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

Chaos I cast two votes in favor of James Gleick in the debate [reviews section, Mathematical Intelligencer, vol. 10, no. 1, 1989] with John Franks. He gets my research mathematician's vote because, as I tried to argue in my recent article in the Notices of the American Mathematical Society ("Mathematics without Theorems?", December 1988, pages 1480-1482), mathematics is more than proving theorems. As the study of the various abstract structures and transformations we perceive in the world, mathematics certainly involves a great deal of theorem proving, but it also involves, and indeed depends upon, a study (from a mathematical standpoint) of various natural phenomena leading both to new applications of existing mathematical tools and to the development of new mathematical tools and "models" of the world. Gleick gets my pedagogic vote because he knew full well that almost all practicing scientists (apart from the pure mathematicians ourselves) have neither the time for, nor the interest in, the precise theorem-proof methodology of (what many mathematicians take to be) the mathematical enterprise. When my own mathematical research led me to start talking to other scientists, I was at first shocked to discover how little they cared about the kinds of things I, like most mathematicians, had spent m y career trying to do (namely, being rigorous and striving for proofs). Doubtless they thought it valuable, just as I value the skills of the mechanic who fixes my Buick from time to time. But they no more wanted to k n o w any of the details than I wanted to learn how my car engine functions. And this is just fellow scientists. Gleick was trying, and s u c c e e d e d b e y o n d a n y o n e ' s wildest d r e a m s (including, doubtless, his own), to reach a far wider audience. By an odd coincidence, Gleick's book appeared in the USA at about the same time my own "popular" book Mathematics: The New Golden Age came out in England, both carrying the same picture on the cover, and published by t h e "sister" companies Viking (USA) and Penguin (Britain). (Even more curious, I decided

to write this book rather than one on "chaos" that the Penguin editor had suggested to me.) Aiming at a "mathematically sophisticated" audience, I chose to include formulas, statements of theorems, and descriptions of proofs, but avoided like the plague any attempts at rigor, and make no apologies for that. Aiming at a still wider group, Gleick very wisely steered well clear of any whiff of real "mathematics" as it is perceived by most people. By reaching that wide group so spectacularly, he surely deserves our loud praise, not criticism for failing to produce a book he did not set out to write. As a profession, we have in the past had precious few outside supporters. Let's not turn against the few we n o w seem to be acquiring. Keith Devlin Department of Mathematics Stanford University Stanford, CA 94305 USA

Chaos, The review [Mathematical Intelligencer, vol. 10, no. 1, 1989] of Gleick's book Chaos: Making a New Science by John Franks reflects the somewhat narrow view that it is not enough for a book to describe mathematics, it must "explain" mathematics, and that means the careful stating of theorems and who proved what and when. Many people are interested in a description of mathematics, in the ideas and concepts of mathematics even if stated only informally or intuitively. Indeed, these people might want to use these concepts for framing and thinking about their own disciplines. And, some people may only be interested in a good story. Is it wrong for a book to try to meet these needs and to succeed? The situation is s o m e w h a t parallel to what can happen with a foreign language. A person is interested in learning a language: perhaps the student wants to visit a country where the language is spoken, perhaps the student needs to read a book in the language, or perhaps the student seeks to speak the language just for fun or for something to do. Occasionally

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a teacher is encountered who insists that the student learn the literature, the history, and culture of that language, arguing that this is the only proper w a y to study a language. Such a view is not without merit, but few would agree that is the only w a y to learn a language. I hope this analogy is precise enough to make my point. Namely, there should be room for a lot of books on mathematics doing all kinds of things, and that a book should be judged on how well it does what it tries to do, not what you believe it should do. Mathematicians often complain that the public does not appredate or understand what they do and that they are doing mathematics or what that even means. Books such as this one by Gleick help to change that. If it doesn't tell the story you want, the w a y you want, then write your own book!

Ronald G. Douglas Department of Mathematics SUNY Stony Brook, NY 11794 USA See the Opinion column of this issue of the Mathematical Intelligencer for additional discussion of the issues raised by Chaos and its review.

,Kramers's T h e s i s Advisor? In David Rowe's interview of Dirk Jan Struik (Mathematical Intelligencer, vol. 11, no. 1, 1989) the latter is quoted as saying that Hans Kramers took his Ph.D. under Niels Bohr at Copenhagen. According to the definitive biography of Kramers by Max Dresden, Kramers took his Ph.D. at Leiden in 1916 with Ehrenfest (see Max Dresden, H. A. Kramers: Between Tradition and Revolution, Springer-Verlag, 1987, p. 94). Kramers then went to Copenhagen to study with Bohr. In May 1919 Kramers returned to Leiden for his thesis defense. Kramers's official thesis advisor was Ehrenfest, but the work was on quantum theory in the Bohr style and Bohr was on hand at the thesis defense and asked questions (Dresden, p. 113).

Henry Heatherly Department of Mathematics The University of Southwestern Louisiana Lafayette, LA 70504 USA 9D i r k Struik R e p l i e s Mea culpa. I should have been more careful, since I still have Kramers's dissertation (Intensities of spectral lines) a m o n g my books. It is indeed a Leiden dissertation under Ehrenfest, but the text was written under Bohr's direction and published by the Danish Academy (1919), and has been reproduced in Kramers's Collected Scientific Papers, pages 3-108. What happened in 1916 is that Kramers passed his

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doctoral examination, also at Leiden. I remember it well, since I took the same examination on the same day. This gave us the title of doctorandus. I also rem e m b e r that on the day before the examination Kramers and I with some friends took a boat ride, ending up at the hospitable home of the Ehrenfests, who treated us to strawberries. On another matter, I read with consternation on page 17 of my interview that gentle Herman Weyl is supposed to have wished bodily harm on a colleague. Let us kill this story in the bud before it becomes part of mathematical lore. It was Schouten himself who, in later years referring to his dissertation of 1914 with its bewildering notation, remarked within my hearing at some German mathematical gathering that he would like to beat up the author of that dissertation ("Den Mann, der das geschrieben hat, m6chte ich erdrosseln").

Dirk Struik 52 Glendale Road Belmont, MA 02170 USA

a teacher is encountered who insists that the student learn the literature, the history, and culture of that language, arguing that this is the only proper w a y to study a language. Such a view is not without merit, but few would agree that is the only w a y to learn a language. I hope this analogy is precise enough to make my point. Namely, there should be room for a lot of books on mathematics doing all kinds of things, and that a book should be judged on how well it does what it tries to do, not what you believe it should do. Mathematicians often complain that the public does not appredate or understand what they do and that they are doing mathematics or what that even means. Books such as this one by Gleick help to change that. If it doesn't tell the story you want, the w a y you want, then write your own book!

Ronald G. Douglas Department of Mathematics SUNY Stony Brook, NY 11794 USA See the Opinion column of this issue of the Mathematical Intelligencer for additional discussion of the issues raised by Chaos and its review.

,Kramers's T h e s i s Advisor? In David Rowe's interview of Dirk Jan Struik (Mathematical Intelligencer, vol. 11, no. 1, 1989) the latter is quoted as saying that Hans Kramers took his Ph.D. under Niels Bohr at Copenhagen. According to the definitive biography of Kramers by Max Dresden, Kramers took his Ph.D. at Leiden in 1916 with Ehrenfest (see Max Dresden, H. A. Kramers: Between Tradition and Revolution, Springer-Verlag, 1987, p. 94). Kramers then went to Copenhagen to study with Bohr. In May 1919 Kramers returned to Leiden for his thesis defense. Kramers's official thesis advisor was Ehrenfest, but the work was on quantum theory in the Bohr style and Bohr was on hand at the thesis defense and asked questions (Dresden, p. 113).

Henry Heatherly Department of Mathematics The University of Southwestern Louisiana Lafayette, LA 70504 USA 9D i r k Struik R e p l i e s Mea culpa. I should have been more careful, since I still have Kramers's dissertation (Intensities of spectral lines) a m o n g my books. It is indeed a Leiden dissertation under Ehrenfest, but the text was written under Bohr's direction and published by the Danish Academy (1919), and has been reproduced in Kramers's Collected Scientific Papers, pages 3-108. What happened in 1916 is that Kramers passed his

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doctoral examination, also at Leiden. I remember it well, since I took the same examination on the same day. This gave us the title of doctorandus. I also rem e m b e r that on the day before the examination Kramers and I with some friends took a boat ride, ending up at the hospitable home of the Ehrenfests, who treated us to strawberries. On another matter, I read with consternation on page 17 of my interview that gentle Herman Weyl is supposed to have wished bodily harm on a colleague. Let us kill this story in the bud before it becomes part of mathematical lore. It was Schouten himself who, in later years referring to his dissertation of 1914 with its bewildering notation, remarked within my hearing at some German mathematical gathering that he would like to beat up the author of that dissertation ("Den Mann, der das geschrieben hat, m6chte ich erdrosseln").

Dirk Struik 52 Glendale Road Belmont, MA 02170 USA

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

Chaos, Rigor, and Hype Morris W. Hirsch Gleick's book Chaos [reviewed by John Franks, Mathematical Intelligencer, vol. 11, no. 1, 1989] captures vividly and faithfully the personalities of the researchers, the atmosphere they worked in, and the spirit of the times (as far as I can tell--I don't know all the people in the book), as well as skillfully expounding many fascinating themes of current research. But to my mind the book has one great defect: It doesn't do justice to the rigorous mathematics underlying a great deal of the research in nonlinear dynamical systems. In fact a nonmathematician or even a mathematician unfamiliar with the material would find it hard to tell that rigorous mathematical p r o o f - as contrasted with conjecture, heuristic, experiment, and computer simulations--played a vital part in this research. This book severely underrates the importance of rigorous mathematics and its influence on the understanding of chaotic dynamics. It is not easy to learn from it that m a n y important chaotic systems, including the earliest and the most influential examples, were first identified and explored not by computer simulation, and not by physical experiment, but by mathematical proof (PoincarG Birkhoff, Levinson, Smale, Anosov, Kolmogorov, Arnold, M o s e r . . . ) . These and many other mathematicians achieved by rigorous mathematical analysis crucial insights into what is now called chaos. It is difficult to imagine that what they discovered could have been found through any kind of experimentation any more than the existence of irrational n u m b e r s - - w h i c h was even more astonishing when it occurred--could have been discovered by computation. By ignoring this, Gleick missed an opportunity to attack the public's profound ignorance about the role and nature of mathematics; he has also misrepresented a chapter in the history of mathematics. Let me illustrate what I m e a n with 6

Smale's work on horsehoes, which is far more important than the book suggests. In discussing the horseshoe Gleick writes of Smale's "intuition," claiming that he "turned his ideas about visualizing global behavior into a new kind of model"; he "put his horseshoe through an assortment of topological paces," and "the horseshoe provided a neat visual analogue of the sensitive dependence on initial conditions that Lorenz would discover in the atmosphere a few years later''; "Smale's horseshoe stood as the first of m a n y new geometrical shapes that gave mathematicians and physicists a new intuition about the possibilities of motion. In some ways it was too artificial to be useful . . . . " But the only mention of mathematics in this discus-

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sion is "mathematics aside"! Thus the reader learns that Smale had a new "model" (of what?), but it was merely an "analogue" of what Lorenz would actually "discover." The horseshoe is "an enduring image," but it was "artificial"; the only reason it is famous is because it caused a "paradigm shift." This is profoundly misleading. It conveys little of what Smale accomplished, what its importance is, w h y it is has been so influential. The horseshoe is important because Smale proved very interesting things about it, and these theorems tell us important facts about many other dynamical systems, facts that could

not have beenfound by intuition, simulation, or experiment. In these theorems and facts lies the importance of Smale's work. The "paradigm shift" was due chiefly to them, not merely to a " n e w kind of model,'" "a neat visual analogue," or a " n e w geometrical shape" that was "too artificial to be useful." Smale proved the following things about the horseshoe map: 1. He proved it is chaotic. For example, arbitrarily near any point there are periodic points with arbitrarily high periods, and there is also a point whose orbit comes arbitrarily close to every other p o i n t - - a precise and extreme form of sensitive dependence of long-term behavior on initial conditions. 2. He proved the horseshoe is structurally stable. This is not something that can be discovered or verified from calculations or simulations. 3. He proved that any system having a "transverse homoclinic orbit" (a concept due to Poincar6) must contain a horseshoe as a subsystem; such systems are thereby proved chaotic. This is a profound result because in many systems coming from physics, biology, etc., it is comparatively easy to demonstrate the existence of such orbits. This result has been used many times to prove that horseshoes are embedded in many particular systems, thus revealing their chaotic nature. 4. He p r o v e d that the chaotic dynamics of the horseshoe is isomorphic to the dynamics of the shift map in the space of bi-infinite sequences of zeroes and ones (so-called "symbolic dynamics"). Now, the dynamics of the shift map are quite transparent; for example, it is easy to see that periodic orbits are dense. Smale's isomorphism immediately opened up the possibility of similarly analyzing other chaotic systems. A great deal of research has since been done on this, greatly enlarging our understanding of chaotic dynamics. 5. He proved horseshoes exist in all dimensions greater than or equal to 2: Therefore, structurally stable chaotic systems exist in great abundance in those dimensions (for flows one more dimension is needed). This was new and striking information. Earlier examples of chaotic systems, such as Birkhoff's, were strictly 3-dimensional.

