NO. 3, 1992 7 1
we go from curves to surfaces and on to higherdimensional spaces, and is especially great in geometry and in physics. The notion of curvature of a surface is due to Gauss. It is a rather natural extension of the notion of curvature of a plane curve. Gauss introduced the notion of the intrinsic geometry of a surface. This is based on the notion of a geod e s i c - t h e surface counterpart of a straight line in the plane. Gauss's great discovery was that curvature is one of the intrinsic properties of a surface, that is, it can be expressed in terms of the first fundamental form of the surface and its derivatives. The question of "rigid" motions on a surface leads to an interest in surfaces of constant curvature. The chapter ends with the Gauss-Bonnet theorem. The full significance of this remarkable theorem is made clear in Chapter 19.
unit disk model in 1854, but neither of them noticed the connection with hyperbolic geometry.) Beltrami extended his idea to n dimensions. In particular, the upper half, z > 0, of ordinary (x, y, z)-space with distance ds = V d x 2 + dy2 + dz2
became a model for hyperbolic 3-space. Here "lines" are semicircles orthogonal to z = 0, "planes" are hemispheres orthogonal to z = 0, and horospheres are spheres tangent to z = 0 together with the planes z = constant. Since on the latter the hyperbolic metric is determined by V d x 2 + dy 2 + dz 2 ds =
Chapter 17: Non-euclidean geometry. Here we take for granted the early phase of the development of noneuclidean geometry and focus on the contributions of Beltrami. While Bolyai and Lobachevski may be said to have approached hyperbolic geometry from the axiomatic side, Beltrami approached it from the side of differential geometry. First Beltrami f o u n d that surfaces that can be mapped into the plane with preservation of geodesics (i.e., their geodesics go over into straight lines in the plane) must have constant curvature. In particular, a hemisphere can be mapped by a central projection onto the plane, and a negatively curved surface S onto a portion of the unit disk. Beltrami's brilliant idea was to extend the image of S with its notion of line and distance to the whole unit disk, and thereby create the first model of the hyperbolic plane, which is known, for valid reasons, as the projective model. The projective model of the hyperbolic plane distorts angles as well as lengths. Beltrami went on to create various conformal models of the hyperbolic plane. To this end he first p r o j e c t e d his projective model "'upward" to a hemisphere, and then used two different central projections of his conformal hemispheric model to obtain the two familiar conformal "Poincar6 models" of the hyperbolic plane. Distance is particularly easy to express in the halfplane model. In fact, the distance ds between (x, y) and (x + dx, y + dy) is given by as = V'dx= +
Y (This formula was obtained by Liouville in 1850. Riemann obtained the distance formula for the conformal 72
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constant
Beltrami obtained a remarkably straightforward proof of Wachter's earlier discovery that the intrinsic geometry of the horosphere is euclidean.
Chapter 18: Group theory. Groups first turned up in mathematics as groups of permutations of a finite set. The abstract definition of a group is empty, in the sense that every group can be thought of as a group of permutations. Galois groups are permutation groups that arise in an algebraic context. Polyhedral groups are geometric groups, but they turn out to be the natural permutation groups A 4, $ 4 , and A s. Starting from this observation, Klein related the theory of equations to the symmetries of the regular polyhedra and functions of a complex variable. The complex variable appears w h e n the regular polyhedra are replaced by regular tessellations of the sphere C U % and their symmetries by linear fractional transformations (see Section 17.6). Klein showed that, with trivial exceptions, all finite groups of linear fractional transformations come from the symmetries of the regular polyhedra in this way. The regular polyhedra were also the source of another approach to groups: presentation by generators and relations. For details see Section 18.6. Finally, Klein used the group concept to define a geometry as the totality of invariants of a group of transformations of a set.
Chapter 19: Topology. General topology is the study of groups of homeomorphisms of very general (topological) spaces. When the space is R" or a subset of R n, then we have geometric topology. The simplest significant part of geometric topology is surface topology.
