No title

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 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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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 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

,,

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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)

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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 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

Graduate Texts in Mathematics, Volume 123 Readings in Mathematics

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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 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,~ 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

THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992

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

THE MATHEMATICAL INTELL1GENCER VOL. 14, NO. 3, 1992

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