These discoveries and the subsequent work they inspired gave strong impetus to the now prevalent belief that chaos is a common phenomenon. They are different

in kind from the simulations of Lorenz and others, have had a different kind of influence, and have led to different insights. Today finding horseshoes is one of the chief ways of analyzing the dynamics of a chaotic system and one of the very few ways of rigorously demonstrating chaos. Far from being merely an artificial image, the horseshoe is a natural source of chaos and a basic tool for investigating it in all fields of applied dynamics. Many of the new insights into nonlinear dynamics arising from the work of many mathematicians would probably not have been discovered (or appreciated, if they had been discovered) without their rigorous mathematical foundations. Moreover, these discoveries are not isolated facts; many of them are parts of extensive theories that have been shown to apply to many fields of science. In fact, they have also been

M a n y important chaotic systems, including the earliest and the m o s t i n f l u e n t i a l examples, were first identified and explored not by computer simulation, and not by physical experiment, but by mathematical proof. used in reverse to obtain new theorems about nonchaotic systems. In contrast, the insight provided by a simulation such as Lorenz's, no matter how striking, does not extend very far to other systems. I do not mean to denigrate the importance and influence of simulations and intuitions such as that of Lorenz and many others. The chaotic phenomena they have discovered are important, seemingly quite different from horseshoes, and apparently less tractable to rigorous analysis. Curiously, the horseshoe has sharpened appreciation of the Lorenz system, because the latter seems highly resistant to a rigorous analysis. I believe that only recently has it been proved that it has periodic orbits of arbitrarily high period. The horseshoe is a tame kind of chaos; compared to it the Lorenz system seems quite wild. I've discussed Smale's work because I know it well and it's extremely important. But there are many other examples of rigorous mathematics leading to insights about chaos. For example, Birkhoff proved what Poincar6 had conjectured: the existence in the restricted planar 3-body problem of infinitely many periodic orbits. Gleick mentions m a n y names, but again he fails to distinguish between computer simulation (e.g., by Ulam) and rigorous mathematics (e.g., by Kolmogorov). THE MATHEMATICAL INTELLIGENCER VOL. 1I, NO. 3, 1989

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Both Poincar6 and Birkhoff deserve much more space than Gleick gives them. And w h y isn't poor Levinson's name revealed on page 48? After all, he sent Smale the letter with the "robust and strange" ex-

To call a system chaotic is not to say much more than that it is nonlinear and complicated. ample, with "stability and chaos together"; following which, Smale's disbelief "slowly melted away." The discoverer of such an important example deserves credit. He's mentioned without explanation on page 182, but he is not indexed. Another difficulty with this influential book, as Franks emphasized, is that Gleick hyperbolizes chaos into "a way of doing science," "a method, a canon of beliefs" (p. 38), something with "universal laws" (p. 304). What does this mean? What are these laws? A remark of Ren6 Thorn is pertinent here: "It is to be expected that after the present initial period of wordplay, people will realize that the term 'chaos' has in itself very little explanatory power, as the invariants associated with the present t h e o r y - - L y a p u n o v exponents, Hausdorff dimension, Kolmogorov-Sinai ent r o p y - s h o w little r o b u s t n e s s in the p r e s e n c e of noise" (Behavioraland Brain Sciences, 10 (2) (June 1987), 182). Thom was criticizing the hypothesis of chaotic dynamics in a biological model. Indeed, I have observed many biologists looking to chaos as a magic key to unlock nature's secrets. Inquiry revealed that in fact they had a superficial knowledge of nonlinear dynamics and a respect for its difficulties. They had the impression, however, that there is a "theory of chaos" that would somehow solve their mathematical problems. I

told them that a chaotic system is really bad n e w s - - i f your system seems chaotic, perhaps it is, but y o u probably can neither prove nor disprove this, nor can you accurately simulate its long-term behavior. In any case, to call a system chaotic is not to say much more than that it is nonlinear and complicated. This was not a welcome message. After all, they had heard wonderful things about the wonderful world of chaos. N o w Gleick tells them that chaos is indeed a "method," with "universal laws." Such hype prejudiced a generation of biologists against the truly original ideas in Thorn's catastrophe theory. Nonlinear dynamics is in no danger of a similar fate; but exaggerated claims may send some scientists on a wild goose chase, while discouraging others from seeking the valuable insights that dynamics can offer.

Exaggerated claims m a y send some scientists o n a w i l d goose chase, w h i l e discouraging others from seeking the valuable insights that dynamics can offer. The foregoing complaints are of interest only to mathematicians. The defects I ascribe to the book will not interfere with anyone else's pleasure in reading it. Let me emphasize that, despite my criticisms, I think Gleick's book is really first rate. Anyone who has tried to explain mathematical concepts to nonscientists will appreciate h o w remarkable is its exposition of difficult ideas. But I wish he had given less publicity to the nonexistent science of chaos and more to rigorous mathematics.

Departmentof Mathematics Universityof California Berkeley, CA 94720 USA

James Gleick Replies I was already enjoying Morris W. Hirsch's thoughtful article (preceding), but I really sat up when I got to the last sentence! "'The nonexistent science of chaos... "? It seems to me that some mathematicians have a rather special view of the world of knowledge. They see a landscape, stretching back into the past, formed of proofs, theorems, mathematical results . . . these things are what knowledge is. Though I'm an outsider, I have a real affection for this way of seeing the world. I tried in Chaos to give some feeling for the beauty and the necessity of it. But of course, it's not the only way, and in fact it's a way that most scientists find peculiar, as much as they may respect it. Anyway, if you want to reconstruct any piece of sci8

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entific history, you have to see a landscape that contains not just results, b u t also people, journals, meetings, ideas, conversation. If y o u look at the agendas of recent meetings in various fields, at the contents of journals, at the research choices made by individual scientists, at the bureaucratic realignments of universities and funding agencies, y o u cannot really deny the existence of the intellectual movement I describe in my book. The mathematicians who, in these pages, have pretended not to see what I mean by "a way of doing science" or "a canon of beliefs" or "universal laws of complexity" are averting their eyes from this broader landscape. Meanwhile, I'm a little sad to be portrayed as an antagonist of mathematics. I'm particularly dismayed

that Hirsch feels it n e c e s s a r y to d e f e n d Smale from me. He says Smale is more important than I think, because Smale proved things which I omitted to detail (that any system having a transverse homoclinic orbit must contain a horseshoe as a subsystem, that the chaotic dynamics of the horseshoe are isomorphic to the dynamics of the shift map in the space of bi-infinite sequences of zeroes and ones, etc.). But look at the broader landscape! In this world,

Smale had influence through his published resuits, certainly, but he also left his mark by calling people's attention to an obscure paper of Lorenz and by suggesting research problems to physicists. Smale is more important than Hirsch thinks, partly because Smale proved things, sure, but also because he had the vision to choose a certain line of research at a historic moment, and because he had the personal force to gather around him a school of mathematicians and guide them along a rewarding path. His work was the finest sort of mathematics, but it also happens that Smale's research choices reflected a keen awareness of what physicists knew and did not know; he had (I believe) a particularly good understanding of the odd gulf that had developed between mathematics and physics. And he did something about it. Certainly the horseshoe was important because he proved things with it and about it. If he had not done so, it would have been worthless. But the horseshoe was also important to physicists who did not care about

those proofs. In the bit of history I attempt to describe in Chaos, Smale had influence through his published results, certainly, but he also left his mark by calling people's attention to an obscure paper of Lorenz and by suggesting research problems to physicists like Feigenbaum. Such contributions are undocumented and nearly invisible--they smack of "atmosphere" and "personality" if you w i s h - - b u t they are no less a part of the course of intellectual history. Furthermore--and this seems to be a bitter pill for some of you to swallow--there are times when mathematical proof (essential though it is!) comes, historically, as an afterthought. Lorenz's work had its greatest influence before anyone even could say with certainty that his attractor was chaotic. Lanford's proof of Feigenbaum was ingenious and admirable, but it did little, really, to validate Feigenbaum's breakt h r o u g h - - e x p e r i m e n t s a c c o m p l i s h e d that. Those mathematicians who choose to look only at the documented genealogy of published proofs do their discipline a disservice, I think. It's no wonder they find it

There are times when mathematical proof (essential though it is!) comes, historically, as an afterthought. awkward or unpleasant to assess Benoit Mandelbrot's place: here is a nominal mathematician who, ostentatiously not proving much, has concretely changed the working lives of many thousands of scientists. You should not doubt Mandelbrot's powerful originality and importance in the science of our t i m e - - t h o u g h I know some of you still do. You can think of mathematics strictly as a hermetic enterprise. You don't have to care about the lovely, erratic, fortuitous process by which knowledge from m a t h e m a t i c s springs forth into messier fields of science. You can study knots without caring a b o u t DNA or particle physics, and maybe you should. You can study the history of Smale's horseshoe without bothering to reconstruct precisely how and w h y physicists and other scientists have transformed the way they think about the behavior of dynamical systems. If your idea of scientific history is to limit yourself to the chain of proven mathematical results, then you might conceivably consider the new science of chaos to be nonexistent. But if you believe that something extraordinary has happened in the science of our time, you have to look further to understand scientists' decisions, influences, moments of error or enlightenm e n t - - t o understand the things they discovered and even the things they proved. 11 Garden Place Brooklyn, N Y 11201 USA THE MATHEMATICAL INTELLIGENCER VOL. II, NO. 3, 1989 9

Chaos, Bourbaki, and Poincar6 Benoit B. Mandelbrot

James Gleick's book, Chaos, needs no defense, but John Franks's review [Mathematical Intelligencer, vol. 11, no. 1, 1989] voices, on the topic of fractals, several unwarranted "feelings" that demand correction. A broader comment on Bourbaki and Poincar6 follows. Mr. Franks feels that "fractal geometry is a static theory only marginally related to any kind of dynamics." Wrong. The very definition of Julia sets and Kleinian limit sets concern the dynamics of the backward iteration of rational functions and of Kleinian groups. In more realistic dynamical systems, the boundaries of basins of attraction are usually fractaI, and many attractors are "strange," that is, they are fractals. Typically the invariant measures carried by these attractors are multifractal. Finally, the original definitions of the simplest fractals, such as the Cantor set or the Sierpi~ski curves, strike many observers as "static," hence "inadequate." In fact, there is a nice theorem of J. Hutchinson that can be reinterpreted (together with the Kleinian groups) in " d y n a m i c terms." In a different context, anyone attuned to statistical physics knows of the enormous effort students of fractals devote to the dynamics of the growth of Laplacian fractal aggregates. Mr. Franks feels that fractal geometry "describes better than it explains." True, and it could be taken as a compliment, because fractal geometry describes so m a n y phenomena in fields where explanation is at present a very distant dream. But it is unwarranted if taken as a criticism, because there are many important things that fractal approaches already explain very well, either by probabilistic arguments, or by dynamical-system arguments, or by the grand old equations that belong jointly to mathematics and physics. To take one example, the dynamics of rational iteration and the fractaI geometry of Julia sets are explained when one describes the lovely set to which Douady and Hubbard have attached my name, together with its complement of H u b b a r d rays. A very different example: Percolation clusters, which are random sets of central importance in statistical physics, are explained by a mixture of all kinds of fractaI techniques. Concerning Hausdorff dimension, Mr. Franks feels that fractional values have been emphasized solely to glamorize the concept. Wrong. To harbor this intention would have required more foresight than I possess. In any event, the first reaction of all scientists to my uses of a dimension that can be a fraction was (to 10