Descartes and Euler discovered the Euler polyhedron formula: V - E + F = 2. The first to understand its topological significance was probably PoincarO. It is a special case of a topological invariant called the Euler characteristic • of a surface. Many lines of research led to the question of topological classification of surfaces. They derived from Euler (classification of polyhedra), Riemann (classification of Riemann surfaces of algebraic curves), Poincar6 and Klein (classification of symmetry groups of tessellations; see Section 19.6), and MObius (classification of smooth, closed surfaces in ordinary space). These lines of research converged w h e n it was realized that all these surfaces are generalized polyhedra, or closed surfaces, now described by topologists as compact, connected, and without boundary. The Euler characteristic • of a closed surface S with p holes closed with handles is 2 - 2p. If the p holes in S are closed with MObius strips, then x(S) = 2 - p. If there are p + q holes in S closed with p handles and q MObius strips, then x(S) = 2 - (2p + q). If we know that S is orientable (nonorientable), then all we need to know to assign it to a unique topological class is the value of • The "global form" of the Gauss-Bonnet theorem states that for a closed orientable surface S with Gaussian curvature K we have: total curvature = .f f s KdA = 2-rr• Thus the total curvature determines the topology of a closed orientable surface!
Chapter 20: Sets, logic, and computation. Rather than attempt the impossible task of surveying 20thcentury mathematics, in this final chapter Stillwell focusses on the question, "What is mathematics?" More specifically, he expounds the remarkable insights of 20th-century logicians, which throw completely new light on this question. Following are some of the key issues and observations. 1. Generation of ever higher ordinals and cardinals. 2. Independence of the continuum hypothesis of standard set theory. This suggests the possibility that the notion of "'set" is open to different natural interpretations, like the notion of "straight line." 3. The connection between measure-theoretic axioms and set-theoretic axioms. Here a specific insight is that Lebesgue measurability of all subsets of R is intimately connected with the existence of sets large enough to model the whole of set theory. 4. Algorithms and Turing machines. All possible rules or algorithms for the solution of problems can be realized by Turing machines. GOdel spoke of this as a "kind of miracle." 5. GOdel showed that any consistent axiomatic system that includes ordinary arithmetic contains undecidable propositions. Also---and this is of crucial importance---through an ingenious encoding, one of
the unprovable propositions of such a system can be interpreted as asserting the consistency of the system. 6. The ability to see meaning in the axioms of Peano arithmetic enables us to see the truth of their consistency, and hence to transcend the power of formal proof. We hope that the preceding summary does not give the impression that Stillwell's book covers entirely familiar ground. In fact, we know of no book on mathematics and its history that covers half as much nonstandard material. Even w h e n dealing with standard material, Stillwell manages to dramatize it and to make it worth rethinking. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many. John Fauvel Faculty of Mathematics The Open University Milton Keynes, MK7 6AA England Abe Shenitzer Department of Mathematics & Statistics York University North York, Ontario M3J 1P3 Canada
A History of Non-Euclidean Geometry By B. A. Rosenfeld Berlin: Springer-Verlag, 1989; ix + 471 pp. US $89 (ISBN 0-387-96458-4). Reviewed by John McCleary It has been almost two hundred years since the birth of Nikolai I. Lobachevskii (1792-1856), who shares the credit with J~nos Bolyai and Carl Friedrich Gauss for the discovery of non-Euclidean geometry. A History of Non-Euclidean Geometry appeared in Russian in 1976 to mark the 150th anniversary of Lobachevskii's first lecture in Kazan on the new geometry. To the author, February 23, 1826, the day on which Lobachevskii lectured, marks the beginning of the modern era of mathematics, the "'era of non-Euclidean mathematics," in which notions such as non-Euclidean and projective geometries, the algebra of groups and fields, and the theory of sets are the results of a n e w freedom to construct and apply structures w i t h o u t classical analogues. That Lobachevskii led this change is certainly an overstatement, but his contributions helped focus the work of several of the leaders of the modern era; for example, Cayley, Klein, Weierstrass, and Poincar& The heady goal of this book is to describe the evolution of our concept of geometric space from antiquity to the modern era in the light of non-Euclidean and not Euclidean geometries. Rosenfeld's success in achieving this goal adds much to our understanding of geometry. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 7 3
1733) quadrilaterals. This part of the book is built on