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re-use the famous words of C. Hermite) to "recoil in horror," and to urge me to desist. One of several virtues of my term "fractal dimension" is that it draws attention away from this quantity's being a fraction. Mr. Franks comments about the role of the concept of limit in measuring length. True, but not applicable in the case of Hausdorff dimension. There, the trouble does not reside in a limit, but in an infimum that is meaningless in physics and is often unmanageable in mathematics (e.g., for the graph of the Weierstrass non-differentiable function). Hence the need for alternative notions of dimension, generically called fractal. (These quantities, furthermore, can take distinct values, not only in the old mathematical examples, which are contrived, but also in many new examples, which are physically meaningful.) Incidentally, no one objects w h e n Hausdorff dimer~sion is called by its name. But some eyebrow rcLai~'When this term is used to denote a different fractal dimension or some freshly minted, and/or ill-specified, new variant of the same general idea. Mr. Franks also reaches outside of mathematics. He feels that the connection between fractal dimension and the "ordinary concept of dimension" is "'somewhat tenuous." I am not an expert on what is or is not "ordinary," but various notions of dimension that need not be integer-valued have become commonplace in physics, and fractal dimension has taken its place among them. Let us n o w move on beyond small details to some interesting issues. The study of chaos and of fractals,

and the review of J. Gleick's book by Mr. Franks ought to provoke a discussion of the profound differences that exist, therefore must be recognized and accepted, between the "top d o w n " approach to knowledge and the various " b o t t o m u p " or "self-organizing" approaches. The former tend to be built around one key principle or structure, that is, around a tool. And they

For Bourbaki, the fields to encourage were f e w in number, and the fields to discourage or suppress were many. rightly feel free to modify, narrow down, and clean up their own scope by excluding everything that fails to fit. The latter tend to organize themselves around a class of problems. A constant tension between these two approaches has proven to be fruitful, and one must not judge one by the criteria of the other, as Mr. Franks does in his review. The top d o w n approach becomes typical of most parts of mathematics, after they have become mature and fully self-referential, and it finds its over-fullfillment and destructive caricature in Bourbaki. I have indeed written that I have left France because of their stifling influence. Mr. Franks criticizes them solely for bad pedagogical style and defends them as threatened defenders of rigor. I feel that the matter has never reduced to pedagogy, and rigor was never attacked in serious discussions of Bourbaki. The serious issues were intellectualstrategy, in mathematics and beyond, and raw politicalpower. An obvious manifestation of intellectual strategy concerns "taste." For Bourbaki, the fields to encourage were few in number, and the fields to discourage or suppress were many. They went so far as to exclude (in fact, though perhaps not in law) most of hard classical analysis. Also u n w o r t h y was most of sloppy science, including nearly everything of future relevance to chaos and to fractals. N o w to the mathematicians' old and new choices of role models. Do not forget that, for Bourbaki, Poincar6 was the devil incarnate. They boasted that they were needed to clean up the "'mess" he had left behind, thus resuming an old concern among French mathematicians. In the 1880s, Hermite kept writing to Mittag-Leffier to complain about the inability of young Poincar6 to complete a proof. For students of chaos and fractals, Poincar6 is, of course, God on Earth. For emphasis, recall the classic quote from PoincarL's address to the Fourth International Congress of Mathematicians (Rome, i908). Having noted that the previous century had driven toward abstraction and absolute rigor, Poincar6 proclaimed that "Mathematics must return upon i t s e l f . . , and in returning upon itself, it goes back to the study of the h u m a n mind

which created it rather than to those creations which borrow the least bit from the external w o r l d . . , and the better, in consequence, does the mind come to know itself. But it is to the opposite s i d e - - t h e side of nature--against which we must direct the main corps of our army.'" Thus, Mr. Franks's apocryphal story, which makes Poincar6 represent mathematics against Einstein, is odd indeed! Clearly, one strength of the recent studies of chaos is that they combine the heritage of Poincar6 at his sloppiest yet most vigorous, and the heritage of an already purified Poincar6. Overall, it is a slowly self-organizing environment, in which only a few parts have already acquired a top down structure. Within the dichotomy we are discussing, certain statements by Mr. Franks are revealing. When disagreeing with Mr. Gleick's characterization of Chaos as a "recent scientific revolution" that did not arise in the mathematical mainstream, Mr. Franks wishes to put the accent, not on the unifying problems, but on a unifying tool (the next best thing to a mathematical structure), namely the computer. Mr. Franks also reveals that he had once been infatuated with catastrophe theory. Ren6 Thom has of course been critical of Bourbaki, and his vivid love for geometry recommends him to every geometer. However, read his recent outburst of scorn for measurement and experiment (as published in Le D~batand other places). It fits

For students of chaos and fractals, Poincar~ is, of course, God on Earth. into a widespread view of catastrophe theory as a quixotic attempt to bring whatsoever extreme of the top down method, without any intermediary stage, into the messiest and most elusive of all the sciences. Was a swing from Poincar6 to Bourbaki and back to Poincar6 normal and preordained, a matter of healthy overall development? There is a widely held cyclic view that history is merely a harmonic pendulum that will swing back and forth, from boom to bust, from anarchy to tyranny . . . . Of course, such labels chop the history of mathematics into manageable chunks, to echo Lord Keynes's remark that business cycles help divide books of economic history into thinner tomes. But Keynes also thought that business cycles have no predictive value, and his opinion should also apply to the cyclic view of mathematics. One of the best effects of chaos theory and of the current revival of Poincar6 might be that crude pendulum-model thinking may at long last be put to final rest. In any event, it is easy to argue that the main reason w h y Bourbaki arose was not internal to mathematics, but externally motivated by a few brilliant persons and by their responses to various aspects of France after THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

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World War I and then after World War II. In the France of the 1940s and 1950s, ideologies reigned and seemed to lead my most brilliant contemporaries to serve one of a small list of all-devouring causes. When J. P. Kahane reminisced on that period a few years ago (in Le Monde), he wrote that he had been saved from Bourbaki (and le formalisme ~ la fran~aise) by his devo-

When J. P. Kahane reminisced on the period, he wrote that he had been saved from Bourbaki (and le formalisme/l la fran~aise) by his devotion to Marx. Others picked a religion or a nationalism. tion to Marx. Others picked a religion or a nationalism. (How did I avoid picking any paramount cause? I may never understand that.) It is even more obvious that the present "move of the pendulum" in a new direction had many causes, including the computer, that are external to mathematics. Last but not least, Bourbaki showed extraordinarily wide-reaching concern with political influence across the age groups and across the disciplines. Power to school the children (even though they disown today the French "new math" of yesterday), to educate the young to have the "correct" taste, to cow the old, and

to shun the strong-willed heretics. And "export" of their standards of rigor and taste where they do not belong has done untold harm. Mr. Franks criticizes fractal geometers because they fail to define this field in clear mathematical fashion. Had they claimed it to be a settled, top down theory, this w o u l d have been a dreadful failure. But in the case of a youthful trend not far along a path to self-organization, this is a non-issue. In any event, the availability of a full-fledged top d o w n theory has never been a prerequisite for a happy and useful life. Take probability theory; old as it is, it has not yet reached this stage, and altogether prospers best when it makes full use of a messy mixture of structures: combinatorics, measure theory (yes, of course), and old-fashioned hard analysis. Similarly, today's fractal geometry prospers by being an assemblage of well-defined chapters, namely the studies of self-similar sets and measures, self-affine sets (very hard and full of open problems), self-squared sets (the study of the map z z 2 + c), self-inverse sets (Kleinian limit sets), self-etc. Only after we determine which groups of transformations one should or should not allow in a "proper" fractal, will choosing a definition and preparing a top down presentation become worthwhile tasks. Thomas J. Watson Research Center IBM P.O. Box 218 Yorktown Heights, NY 10598 USA

Comments on the Responses to my Review of Chaos John Franks The several responses to my review raise some interesting questions. What does "doing mathematics" mean? Is it possible or desirable to give an honest explanation of its meaning to a general audience? H o w important is the role of theorem-proving in doing mathematics? It is wrong to try, as James Gleick does in these pages, to make a dichotomy between discovery and proof in mathematics. Usually the discovery is the proof, or at least it is inextricably tied to it. Very rarely is a proof a historical afterthought. Almost always it is a proof, or the process of finding it, that creates new mathematical knowledge. A proof is not some kind of super spelling checker that merely validates mathematical facts. Most mathematicians would consider proofs to be the central content of mathematical knowledge. Who would be satisfied if God were to announce that the Riemann Hypothesis is true, but deny us the proof? 12

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Let me paraphrase an old joke about money. In "doing mathematics" proving theorems isn't everything, but it's w a y ahead of whatever is in second place. Proving a theorem is one of the most creative acts of which the human mind is capable. Most mathematicians find it an exhilarating experience. It is often exciting, sometimes even thrilling. There is also pain and disappointment w h e n a putative theorem falls through. Several years ago a well-known mathematician caused a minor flap by comparing mathematics to sex (in the pages of the Bulletin of the American Mathematical Society). The propriety of his simile may have been questionable, but I prefer it to the comparison in Professor Devlin's letter in response to my review, w h i c h likens p r o v i n g a t h e o r e m to a m e c h a n i c working on a Buick. Mr. Gleick is probably right when he suggests that m a n y m a t h e m a t i c i a n s consider the concept of a "mathematician .... o s t e n t a t i o u s l y not proving

much" as something of a self-contradiction. A view commonly held by mathematicians is that a mathematician is someone who creates mathematics and that, for the most part, creating mathematics means discovering and proving theorems. There is no intent here to denigrate those who use mathematics but prove no theorems, especially those who use it in creative ways. But there is a difference between a composer and a musician. And mathematicians take great pride in their c r a f t - - e v e n to the extent of believing the theorem may outlive the application, as the musical composition outlives the performer. Mr. Gleick considers this a myopic view of the history of science; I acknowledge it is not an unbiased one. I hope, at least, he realizes it is not a petty jealousy directed at a single individual. By and large, mathematicians have done a terrible job of communicating to the general public what we are doing (or what we think we are doing). Professor Devlin says, "Gleick very wisely steered well clear of

A proof is not some kind of super spelling checker that merely validates mathematical facts. any whiff of real 'mathematics' as it is perceived by most people." I saw, instead, a missed opportunity. Mr. Gleick is a rare find for the scientific community - - a n exceptionally talented writer with an interest in science. I do not believe he i s a n antagonist of mathematics, quite the contrary. But I wish he had chosen to give his readers "a whiff of real mathematics" as it is perceived by mathematicians. Obviously this does not mean citing theorems and proofs; and it certainly does

not mean reciting a list of who did what when. It does mean explaining the mathematician's perspective on mathematical creativity as a process by which new knowledge is attained. Professor Hirsch does not suggest this process is the only legitimate topic in a history of chaos, but he is correct in saying it played a major role, which Gleick neglected in his book. Mr. Gleick tells us that Lanford's proof w a s n ' t needed to validate the discoveries of Feigenbaum. But, by the same token, applications to DNA w e r e n ' t needed to validate the theorems of knot theorists. It is a wonderful thing when a branch of mathematics suddenly becomes relevant to new discoveries in another science; both fields benefit enormously. But many (maybe most) areas of mathematics will never be so fortunate. Yet most mathematicians feel their work is no less valid and no less important than mathematics that has found utility in other sciences. For them it is enough to experience and share the beauty of a new theorem. New mathematical knowledge, like knowledge of subatomic particles or knowledge of Mars, is an end in itself! This is part of the "whiff of real mathem a t i c s " we n e e d to c o m m u n i c a t e to the general public.