POSTULATE V: That, if a straight line falling Rosenfeld's extensive scholarship on the medieval Ison two straight lines makes the interior an- lamic geometers; it makes the book unique among surgles on the same side less than two right an- veys of the history of non-Euclidean geometry. The problem of proving the Parallel Postulate drew gles, the two straight lines, if produced indefthe attention of European geometers when Latin ediinitely, meet on that side on which the angles tions of Euclid with commentaries by Arabic geometers are less than the two right angles. became available in the sixteenth century. The most In its prehistory geometry was a practical matter of land measure and stellar motion. To the Greeks of the Golden Age these and other practical matters were the stuff of contemplation and philosophy. Centuries of systematic reorganization culminated in Euclid's Elements, which has stood as a model of pure mathematical discourse for m a n y generations. To its author and early readers, however, the Elements provided an idealized description of the space around them. From this viewpoint it is natural to understand the objections to Euclid's Parallel Postulate by early critics. The phrase "if produced indefinitely" strains an intuition based on constructions with compass and straight-edge. Furthermore, Euclid studiously avoided using the Parallel Postulate for the first 28 Propositions of Book I. The first application was to prove 1.29, the converse of 1.27 and 1.28. Several of the previous Propositions are related to their neighbors as converses with proofs that simply observe the contradiction to the earlier statement were the converse false. Proposition 29 does not yield to this logic, and against the backdrop of the development of Book I up to 1.29 it is disturbing to introduce such an unnatural statement as the Parallel Postulate. To eliminate this blemish on Euclid's great work, subsequent generations of geometers sought a proof of the Parallel Postulate from the other assumptions, or they tried to replace it with a more selfevident assumption. Ptolemy and Proclus, the earliest noted critics of Euclid, are among the few whose work did not perish in the fires of Alexandria. Most of our present knowledge of Greek mathematical texts is owed to another high culture, that of medieval Islam. Arabic geometers studied astronomy and time-keeping, which led them to develop sophisticated theorems in spherical geometry. Their translation of the Elements and of the intervening criticism kept alive these ideas and led them to apply considerable ingenuity to improving Euclid. The best known (through the accounts of Heath [2] and Bonola [1]) among these geometers are al-Nayrizi, Thabit ibn Qurra, and Na.sir al-Din al-T.Qsi. Rosenfeld discusses their work along with the work of m a n y others who contributed to a rich and previously ignored mathematical life. Among those who attempted proofs of the Parallel Postulate are Ibn al-Haytham (965-1041) and "Umar Khayyam (1027-1123, of Ruba'iyat fame), who developed theories of parallels based on the figures now called Lambert (1728-1777) and Saccheri (166774
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celebrated attempts are due to Wallis (1616-1703), Saccheri, Lambert, and Legendre (1752-1833). (Newton also sought to improve Euclid but he focused on Book II.) During the century before Lobachevskii's birth, European mathematics changed significantly. Calculus had arrived, and there was a tremendous push to develop and extend this powerful tool. In previous centuries mathematicians were occupied with ideas derived from classical geometry and arithmetic, along with primitive ideas of motion. By the beginning of the nineteenth century such methods and problems as the Parallel Postulate were no longer part of the mainstream of mathematics--analysis and its problems were the dominating concerns. Lobachevskii was part of this world of mathematics; he contributed papers on algebra and trigonometric series and taught courses on analytic topics and mechanics. His foremost interest was geometry, however, and though this placed him outside the mainstream, his work is inseparable from the interests of the time.
In geometry I find certain imperfections which I hold to be the reason w h y this science, apart from transition into analytics, can as y e t make no advance from that state in which it has come to us from Euclid.
In the first paper he submitted to the Petersburg Academy, On the principles of geometry (1829), Lobachevskii included applications of his new geometry to the computation of two definite integrals. Rosenfeld recounts the immediate reactions to Lobachevskii's work: the reviewer for the Academy, M. V. Ostrogradski/, ignored the geometry and concentrated on the integrals, and he observed that one was already known and the other was false. The criticism led to lampoons in Petersburg literary journals such as Son of the Fatherland (1834): Glory to Mr. Lobachevskii who took upon himself the labor of revealing, on one hand, the insolence and shamelessness of false new inventions, and on the other the simpleminded ignorance of those who worship their new inventions.