Department of Mathematics Northwestern University Evanston, IL 60208 USA

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Allen Shields* Banach Algebras, 1939-1989 In a remarkable series of three notes written fifty years ago in the Doklady Akad. Nauk, I. M. Gel'fand [1939] sketched the theory of commutative Banach algebras together with its principal applications. This was apparently his doctoral dissertation (in the bibliography of Silov [19401] the dissertation title is given as: The theory of normed rings). According to Mat XL, vol. II, Gel'fand was born 20 August 1913 in the Odessa region; he received the degree of Candidate in 1935 and the degree of Doctor in 1940. The Candidate of Science degree is equivalent to an American Ph.D., while the Doctorate is a more advanced degree. As happens only very rarely in mathematics, this new theory was essentially the creation of one person. Just preceding these Doklady notes, there was a joint paper by Gel'fand and Kolmogorov [1939] entitled On rings of continuous functions on topological spaces. In it they prove that if S1 and S2 are completely regular spaces, and if their rings of real-valued continuous functions are algebraically isomorphic as rings, then S1 and S2 are h o m e o m o r p h i c . The p r o o f begins by showing that there is a one-to-one correspondence between the maximal ideals in the ring of functions and the points in the underlying space. The space is recovered by introducing a suitable topology on the set of maximal ideals. Thus one has a hint of the Gel'fand theory of commutative Banach algebras in this context. A word about terminology. Gel'fand always spoke of "normed rings," although they were really algebras over the field of complex numbers. In Naimark [1956] they are called "Banach rings." The term "Banach algebra" seems to have been used first by Ambrose [1945]. In the first of Gel'fand's three notes the basic definitions and theorems are stated; no proofs are given. Eight theorems are stated, the last being the following theorem on automatic continuity: Let A,B be complete (commutative) normed rings, B having the additional property that the intersection of all its maximal ideals is {0}. Then any homomorphism of A onto B is continuous. (The non-commutative analog of this is much harder; it was finally proved by B. E. Johnson [1967].) * C o l u m n editor's address: D e p a r t m e n t of Mathematics, University of Michigan, A n n Arbor, MI 48109-1003 USA

The second note proves the theorem that if an element never vanishes (when viewed as a function on the space of maximal ideals), then it has an inverse. This is used to prove Wiener's theorem that if an absolutely convergent trigonometric series never vanishes, then the reciprocal has an absolutely convergent series. (Wiener [1932] p r o v e d this originally as a lemma needed in the proof of his Tauberian theorem.) Gel'fand points out that the method can also be used for absolute convergence with weights and for absolutely convergent Fourier-Stieltjes integrals on the line (thereby generalizing a result of Wiener and Pitt). He also considers absolutely convergent series, with weights, on general abelian groups (with the discrete topology). This paper had a very great impact on mathematicians the world over and helped lead to the rapid acceptance and development of the new theory. The third note deals with the ring of almost periodic functions in the sense of Harald Bohr. Gel'fand proves that this ring is isometrically isomorphic to the ring of all continuous functions on the (compact) group of characters of the additive group of real numbers (with the discrete topology), nowadays often called the Bohr group. Gel'fand [1941] contains the proofs of the theorems announced in the first note, together with the functional calculus for analytic functions of one complex variable. In the definition of a normed ring he assumes that there is an element e # 0 that is a multiplicative identity, and that multiplication is continuous in each variable separately. By regarding the elements as multiplication operators on the underlying Banach space and using the operator norm, he shows that there is an equivalent norm in which Ji e II = 1 and II xy I] ~ II x It II y II for all x, y. This paper runs from page 3 to page 24 in the Matem. Sbornik. Immediately following it are four additional papers by Gel'fand, the first one jointly with ~ilov (pages 25-39). It is on different ways to topologize the set of maximal ideals. It also discusses the embedding of a completely regular topological space S in a compact space, by means of the maximal ideal space of the ring of bounded continuous real functions on S. The next paper is on ideals and primary ideals in

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normed rings (pages 41-47). A generalized nilpotent (or quasinilpotent) element in a normed ring is an element x such that II x" ItTM ~ 0. (This is equivalent to saying that x, viewed as a function on the space of maximal ideals, is the constant function 0.) The first theorem states that if x is a generalized nilpotent and if [[ (e - ~x) -1 [I <~ c(r/cos~)', ~ = rei% q~~ ~r/2, then x "+1 = 0. The proof uses a Phragm6n-Lindel6f theorem in a form due to Nevanlinna. Then Gel'fand gives criteria for each closed ideal to be the intersection of maximal ideals. A primary ideal is defined to be a closed ideal contained in only one maximal ideal. For a given nonnegative integer n, the last theorem gives a criterion, for a special class of normed rings, that each maximal ideal contain at most n primary ideals. Next comes a paper (pages 49-50) entitled On the theory of characters of abelian topological groups. THEOREM 1. If y is a generalized nilpotent, if x = e - y, and if ll x" l] <<-M for n = 0,+-1,+2 . . . . , then y = O. Note that x is invertible since, viewed as a function on the maximal ideal space, it is the constant function 1 and thus never vanishes. The proof uses the first theorem of the preceding paper (see above). This theorem attracted a lot of attention, and generalizations were published by Hille [1944] and Stone [1948]. Hille showed that the conclusion remains true under the weaker hypothesis that n-lIr x, H~ 0 (n ~ + ~). Stone generalized this further: yk = 0 if and only if II xn r[ = o(I n Ik). Also, if o is replaced by O then the condition is necessary and sufficient for yk+l = O. Recent improvements in this result will be found in Pytlik [1987]. Silov [1950] pointed out that all of these results could be obtained from the last theorem of the preceding paper of Gel'fand on bounding the number of primary ideals contained in a maximal ideal (see above). Gel'fand states that the method of proof of Theorem 1 will also show that if y is any element in a normed ring (y need not be a generalized nilpotent), and i f l l e + y ] ] = ] [ e - y l ] = 1, t h e n y = 0. Usingthis one can show that e is an extreme point of the unit ball. THEOREM 2. Let R be a commutative normed ring and let G be a bounded subset that is a multiplicative group. Then for each two distinct elements x 1 and x 2 of G there is a continuous character of G that takes different values at x 1 and x 2. The last paper in this remarkable series (pages 51-66) is entitled On absolutely convergent trigonometric series and integrals. This paper expands the material in the second Doklady note of 1939 and provides full proofs. The first phase of development of the theory was summarized in a long expository article, Gel'fand, 16

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Raikov, and ~ilov (GRS) [1946] had written in 1940; publication was delayed by the war. Subsequently, GRS [1960], the article was expanded and rewritten as a book. Among the major contributors to the theory was G. E. Silov. In particular he is remembered for the Silov boundary and the Silov idempotent theorem. For a commutative normed ring he proved the existence of a smallest closed subset (now called the ~ilov boundary) of the maximal ideal space with the property that every element in the ring, when viewed as a function on the maximal ideal space, attains its maximum modulus on this set. This was published in GRS [1946], where it was credited to him. ~ilov showed that the b o u n d a r y arises in another context. The Gel'fand theory shows that multiplicative linear functionals on a commutative Banach algebra are automatically continuous and have norm 1. Further, there is a one-to-one correspondence between the maximal ideals and the kernels of multiplicative linear functionals. The HahnBanach theorem guarantees that continuous linear functionals can be extended from a closed subspace of a Banach space to the whole space, preserving the norm. However, if one algebra is contained as a closed subalgebra in another, with the same identity, there is no-guarantee that a multiplicative linear functional on the smaller algebra can be extended to be multiplicative on the larger algebra. Silov showed that multiplicative functionals corresponding to maximal ideals in the boundary of the smaller algebra can always be so extended. The ~ilov idempotent theorem states that the maximal ideal space of a commutative Banach algebra is disconnected if and only if the identity is the sum of two nontrivial idempotents, whose product is 0. This happens if and only if the algebra is the direct sum of two closed ideals, each of which has an identity. In this case the maximal ideal space of the original algebra may be identified with the disjoint union of the maximal ideal spaces of the two subalgebras. This result is in Silov [1953]. For the special case of a symmetric algebra (to each x in the algebra there corresponds an element x* in the algebra such that, viewed as functions on the maximal ideal space, each is the complex conjugate of the other) the result had been obtained earlier by Gel'fand [1941]. In addition we mention two papers of Gel'fand and N a i m a r k [1943] a n d [1948]. T h e y p r o v e d t h a t a normed ring with an involution can be isometrically *-isomorphically embedded into the ring of bounded linear transformations on Hilbert space. If the ring is commutative, then it is isometrically *-isomorphic to the ring of all continuous functions on the maximal ideal space. They needed the assumptions: J[ x*x il = Jpx* Jl II x tt, I1x 1t = tl x* Pf, and e + x*x is invertible, for all x. They conjectured that the last two were consequences of the first. This was e v e n t u a l l y settled

a r o u n d 1960 by J. Glimm, R. Kadison a n d T. Ono. (The reader had best consult MR 22 #5906, 5905, and 47 #2379 for bibliographic data and priorities.) The author is indebted to R. B. Burckel for the reference to R. Doran and V. Belfi [1986]. In this beautiful book the reader will find a history together with the most efficient modern proofs of the Gel'fand-Naimark t h e o r e m s u n d e r still w e a k e r h y p o t h e s e s , a n d pagesize p h o t o s of the two men.

Gel'fand must have been a strong candidate for a Fields Medal on the basis of this work and his work on group representations. Gel'fand must have b e e n a strong candidate for a Fields Medal on the basis of this work and his w o r k on g r o u p representations. The first Fields medals were given (to L. Ahlfors and J. Douglas) at the International Congress of Mathematicians in 1936. The next congress had been scheduled for 1940, but because of the war it was p o s t p o n e d , and was finally held in 1950 in C a m b r i d g e , M a s s a c h u s e t t s . U n f o r t u n a t e l y , the Cold War was t h e n in full swing, and this m a y have w o r k e d against Gel'fand's chances; at any rate, he did not receive a medal. Shortly before the o p e n i n g of the C o n g r e s s a telegram was received from the Soviet a c a d e m y stating t h a t n o m a t h e m a t i c i a n s f r o m the USSR w o u l d attend the Congress. Gel'fand was p r o m o t e d to professor at M o s c o w University in 1943; in addition he had held a position at the Mathematics Institute of the A c a d e m y of Sciences since 1939. I. M. Yaglom told me that a n u m b e r of Jewish mathematicians, including Gel'fand and Yaglom, were dismissed from the university; I believe it was in 1949. The list was p o s t e d a n d instead of m e r e l y listing initials for the first n a m e a n d patronymic these were written out in full: Israel Moiseevi~ Gel'fand, Israel Moiseevi~ Yaglom. O n the other hand, Gel'fand was a w a r d e d a Stalin Prize in 1951, which was a high h o n o r and included a substantial a m o u n t of m o n e y . There is one more result about general commutative Banach algebras that we wish to mention, namely, the local m a x i m u m m o d u l u s principle of Rossi [1960]. By the Gel'fand t h e o r y w e m a y view the elements of the algebra as continuous functions on the maximal ideal space. If U is an o p e n subset of this space, disjoint from the Silov b o u n d a r y , t h e n the maximum, on the closure of U, of the absolute value of each function in the algebra is attained on the b o u n d a r y of U. Subsequent w o r k t e n d e d to be on uniform algebras (closed subalgebras of the continuous functions on a compact H a u s d o r f f space, separating the points of the space a n d c o n t a i n i n g the constants). This is a rich t h e o r y with m a n y applications, b u t that is a n o t h e r story; see, for example, Gamelin [1969].

Bibliography MR = Mathematical Reviews, ZBL = Zentralblatt fiir Mathematik. Doklady = Doklady Akad. Nauk SSSR. W. Ambrose [1945], Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57, 364-386. MR 7, 126. R. S. Doran and V. A. Belfi [1986], Characterizations of C*-AIgebras. The Gel'fand-Na~mark Theorems, Marcel Dekker, New York. MR 87k:46115. T. W. Gamelin [1969], Uniform algebras, Prentice Hall, Englewood Cliffs, N.J. MR 53 #14137. I. M. Gel'fand [1939], On n o r m e d rings, Doklady 23, 430-432. --. To the theory of normed rings. II. On absolutely convergent trigonometric series and integrals, Doklady 25, 570-572. MR 1, 330. --. To the theory of normed rings. III. On the ring of almost periodic functions, Doklady 25, 573-574. MR 1, 331. --. [1941], Normierte ringe, Mat. Sbornik 9 (51), 3-24. MR3, 51. I. M. Gel'fand and A. N. Kolmogorov [1939], On rings of continuous functions on topological spaces, Doklady 22, 11-15. I. M. Gel'fand and M. A. Naimark [1943], On the embedding of normed rings into the ring of operators on Hilbert space, Matem. Sbornik 12, 197-213. MR 5, 147. --. [1948], Normed rings with involution and their representations, Izvest. Akad. Nauk, SSSR, Mat. 12, 445-480; (Russian). MR 10, 199. I. M. Gel'fand, D. A. Raikov, and G. E. ~ilov [1946], Commutative normed rings, Uspehi Matem. Nauk 1, no. 2 (12), 48-146; (Russian). MR 10, 258. --. [1960], Commutative normed rings. Gos. Izdat. Fiz.Mat. Lit., Moscow; (Russian). MR 23 #A1242. E. Hille [1944], On the theory of characters of groups and semi-groups in normed vector rings, Proc. Nat. Acad. Sci. USA 30, 58-60. MR 5, 189 B. E. Johnson [1967], The uniqueness of the (complete) norm topology, Bull. A.M.S. 73, 537-539. MR 35 #2142. Mat XL [1959], Mathematics in the USSR for 40 years, 1917-1957, vol. II. M. A. Naimark [1956], Normed rings, Gos. Izdat. Tekh.-Teor. Lit., Moscow; (Russian). MR 19, 870. T. Pytlik [1987], Analytic semigroups in Banach algebras and a theorem of Hille, Colloq. Math. 51, 287-294, MR 88h:47061. H. Rossi [1960], The local maximum modulus principle, Ann. of Math. 72, 1-11. MR 22 #8317. G. E. Silov [19401], Sur la th6orie des ideaux dans les anneaux norm6s de fonctions, Doklady 27, 900-903. MR 2, 224. --. [19402], On the extension of maximal ideals, Doklady 29, 83-84. MR 2, 314. --. [1950], On a theorem of Gel'fand and its generalizations, Doklady 72, 641-644. MR 12, 111. 9 [1953], On the decomposition of a commutative normed ring into the direct sum of ideals, Mat. Sbornik 32, 353-364; (Russian). MR 14, 884. --. [1957], On certain problems in the general theory of commutative normed rings, Uspehi Mat. Nauk 12, no. 1, 246-249. MR 18, 912. Letter to the editor, ibid. 12, no. 5, 270. MR 19, 969. M. H. Stone [1948], On a theorem of P61ya, J. Indian Math. Soc., N.S. 12, 1-7. MR 10, 308. N. Wiener [1932], Tauberian theorems, Ann. of Math. (2) 33, 1-100. ZBL 4, 59. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

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The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the European Editor, Ian Stewart.

Thomas Bayes: A Memorial A. I. D a l e

Modern statistical theory and practice owe a great deal (if only in controversial contribution) to a relatively obscure nonconformist minister of eighteenth-century England. Indeed, were it not for one work in which he briefly shone velut inter ignes Lunae minores, Thomas Bayes would probably be little remembered today. Little is known of Bayes's personal (or indeed his public) life, and this uncertainty extends to his immediate family also. His paternal great-uncle Samuel Bayes was ejected by the Act of Uniformity of 1662 from a living in Derbyshire, and removed to Manchester. This act, passed by the anti-Puritan parliament after the restoration of Charles II, provided that all ministers not episcopally ordained or refusing to conform should be deprived of their livings on St. Bartholomew's Day. Joshua Bayes, Thomas's father, was born in 1671 and m a y have received his early schooling at the G r a m m a r School in Manchester. Thereafter he attended Richard Frankland's nonconformist academy, and on concluding his studies there, he proceeded to London, where, on the 22nd of June 1694, he was one of seven candidates to be ordained during the first public ceremony of such nature among dissenters in the city after the Act of Uniformity. Thomas Bayes, the eldest of seven children of Joshua and his wife Ann, was born in 1701 or 1702 (the latter date seems generally favoured, but the present epitaph on the family vault merely gives his age at death, in April 1761, as 59). Of his early childhood little is known. While some sources assert that he was privately educated, others believe he received a liberal 18

education for the ministry: the two views are perhaps not altogether incompatible. It is possible that Thomas attended the Congregational Fund Academy in Tenter Alley, w h e r e John Eames (later one of Bayes's sponsors for a Fellowship of the Royal Society) was assistant tutor in classics and science (and later theological tutor), though this is but conjecture. Following in his ancestral footsteps, Thomas sought ordination as a nonconformist minister, this ordination taking place in or before 1727. In 1728 Bayes succeeded John Archer as minister at the meetinghouse, Little Mount Sion, in Tunbridge Wells. Bayes's minor works include a pamphlet of 1731 entitled Divine Benevolence, or an attempt to prove that the Principal End of the Divine Providence and Government is the Happiness of his Creatures, a tract on the doctrine of

fluxions, written in defence of Newton, and a posthumous paper on divergent series. The second work, which, like the first, was published anonymously, was apparently the sufficient cause of his election as a Fellow of the Royal Society in 1742. In April 1761 Thomas Bayes died and is interred in the family vault in Bunhill Fields, by Moorgate, London. The date of his death is given variously as the 7th, the 14th and the 17th; the date on the present restored vault is barely decipherable as a "7": whether a "1" preceeds it or not is unclear. Bayes's papers were demised to the Rev. William Johnstone, his successor at Little Mount Sion. He in turn called in Richard Price, who forwarded Bayes's A n Essay towards solving a Problem in the Doctrine of Chances to John Canton. This essay, containing the re-

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3 9 1989 Springer-Verlag New York

The Bayes family vault

sult fundamental to modern Bayesian statistics, was published in 1764 in the Philosophical Transactions (volume 53 for 1763). Bunhill Fields was the dissenters' cemetery. Here, among others, are buried Richard Price and his wife Sarah, Daniel Defoe, and John Bunyan. The Bayes family vault, having fallen into disrepair, was restored in 1969 "with contributions received from statisticians t h r o u g h o u t the w o r l d . " It is once again in a sorry state, as are many of the headstones and vaults in this burying-ground.

Appendix In one form or a n o t h e r a result k n o w n as Bayes's Theorem appears in most texts containing a discussion of finite probability. One version runs as follows: let C1, C2. . . . . Cn be a sequence of mutually exclusive events. If Hi: the

Ci are exhaustive

then for any event C, n

PF[CilC] = Pr[CIQ]Pr[Q]/~ PF[CICj]Pr[Cj] 1 for each i ~ {1,2 . . . . . n}. Here each "given" event (i.e., each event appearing on the right-hand side of a vertical line) is supposed to be of positive probability. Formulations of this result vary, the hypothesis H 1 being sometimes replaced by either of the following:

n

H 2 : C is any event such that C C ~.J Ci; 1

H a : Pr

Ci = 1.

A sufficient condition for the drawing of the desired con-

clusi~ is easily seen t~ be that Pr[C N 0= 0 Ci] "1 Bayes himself did not give this discrete result: it appears to be due to Laplace, and is a fairly trivial consequence of the formula for conditional probability, viz., Pr[AIB ] = Pr[ANB] / Pr[B]. The problem Bayes in fact addressed in his essay of 1764 runs as follows:

Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named. [p. 376]. The solution, to be found in Bayes's tenth proposition, runs in modern notation as follows:

Pr[o~ ~ ~ ~ ~IP happenings and q failures of the u n k n o w n event] 1

= ~xP(l

x)qdX/~o XP(l _

Department of Mathematical Statistics University of Natal Durban 4001 South Africa THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989 1 9

Reminiscences of Mathematics at Chicago* Marshall H. Stone

In 1946 I moved to the University of Chicago. An im- quickly b e c a m e a brilliant center of mathematical portant reason ,for this move was the opportunity to study and research. Among its early students were participate in the rehabilitation of a mathematics de- such mathematicians as Leonard Dickson, Oswald partment that had once had a brilliant role in Amer- Veblen, George Birkhoff, and R. L. Moore, destined to ican mathematics but had suffered a decline, acceler- f u t u r e p o s i t i o n s of l e a d e r s h i p in r e s e a r c h a n d ated by World War II. During the war the activity of teaching. Some of these students remained at Chicago the department fell to a low level and its ranks were as members of the faculty. Messrs. Dickson, Bliss, depleted by retirements and resignations. The admin- Lane, Reid, and Magnus Hestenes were among them. Algebra, functional analysis, calculus of variations, istration may have welcomed some of these changes, because they r e m o v e d persons w h o had o p p o s e d and projective, differential geometry were fields in some of its policies. Be that as it may, the university which Chicago obtained special distinction. With the resolved at the close of the war to rebuild the depart- passage of time, retirements and new appointments had brought a much increased emphasis on the calment. The decision may have been influenced by the plans culus of variations and a certain t e n d e n c y to into create new institutes of physics, metallurgy, and bi- breeding. When such outstanding mathematicians as ology on foundations laid by the university's role in E. H. Moore or Wilczynski, a brilliant pioneer in prothe Manhattan Project. President Hutchins had seized jective differential geometry, retired from the departthe opportunity of retaining many of the atomic scien- ment, replacements of comparable ability were not tists brought to Chicago by this project, and had suc- found. Thus in 1945 the situation was ripe for a receeded in making a series of brilliant appointments in vival. A s e c o n d , and p e r h a p s e v e n more important, physics, chemistry, and related fields. Something similar clearly needed to be done when the university reason for the move to Chicago was my conviction started filling the vacancies that had accumulated in that the time was also ripe for a fundamental revision mathematics. Professors Dickson, Bliss, and Logsdon of graduate and undergraduate mathematical educahad all retired fairly recently, and Professors W. T. tion. The invitation to Chicago confronted me with a very Reid and Sanger had resigned to take positions elsewhere. The five vacancies that had resulted offered a difficult question: "Could the elaboration of a modernsplendid challenge to anyone mindful of Chicago's ized curriculum be carried out more successfully at great contribution in the past and desirous of ensuring Harvard or at Chicago?" When President Hutchins invited me to visit the its continuation in the future. When the University of Chicago was founded under university in the summer of 1945, it was with the purthe presidency of William Rainey Harper at the end of pose of interviewing me as a possible candidate for the the nineteenth century, mathematics was encouraged deanship of the Division of Physical Sciences. After and vigorously supported. Under the leadership of two or three days of conferences with department Eliakim Hastings M o o r e , Bolza, and M a s c h k e it heads, I was called to Mr. Hutchins's residence, where he announced that he would offer me not the deanship but a distinguished service professorship in the Department of Mathematics. * This article by Marshall Stone, along with the accompanying comThe negotiations over this offer occupied nearly a mentary by Felix Browder, was originally published in the University of Chicago Magazine, August 1976. Copyright 1976 by the University year, during which I sought the answer to the quesof Chicago Magazine. Reprinted by permission. tion with which it confronted me. It soon became clear 20 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3 9 1989Springer-Verlag New York