There is a certain irony in this rejection; it was the analytic features of his work that helped convince Lobachevskii of its consistency. He derived from nonEuclidean assumptions trigonometric formulas which were consistent with formulas found in spherical trigonometry, but for a sphere of negative radius 9 The denial of the Parallel Postulate did not lead to a contradiction in the analytic formulas from which much of the geometry could be derived. In the opening paragraph of "Geometric Investigations on the Theory of Parallels" (see [1]), Lobachevskii emphasizes these developments when he writes, In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it has come to us from Euclid. It w a s the analytics that gave him a n d his codiscoverer, J~inos Bolyai, the confidence to make public their new world. The trigonometric formulas play another role in Lobachevskii's work. The parameter in his formulas, related to the c u r v a t u r e of a constant-negativecurvature surface, can be bounded by measurements in the physical world. In particular, the greater the lower bound on this constant, the closer to Euclidean is our world of perception. Astronomical calculations of the parallax of Sirius allowed Lobachevskii to estimate a lower bound for this constant. At the scale of the Earth, non-Euclidean geometry and Euclidean geometry would be imperceptibly different. Thus the role of geometry, to describe the space around us, would be consistent in non-Euclidean geometry with our senses 9 Lobachevskii saw this as concrete proof of the correctness of Euclidean geometry; he writes, 9 . . one can imagine to what extent this difference, on which our theory of parallels is based, supports the accuracy of all calculations of ordinary Geometry and lends support to the attitude of regarding the principles of the latter to have been, presumably, rigorously established9
Lobachevskii realized (see [4]) that his work was not done; consistent trigonometric formulas were not enough to determine the existence of a non-Euclidean geometry. The problem of existence was solved by E. Beltrami (1835-1900), applying more powerful analysis in the form of differential geometry, in particular the notion of curvature, which was introduced by Gauss and developed by Riemann. The final steps are found in the investigation of surfaces of constant negative curvature begun by Minding (1806-1885), a student of Gauss, and carried out by the Italians Dini (1845-1918), Codazzi (1824-1873), and Beltrami. These led to Beltrami's 1868 paper "Saggio d'interpretazione della
geometria noneuclidea," which contains his famous model for the geometry of Lobachevskii and firmly establishes the relative consistency of non-Euclidean geometry with Euclidean. The d i s c o v e r y and later a s s i m i l a t i o n of n o n Euclidean geometry would not have been possible without the parallel development of various not Euclidean geometries: spherical, affine, and projective geometries, and algebraic geometry. In the modern era these currents mix to produce modern geometry with its topological, algebraic, and analytic facets9 The unification of such varied phenomena can be expressed by the algebraic theory of transformation groups and by the m o d e m theory of manifolds. The interplay of these developments and their generalizations led to the theory of Lie groups and Lie algebras and the geometric foundations of general relativity. Rosenfeld artfully sketches the history of these notions from their prehistory to recent times. Where some look at geometry and see algebra, Rosenfeld and his co-workers see geometry in algebra; of particular note are the chapters closing the book, on groups of transformations and applications of algebras, where he describes some of his own work on realizing certain groups as groups of transformations of non-Euclidean spaces. The non-Euclidean geometry of Lobachevskii is still thriving today, as are the many branches of not Euclidean geometry. One can imagine February 23, 2026 and the appearance of The Further History of Non-Euclidean Geometry with chapters on Thurston's Geometrization Theorem, applications of non-Euclidean geometry to the comparison of algorithms (work of Tarjan and Thurston), and the discovery of fake four-dimensional Euclidean spaces (Donaldson). Rosenfeld's history of non-Euclidean geometry from its prehistory through its first 150 years will be there as a point of departure-one rich with detail and insight.
R e f e r e n c e s
1. Bonola, R., Non-Euclidean Geometry, New York: Dover Publications (1955). 2. Euclid, The Thirteen Books of The Elements, translated and edited by Sir Thomas L. Heath, New York: Dover, second edition (1956). 3. Gray, J., "Non-Euclidean geometry--a reinterpretation," Hist. Math., 6 (1979), 236-258. 4. Gray, J., "The discovery of non-Euclidean geometry," in Studies in the History of Mathematics, ed. Esther R. Phillips, Washington, DC, MAA Publ. (1987).
Department of Mathematics Vassar College Poughkeepsie, NY 12601 USA THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 7 5
Robin Wilson*
Greek Mathematics I Pythagoras (c.580-500 B.C.), a semilegendary figure, was reputedly born on the Aegean island of Samos. After some years of travelling in Egypt and Asia Minor, he migrated to the Greek seaport of Crotona, now in Italy, where he founded the so-called Pythagorean school. This was a closely knit brotherhood whose purpose, according to later writers, was to further the study of mathematics, philosophy, and the natural sciences. Particular emphasis was laid on the four "mathematical arts": arithmetic (the theory of numbers), geometry, astronomy, and music--and on the development of deductive logical reasoning. It is not certain who first proved Pythagoras's theorem, but the result was certainly familiar to the Chinese, and Pythagorean triples were known to the Babylonians more than a thousand years earlier. These stamps, from a variety of countries, feature Pythagoras and his theorem. Included is a set of four Greek stamps, issued in 1955 to commemorate the life of Pythagoras.
* C o l u m n editor's a d d r e s s : Faculty of M a t h e m a t i c s , T h e O p e n University, Milton K e y n e s , MK7 6AA E n g l a n d . 76 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 3 9 1992 Springer VeflagNew York