that the situation at Harvard was not ripe for the kind of change to which I hoped to dedicate my energies in the decade following the war. However, it was by no means clear that circumstances would be any more propitious at Chicago than they seemed to be at Harvard. In c o n s u l t i n g some of m y friends a n d colleagues, I was advised by the more astute among them to come to a clear understanding with the Chicago administration concerning its intentions. There are those who believe that I went to Chicago to execute plans that the administration there already had in mind. Nothing could be further from the truth. In fact, my negotiations were directed toward developing detailed plans for reviving the Chicago Departm e n t of Mathematics and obtaining some kind of commitment from the administration to implement them. Some of the best advice given me confirmed my own instinct that I should not join the University of Chicago unless I were made chairman of the department and thus given some measure of authority over its development. Earlier experiences had taught me that administrative promises of wholehearted interest in academic improvements were too often untrustworthy. I therefore asked the University of Chicago to commit itself to the development program that was under discussion, at least to the extent of offering me the chairmanship. This created a problem for the university, as the department had to be consulted about the matter, and responded by voting unanimously that Professor Lane should be retained in the office. As I was unwilling to move merely on the basis of a promise to appoint me to the chairmanship at some later time, the administration was brought around to arranging the appointment, and I to accept it. Mr. Lane, a very fine gentleman in every sense of the word, never showed any resentment. Neither of us ever referred to the matter, and he served as an active and very loyal member of the department until he retired several years later. I was very grateful to him for the grace and selflessness he displayed in circumstances that might have justified a quite different attitude. Even though the university made no specific detailed commitments to establish the program I had proposed during these year-long negotiations, I was ready to accept the chairmanship as an earnest of forthcoming support. I felt confident that with some show of firmness on my part the program could be established. In this optimistic spirit I decided to go to Chicago, despite the very generous terms on which Harvard wished to retain me. Regardless of what, m a n y seem to believe, rebuilding the Chicago Department of Mathematics was an uphill fight all the way. The university was not about to implement the plans I had proposed in our negotiations without resisting and raising objections at every step. The department's loyalty to Mr. Lane had

the fortunate consequence for me that I felt released from any formal obligation to submit my recommendations to the department for approval. Although I consulted my colleagues on occasion, I became an autocrat in making my recommendations. I like to think that I am not by nature an autocrat, and that the later years of my chairmanship provided evidence of this belief. At the beginning, however, I took a strong line in what I was doing in order to make the department a truly great one. The first recommendation sent up to the administration was to offer an appointment to Hassler Whitney. The suggestion was promptly rejected by Mr. Hutchins's second in command. It took some time to persuade the administration to reverse this action and to

Marshall H. Stone

make an offer to Professor Whitney. When the offer was made, he declined it, and remained at Harvard for a short time before moving to the Institute for Advanced Study. The next offer I had in mind was one to Andr6 Weil. He was a somewhat controversial personality, and I found a good deal of hesitation, if not reluctance, on the part of the administration to accept my recommendation. In fact, while the recommendation eventually received favorable treatment in principle, the administration made its offer with a substantial reduction in the salary that had been proposed. I was forced to advise Professor Weil, who was then in Brazil, that the offer was not acceptable. When he declined the offer, I was in a position to take the matter up at the highest level. Though I had to go to an 8 A.M. appointment suffering from a fairly high fever, in order to discuss the appointment with Mr. Hutchins, I was rewarded by his willingness to renew the offer on the terms I had originally proposed. Professor Weil's acceptance of the improved offer was an important event in the THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

21

history of the University of Chicago and in the history of American mathematics. My conversation with Mr. Hutchins brought me an unexpected bonus. At its conclusion he turned to me and asked, "When shall we invite Mr. Mac Lane?" I was happy to be able to reply, "Mr. Hutchins, I have been discussing the possibility with Saunders and believe that he would give favorable consideration to a good offer whenever you are ready to make it." That offer was made soon afterward and was accepted. There were other appointments, such as that of Professor Zygmund, that also went smoothly, but what would happen in any particular case was always unpredictable.

Hand-to-Mouth Budgeting One explanation doubtless was to be found in the university's hand-to-mouth practices in budgeting. This would appear to have been the reason w h y one evening I was given indirect assurances from Mr. Hutchins that S. S. Chern would be offered a professor-

22

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

ship, only to be informed by Vice-President Harrison the next morning that the offer would not be made. Such casual, not to say arbitrary, treatment of a crucial recommendation naturally evoked a strong protest. In the presence of the dean of the Division of Physical Sciences I told Mr. Harrison that if the appointment were not made, I would not be a candidate for reap-

Earlier experiences had taught me that administrative promises of wholehearted interest in academic improvements were too often untrustworthy. pointment as chairman when my three-year term expired. Some of my colleagues who were informed of the situation called on the dean a few hours later to associate themselves with this protest. Happily, the protest was successful, the offer was made to Professor C h e r n , a n d he a c c e p t e d it. This was the stormiest incident in a stormy period. Fortunately the period was a fairly short one, and at the roughest

history of the University of Chicago and in the history of American mathematics. My conversation with Mr. Hutchins brought me an unexpected bonus. At its conclusion he turned to me and asked, "When shall we invite Mr. Mac Lane?" I was happy to be able to reply, "Mr. Hutchins, I have been discussing the possibility with Saunders and believe that he would give favorable consideration to a good offer whenever you are ready to make it." That offer was made soon afterward and was accepted. There were other appointments, such as that of Professor Zygmund, that also went smoothly, but what would happen in any particular case was always unpredictable.

Hand-to-Mouth Budgeting One explanation doubtless was to be found in the university's hand-to-mouth practices in budgeting. This would appear to have been the reason w h y one evening I was given indirect assurances from Mr. Hutchins that S. S. Chern would be offered a professor-

22

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

ship, only to be informed by Vice-President Harrison the next morning that the offer would not be made. Such casual, not to say arbitrary, treatment of a crucial recommendation naturally evoked a strong protest. In the presence of the dean of the Division of Physical Sciences I told Mr. Harrison that if the appointment were not made, I would not be a candidate for reap-

Earlier experiences had taught me that administrative promises of wholehearted interest in academic improvements were too often untrustworthy. pointment as chairman when my three-year term expired. Some of my colleagues who were informed of the situation called on the dean a few hours later to associate themselves with this protest. Happily, the protest was successful, the offer was made to Professor C h e r n , a n d he a c c e p t e d it. This was the stormiest incident in a stormy period. Fortunately the period was a fairly short one, and at the roughest

times Mr. Hutchins always backed me unreservedly. As soon as the department had been brought up to strength by this series of new appointments, we could turn our attention to a thorough study of the curricu l u m and the requirements for higher degrees in mathematics. The group that was about to undertake the task of redesigning the department's work was magnificently equipped for what it had to do. It included, in alphabetical order, Adrian Albert, R. W. Barnard, Lawrence Graves, Paul Halmos, Magnus Hestenes, Irving Kaplansky, J. L. Kelley, E. P. Lane, Saunders Mac Lane, Otto Schilling, Irving Segal, M. H. Stone, Andr6 Weil, and Antoni Z y g m u n d . Among them were great mathematicians, and great teachers, and leading specialists in almost e v e r y branch of pure mathematics. Some were new to the university, others familiar with its history and traditions. We were all resolved to make Chicago the leading center in mathematical research and education it had always aspired to be. We had to bring great patience and open minds to the time-consuming discussions that ranged from general principles to detailed

mathematical questions. The presence of a separate and quite independent college mathematics staff did not relieve us of the obligation to establish a new undergraduate curriculum beside the new graduate program. Two aims on which we came to early agreement were to make course requirements more flexible and to limit examinations and other required tasks to those having some educational value. This streamlined program of studies, the unusual distinction of the mathematics faculty, and a rich offering of courses and seminars have attracted many very promising young mathematicians to the University of Chicago ever since the late 1940s. The successful coordination of these factors was reinforced by the concentration of all departmental activities in Eckhart Hall with its offices (for faculty and graduate students), classrooms, and library. As most members of the department lived near the university and generally spent their days in Eckhart, close contact between faculty and students was easily established and maintained. (This had been foreseen and planned for by

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

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Professor G. A. Bliss w h e n he counseled the architect would be possible to bring about closer cooperation e n g a g e d to build Eckhart Hall.) It was one of the than had existed in the past between the Departments reasons why the mathematical life at Chicago became of Mathematics and Physics. Circumstances were unfavorable. The university felt so spontaneous and intense. By helping create conditions so favorable for such mathematical activity, Pro- little pressure to increase its offerings in applied mathfessor Bliss earned the eternal gratitude of his univer- ematics. It had no engineering school, and rather resity and his department. Anyone who reads the roster cently had even rejected a bequest that would have of Chicago doctorates since the later 1940s cannot but endowed one. Several of its scientific departments ofbe impressed by the prominence and influence many of them have enjoyed in American--indeed in world Although I consulted my colleagues on o c c a --mathematics. It is probably fair to credit the Chicago s i o n , I became an autocrat in making m y recprogram with an important role in stimulating and guiding the d e v e l o p m e n t of these mathematicians o m m e n d a t i o n s . during a crucial phase of their careers. As I have described it, the Chicago program made fered courses in the applications of mathematics to one conspicuous omission--it provided no place for specific fields such as biology, chemistry, and meteoapplied mathematics. During my correspondence of rology. The Department of Physics and the Fermi In1945-46 with the Chicago administration I had in- stitute had already worked out an entirely new prosisted that applied mathematics should be a concern of gram in physics and were in no mood to modify it in the department, and I had outlined plans for ex- the light of subsequent changes that might take place panding the department by adding four positions for in the Mathematics Department. However, m a n y students of physics elected adprofessors of applied subjects. I had also hoped that it

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vanced mathematics courses of potential interest for t h e m - - f o r example, those dealing with Hilbert space or operator theory, subjects prominently represented among the specialties cultivated in the Mathematics Department. On the other hand, there was pressure for the creation of a Department of Statistics, exerted particularly by the economists of the Cowles Foundation. A committee was appointed to make recommendations to the administration for the future of statistics with Professor Allen Wallis, Professor Tjalling Koopmans, and myself as members. Its report led to the creation of a Committee on Statistics, Mr. Hutchins being firmly opposed to the proliferation of departments. The committee enjoyed powers of appointment and eventually of recommendation for higher degrees. It was housed in Eckhart and developed informal ties with the Department of Mathematics. At a somewhat later time a similar committee was set up to bring the instruction in applied mathematics into focus by coordinating the courses offered in several different d e p a r t m e n t s and eventually recommending higher degrees. Long before that, h o w e v e r , the D e p a r t m e n t of Mathematics had sounded out the dean of the division, a physicist, about the possibility of a joint appointment for Freeman Dyson, a young English physicist then visiting the United States on a research grant. We had invited him to Chicago for lectures on some brilliant work in number theory that had marked him as a mathematician of unusual talent. We were impressed by his lectures and realized that he was well qualified to establish a much needed link between the two departments. However, Dean Zachariasen quickly stifled our initiative with a simple question, " W h o is D y s o n ? " Dyson soon became a permanent member of the Institute of Advanced Study. By 1952 I realized that it was time for the Department of Mathematics to be led by someone whose moves the administration had not learned to predict. It was also time for the department to increase its material support by entering into research contracts with the government. Fortunately there were several colleagues w h o were more than qualified to take over. The two most conspicuous were Saunders Mac Lane and Adrian Albert. The choice fell first on Professor Mac Lane, w h o served for the next six years. Under the strong leadership of these two gifted mathematicians and their younger successors the department experienced many changes, but flourished mightily and was able to maintain its acknowledged position at the top of American mathematics.

[Marshall Stone died on 9 January 1989.--Ed.] THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989 2 5

Steven H. Weintraub* For the general philosophy of this section, see Volume 9, No. 1 (1987). A bullet (o) placed beside a problem indicates a submission without solution; a dagger (t) indicates that it is not new. Contributors to this column who wish an acknowledgment of their contribution should enclose a self-addressed postcard. Problem solutions should be received by I November 1989.

Problems The Master Spy: Problem 89-5 by Raymond Smullyan (Elka Park, NY, USA)

Greatest common divisor of polynomials over GF(2): Problem 89-6 b y Helmut Meyn (Universit/it E r l a n g e n - N i i r n b e r g , FRG) Define the trinomial Tm(x ) = x 2m + x § 1

The position below was reached in the course of a legal chess game. However, one of the pieces therein (not a pawn) is a spy, i.e., it is of the correct denomination but of the wrong color. Which piece is it?

over GF(2), the field of two elements, and let v(m) be the 2-valuation of m, i.e., 2v(m) divides m but 2v(m) + 1 does not. Show that for any natural numbers m, n the following holds:

gcd (Vm(x), Tn(x))

* C o l u m n editor's address: D e p a r t m e n t of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918 USA 26 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3 9 1989Springer-VerlagNew York

l

Tgcd(m,n)(X)

if v(m) = v(n)

1

otherwise

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

The Greatest Mathematical Paper of All Time A. J. Coleman

You will say that my title is absurd. "Mathematical papers cannot be totally ordered. It's a great pity! Poor old Coleman has obviously gone berserk in his old age." Please read on. If in 1940 you had asked the starry-eyed Canadian graduate student who was lapping up the K-calculus from Alonzo Church in Princeton to name the single most important mathematics paper, without doubt I would have chosen Kurt G6del's bombshell [12] that had rocked the foundations of mathematics a few years before. In 1970, after my twenty years of refereeing and reviewing, if you had posed the same question, without any hesitation I w o u l d have chosen the enormous paper of Walter Feit and John Thompson [11] confirming Burnside's 1911 conjecture [3] that simple finite groups have even order.

Now, in the autumnal serenity of semi-retirement, having finally looked at some of Wilhelm Killing's writings, without any doubt or hesitation I choose his paper dated "Braunsberg, 2 Februar, 1888" as the most significant mathematical paper I have read or heard about in fifty years. Few can contest my choice since apart from Engel, Umlauf, Molien, and Cartan few seem to have read it. Even my friend Hans Zassenhaus, whose LiescheRinge (1940) was a landmark in the subject, admitted over our second beer at the American Mathematical Society meeting in January 1987 that he had not read a word of Killing. Presupposing that my reader has a rudimentary understanding of linear algebra and group theory, I shall attempt to explain the main new ideas introduced in Killing's paper, describe its remarkable results, and suggest some of its subsequent effects. The paper that,

THE MATHEMATICAL 1NTELL1GENCERVOL. 11, NO. 3 9 1989 Springer-Veflag N e w York

29

following Cartan, I shall refer to as Z.v.G.II, was the second of a series of four [18] about Lie algebras. The series was churned out in Braunsberg, a mathematically isolated spot in East Prussia, during a period when Killing was overburdened with teaching, civic duties, and concerns about his family.

The Ahistoricism of Mathematicians Most mathematicians seem to have little or no interest in history, so that often the name attached to a key result is that of the follow-up person who exploited an idea or theorem rather than its originator (Jordan form is due to Weierstrass; Wedderburn theory to Cartan and Molien [13]). No one has suffered from this ahistoricism more than Killing. For example, the so-called "'Cartan sub-algebra" and "Cartan matrix, A = (aij)'" w e r e defined and exploited by Killing. The very symbols aij and e for the rank are in Z.v.G.II. Hawkins [14, p. 290] correctly states: Such key notions as the rank of an algebra, semi-simple algebra, Cartan algebra, root systems and Cartan integers originated with Killing, as did the striking theorem enumerating all possible structures for finite-dimensional Lie algebras over the complex numbers . . . . Cartan and Molien also used Killing's results as a paradigm for the development of the structure theory of finite dimensional linear associative algebras over the complex numbers, obtaining thereby the theorem on semisimple algebras later extended by Wedderburn to abstract fields and then applied by Emmy Noether to the matrix representations of finite groups. In this same paper Killing invented the idea of root systems and of o~root-sequences through/3. He exhibited the characteristic equation of an arbitrary element of the Weyl group w h e n Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born! I have found no evidence that Hermann Weyl read anything by Killing. Weyl's important papers on the representations of semisimple groups [26], which laid the basis for the subsequent development of abstract harmonic analysis, are based squarely on Killing's results. But Killing's name occurs only in two footnotes in contexts suggesting that Weyl had accepted uncritically the universal myth that Killing's writings were so riddled with egregious errors that Cartan should be regarded as the true creator of the theory of simple Lie algebras. This is nonsense, as must be apparent to anyone w h o even glances at Z.v.G.II or indeed to anyone w h o reads Cartan's thesis carefully. Cartan was meticulous in noting his indebtedness to Killing. In Cartan's thesis there are 20 references to Lie and 63 to Killing! For the most part the latter are the theorems or arguments of Killing that Cartan incorporated into his thesis, the first two-thirds of which is essentially a commentary on Z.v.G.II. 30

T H E M A T H E M A T I C A L INTELLIGENCER VOL. 11, N O . 3, 1989

Cartan did give a remarkably elegant and clear exposition of Killing's results. He also made an essential contribution to the logic of the argument by proving that the "Cartan subalgebra" of a simple Lie algebra is abelian. This property was announced by Killing but his p r o o f w a s invalid. In parts, other than II, of Killing's four papers there are major deficiencies which Cartan corrected, notably in the treatment of nilpotent Lie algebras. In the last third of Cartan's thesis, m a n y new and important results are based upon and go beyond Killing's work. Personally, following the value scheme of my teacher Claude Chevalley, I rank Cartan and Weyl as the two greatest mathematicians of the first half of the twentieth century. Cartan's work on infinite dimensional Lie algebras, exterior differential calculus, differential geometry, and, above all, the representation theory of semisimple Lie algebras was of supreme value. But because one's Ph.D. thesis seems to predetermine one's mathematical life work, perhaps if Cartan had not hit u p o n the idea of basing his thesis on Killing's epoch-making work he might have ended his days as a teacher in a provincial lyc6e and the mathematical world would have never heard of him!

The Foothills to Parnassus Before we enter directly into the content of Z.v.G.II, it may be well to provide some background. What we n o w call Lie algebras were invented by the Norwegian mathematician Sophus Lie about 1870 and i n d e p e n d e n t l y by Killing about 1880 [14]. Lie was seeking to develop an approach to the solution of differential equations analogous to the Galois theory of algebraic equations. Killing's consuming passion was non-Euclidean geometries and their generalizations, so he was led to the problem of classifying infinitesimal motions of a rigid body in any type of space (or Raumformen, as he called them). Thus in Euclidean space, the rotations of a rigid body about a fixed point form a group under composition which can be parameterized by three real n u m b e r s - - t h e Euler angles, for example. The tangent space at the identity to the parameter space of this group is a three-dimensional linear space of "infinitesimal" rotations. Similarly, for J . a group that can be paramet~nzed by a smooth manifold of dimension r, there is an r-dimensional tangent space, _z?,at the identity element. If the product of two elements of the group is continuous and differentiable in the parameters of its factors, it is possible to define a binary operation on _z?which we denote by "o," such that for all x, y, z, ~ _E, (x, y) ~ x o y is linear in each factor, xoy +yox

= 0

(1)

Lyzeum Hosianum in 1835.

and x o (y o z) + y o (z o x) + z o (x o y) = 0

(2a)

or equivalently, x o (y o z) = (x o y) o z + y o (x o z).

(2b)

The equation in (2a) is called the Jacobi identity and in the form (2b) should remind you of the rule for differentiating a product. (_z?, +, o) is a Lie algebra with an anti-commutative non-associative product. The Jacobi identity replaces the associativity of familiar rings such as the integers or matrix algebras. Obviously if ~ is a subspace of L such that x, y E ~t x o y E d~, then 9 is a sub-algebra of 2?. Further, if p: (271, +, o) --~ (2?2, + , o) is a homomorphism of one Lie algebra onto another, the kernel of p is not merely a subalgebra but an ideal. For if K = {x E d?l{p(x)=0}, then for any x E K and any y E 271, p(x o y) = p(x) o p(y) = 0. Thus K is not only a sub-algebra but has the property, characteristic of an ideal, that for any x E K, we have y o x E K for every y E d?1. We can then define a quotient algebra 2?1/K isomorphic to 2?2, in a manner analogous to groups with normal subgroups. Thus a Lie algebra 17 whose only ideals are {0} and 27 is homomorphic only to 2? or {0}. Such an algebra is called simple. The simple Lie algebras are the building blocks in terms of which any Lie algebra can be analyzed. Lie recognized rather early that the search for solutions of systems of differential equations would be greatly facilitated if all simple Lie algebras were known. But Lie's attempts to find them ran into the sands very quickly. In his quest for all uniform spatial forms Killing formulated the problem of classifying all Lie algebras

over the reals--a task which in the case of nilpotent Lie algebras seems unlikely to have a satisfactory solution. In particular, he was interested in simple real Lie algebras; as a step in this direction he was led, with the encouragement of Engel, to the problem of classifying all simple Lie algebras o v e r the c o m p l e x numbers. When ~ is an associative algebra--for example the set of n x n matrices over C - - t h e n for X, Y, Z E ~ , we define X o Y = X Y - Y X = IX, Y/--the so-called commutator of X and Y. It is then trivial to show that X o Y satisfies (1) and (2). Thus any associative algebra ( ~ , + , .) can be t r a n s f o r m e d into a Lie algebra (~, +, ~ by the simple expedient of defining X ~ Y = IX, Y]. This immediately leads us to the notion of a linear representation of a Lie algebra (2?, +, o) as a mapping p of _z?into Hom(V), satisfying the following condition: p(x o y) = [O(x), 0(Y)]. Although the definition of a representation of a Lie algebra in this simple general form was never given explicitly by Killing or others before 1900, the idea was implicit in what Engel, and Killing following him, called the adjoint group [15, p. 143] and what we now call the adjoint representation.

In passing, let us note that until about 1930 what we now call Lie groups and Lie algebras were called "continuous groups" and "infinitesimal groups" respectively; see [8], for example. These were the terms Weyl was still using in 1934/5 in his Princeton lectures [27]. However, by 1930 Cartan used the term groupes de Lie [4, p. 1166]; the term Liesche Ringe appeared in the title of the famous article on enveloping algebras by Witt [28]; and, in his Classical Groups, Weyl [1938, p. 260] wrote "In homage to Sophus Lie such an algebra is called a Lie Algebra." Borel [1, p. 71] attributes the term "Lie group" to Cartan and "Lie algebra" to Jacobson. THE MATHEMATICAL 1NTELLIGENCER VOL. 11, NO. 3, 1989

31

He defined k to be the m i n i m u m for x E ~ of the multiplicity of zero as a root of (4). This is now called the rank of d?. But Killing and Cartan used the term rank for the n u m b e r of functionally i n d e p e n d e n t Oi regarded as functions of x E _s Killing noted that O~i(x) are p o l y n o m i a l invariants of the Lie group corresponding to the Lie algebra considered. Though expressed in a rather clumsier notation, he realized that = {h E -s

XPh = 0 for some p}

is a subalgebra of d. This follows from a sort of Leibnitz differentiation rule: Xn(y o z) = Y~[~]Xn-Sy o XSz, for 0 ~ s ~ n. For arbitrary _s if X is such that the dim(~) is a m i n i m u m , the subalgebra is n o w called a Cartan subalgebra. As a Lie algebra itself ~ is nilpotent or w h a t Killing called an algebra of zero rank. For the adjoint representation of ~ on ~ , I(oI-HI~ = (ok for all h E ~ , so all tbi vanish identically. If _z?is simple, ~ is in fact abelian. Killing convinced himself of this by an invalid argument. The filling of this lacuna was a significant contribution by Cartan to the classification of simple Lie algebras over C. It was a stroke of luck on Killing's part that though his a r g u m e n t was mildly defective, his conclusion on this important matter was correct. Assuming that ~f is abelian, it is trivial to show that in the equation i(oi_ HI = (ok II~((o - oL(h)),

Killing as rector, 1897-1898.

For the adjoint representation of _t' the linear space V, above, is taken to be _Z'itself and p is defined by f)(x)z = x o z for every z E ~.

(3)

The reader is urged to verify that with this definition of p, the Jacobi identity (2b) implies that f)(x o y) =

the roots, oL(h), are linear functions of h E ~f. Thus {x E ~*, the dual space of ~ . Following current usage we denote b y / ~ the set of roots ~ that occur in (5). Killing proceeded on the a s s u m p t i o n that all ~ h a d multiplicity one, or that the r - k functions c~(h) were distinct. It follows that for each oLthere is an element e~ E d? such that h o e~ = ~(h) e~ for all h E ~. Then using (2b) it easily follows that for (x, [3, E A

[~(x), ~(v)].

Ja/o (e~ o e~) = (or(h) + 13(h))G, o e,.

Killing Intervenes

Killing had completed his dissertation u n d e r Weierstrass at Berlin in 1872 a n d k n e w all about eigenvalues and w h a t we n o w call the Jordan canonical form of matrices, whereas Lie k n e w little of the algebra of the contemporary Berlin school. It was therefore Killing r a t h e r t h a n Lie w h o a s k e d the decisive question: " W h a t can one say about the eigenvalues of X : = p(x) in the adjoint representation for an arbitrary x E _s Since X x = x o x = 0, X always has zero as an eigenvalue. So Killing looked for the roots of the characteristic equation (a term he introduced!): [(oI-X[

= (or _

~I(X)(or-1

----- ~ b ~ _ l ( X ) ( o

32

--

q- ~ 2 ( X ) ( o r - 2 0.

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

--

. . . (4)

(5)

(6)

This equation is the key to the classification of the root systems/~ that can occur for simple Lie algebras. From (6) we can immediately conclude: (i) e~oel~ # O ~ a (ii) a + 13 ~ ~ e a o e ~

(iii) 0 # e ~ o e ~

E ~o~

+ 13 E /N = 0 + 13 = 0.

It turns out that for every o~ E A, there is a corresponding -c~ E /X such that 0 # ha: = e~ o e_~ E ~. So the n u m b e r of roots is even, say 2m, and r = k + 2 m = dim(_s

In the adjoint representation let E~ correspond to e~, and for a n y e~ # 0 consider the element E~e~ for n E Z +. Starting from (6) we see by induction that

h o E~ea = (~(h) + no~(h))E"~e~.

Thus if E]ea ~ O, f3 + n a ~ A . But vectors with distinct eigenvalues are linearly i n d e p e n d e n t . Thus if 2? is fin i t e - d i m e n s i o n a l t h e r e is a h i g h e s t v a l u e of n for which E"~el~ ~ O. Call it p. Similarly let q be the largest value of n such that E"__~ea ~ O. Thus for a, 13 ~ A there is an n-sequence of roots t h r o u g h 13 of length p + q + 1 - - w h a t Killing called Wurzelreihe: 13 - qa, 13 - ( q - 1 ) a . . . . ,[3,[3 + a . . . . . + pa.

(7)

Because H~ = [E~, E_~], the trace of H= is zero, which implies 2f3(h~,) + (p-q)a(h~,)

= O.

(8)

T h i s , in o u r n o t a t i o n , is e q u a t i o n (7), p. 16, of Z.v.G.II. The d i m e n s i o n of the Cartan subalgebra is n o w called the rank of -/'. For simple Lie algebras this definition a n d Killing's definition of r a n k coincide. That is, for simple Lie algebras k = e. H e n c e dim(~*) = f, so there can be at most f linearly i n d e p e n d e n t roots. Using (8), Killing s h o w e d that there exists a ba'sis B = { a l , a 2 . . . . . a e } of gs w h e r e a i ~ Z~ is such that each ~3 E A has rational c o m p o n e n t s in the basis B. Indeed, the a i { B can be so chosen that a i is a top root in any aj-sequence t h r o u g h it. Thus for each i and j there is a root-sequence %, ai - o9 . . . . .

ai + aila i

Wilhelm Killing in his later years.

(9)

w h e r e aq is a n o n - p o s i t i v e integer. In particular, it turns out that aii -= - 2 .

The Still Point of the Turning World The definition of the integers aq was a turning point in mathematical history. It appears at the top of page 16 of II. By page 33 Killing had found the systems A for all simple Lie algebras over C together with the orders of the associated Coxeter transformations. We continue, using Killing's o w n words taken from the last p a r a g r a p h of his introduction, u n c h a n g e d except for notation: If cti and % are two of these (~roots, there are two integers aij and a# that define a certain relation between the two roots. Here we mention only that together with cti and txj both cti + aqet, and e9 + ajieq and cti + atxjare roots where a is an integer ~etween aij and 0. The coefficients ali are all equal to -2; the others are by no means arbitrary; indeed they satisfy many constraining equalities. One series of these constraints deriveg from the fact that a certain linear transformation, defined in terms of a0, when iterated gives the identity transformation. Each system of these coefficients is simple or splits into simple systems. These two possibilities are distinguished as follows. Begin with any index i, 1 ~< i ~< 2. Adjoin to it all j such that a~i # 0; then

adjoin all k for which an ajk ~ O. Continue as far as possible. Then, if all indices 1,2 . . . . • have been included, the system of a 0 is simple. The roots of a simple system correspond to a simple group. Conversely, the roots of a simple group can be regarded as determined by a simple system. In this way one obtains the simple groups. For each f there are four structures supplemented for f ( {2, 4, 6, 7, 8} by exceptional simple groups. For these exceptional groups I have various results that are not in fully developed form; I hope later to be able to exhibit these groups in simple form and therefore am not communicating the representations for them that have been found so far. In r e a d i n g this, recall that Lie a n d Killing u s e d the t e r m " g r o u p " to i n c l u d e the m e a n i n g w e n o w attribute to "'Lie algebra." His statement is correct as it stands for (~ > 3 but as is a p p a r e n t from his explicit list of simple algebras he k n e w that for f = 1 there is only one i s o m o r p h i s m class and for f = 2 and 3 there are three. Replacing a i by - a i gives rise to integers satisfying aij = 2, aij ~ 0 for i # j, which is currently the usual convention. The "certain linear transformation'" m e n t i o n e d b y Killing is the Coxeter t r a n s f o r m a t i o n discussed below. It is w o r t h noting that in Killing's explicit tables the coefficients for all roots in terms of his c h o s e n basis are integers, so h e came close to obtaining w h a t we n o w call a basis of simple r o o t s / l la THE MATHEMATICAL 1NTELLIGENCER VOL. 11, NO. 3, 1989

33

Dynkin. As far as I am aware, such a basis appeared explicitly for the first time in Cartan's beautiful 1927 paper [4, p. 793] on the geometry of simple groups. The one minor error in Killing's classification was the exhibition of two exceptional groups of rank four. Cartan noticed that Killing's two root systems are easily seen to be equivalent. It is peculiar that Killing overlooked this since his mastery of calculation and algebraic formalism was quite phenomenal. Killing's notation for the various simple Lie algebras, slightly modified by Cartan, is what we still employ: A n den o t e s the i s o m o r p h i s m class c o r r e s p o n d i n g to se(n+ l,C); B n corresponds to so(2n+ I); C n to sp(2n); D n to so(2n). The classes A n, Bn, D n were known to Lie and Killing before 1888. Killing was unaware of the existence of type Cn though Lie knew about it, at least for small values of n. On this point see the careful discussion of Hawkins [15, pp. 146-150]. The exceptional algebra of rank two which we now label G2 was denoted as IIC by Killing. It has dimension 14 and has a linear representation of dimension 7. In a letter to Engel [ 15, p. 156] Killing remarked that G2 might occur as a group of point transformations in five, but not fewer, dimensions. That such a representation exists was subsequently verified independently by Cartan and Engel [4, p. 130]. The exceptional algebras F4, E a, E7, E8 of rank 4, 6, 7, 8 have dimension 52, 78, 133, 248, respectively. The largest of Killing's exceptional groups, E8 of dimension 248, is now the darling of super-string theorists!

Forward to Coxeter For an arbitrary simple Lie algebra of rank n, the dimension is n ( h + l ) , where h is the order of a remarkable element of the Weyl group now called the Coxeter transformation (because Coxeter expounded its properties as part of his study of finite groups generated by reflections or, as they are now called, Coxeter groups [6, 7]). Coxeter employed a graph to classify this type of g r o u p . D u r i n g t h e 1934/5 l e c t u r e s by Weyl at Princeton, he noticed that the finite group of permutations of the roots which played a key role in Killing's argument and which is isomorphic to what we now call the Weyl group is in fact generated by involutions. The notes of Weyl's course [27] contain an Appendix by Coxeter in which a set of diagrams equivalent to those of Table 1 appears. Some years later Dynkin independently made use of similar diagrams for characterizing sets of simple roots so that they are now generally described as Coxeter-Dynkin diagrams. The left-hand c o l u m n of Table 1 encapsulates Killing's classification of simple Lie algebras. By studying the Coxeter transformation for Lie algebras of rank 2, Killing showed [Z.v.G.II, p. 22] that aijaji {0, 1, 2, 3}. There is a one-to-one correspondence between the Cartan matrices of finite-dimensional simple 34

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

Lie algebras and the left-hand column of Table 1. The n nodes of a graph correspond to Killing's indices 1, 2, 3 . . . . . n, or to the roots of a basis or to generators Si of the Weyl group. A triple bond as in G2 means that aijaji = 3. Double and single bonds are interpreted similarly.

On to Kac and Moody! If we use the current convention that aii = 2 and that aij is a non-positive integer if i # j, it is not difficult to see that Killing's conditions imply that d? is a finite-dimensional Lie algebra if and only if the determinant of A = (aq) and those of all its principal minors are strictly positive. Further, Killing's equations (6) [Z.v.G.II. p. 21] imply that A is symmetrisable--that is, there exist non-zero numbers d i such that diaij = djaji. In particular, aii and aii are zero or non-zero together. Almost simultaneously in 1967, Victor Kac [16] in the USSR and Robert Moody [22] in Canada noticed that if Killing's conditions on (aij) were relaxed, it was still possible to associate to the Cartan matrix A a Lie algebra which, necessarily, would be infinite dimensional. The current method of proving the existence of such Lie algebras derives from a short paper of Chevalley [5]. This paper was also basic to the work of m y students Bouwer [2] and LeMire [19], who discussed infinite dimensional representations of finite Lie algebras. Chevalley's paper also initiated the current widespread exploitation of the universal associative enveloping algebras of Lie algebras--a concept first rigorously defined by Witt [28]. Among the Kac-Moody algebras the most tractable are the symmetrisable. The most extensively studied and applied are the affine Lie algebras which satisfy all Killing's conditions except that the determinant [A[ is 0. The Cartan matrices for the affine Lie algebras are in one-to-one correspondence with the graphs in the right-hand column of Table 1, which first appeared in [27].

Wilhelm Killing the Man Killing was born in Burbach in Westphalia, Germany, on 10 May 1847 and died in M~inster on 11 February 1923. Killing began university study in M~inster in 1865 but quickly moved to Berlin and came under the influence of K u m m e r and Weierstrass. His thesis, completed in March 1872, was supervised by Weierstrass and applied the latter's recently developed theory of elementary divisors of a matrix to "Bundles of Surfaces of the Second Degree." From 1868 to 1882 much of Killing's energy was devoted to teaching at the g y m n a s i u m level in Berlin and Brilon (south of M~inster). At one stage w h e n Weierstrass was urging him to write up his research on space structures (Raumformen) he was spending as much as 36 hours

An'O

9

o...

n(n+2) 1

2

3

O~O

9

n-1

An:

9 1

n

1

On O ~ O 1

9 .., O~O

2

3

I n-2

1

01

On

n(2n-1

1

9

Dn:

1

9 1

Ol

O ~ O ~ O . . . O ~ O ~ O

1

n-1

2

2

2

2

1

O1 I

O1

E6 78

I

O~O

2

O2 1

O ~ O ~ O

3

4

5

0

E6:

o ~ o ~ o 1 2

6

grog

2

3

02

07 E7:O~O~O 1 2 133

I 3

1

1

O--O~O

4

5

6

E v# :

I

0

O~O

,, O ~ O ~ O ~ O

1

2

3

4

3

2

1

03

08

I

E8 O ~ O ~ O ~ O ~ O ~ O ~ O 1 2 3 4 5 6 7 248

I

1 E8:0

o m o ~ o 2 3 4

1

9 5

O ~ O ~ O

6

4

2

1

A 1 99 1 3

A 19 2 A I9

O~O 1

1

O~O I

2

14 G2 9

,

F4 9

F 4 9O ~ O : = ~ , = O ~ O 52

1

2--3

9 2

4

F4 9 B1

B - ' O ~ O ~ O " ' O ~ O ~ O

n(2n+lfy1

2

3

n-2

n-1

n

n

0 I

o 1

9 2

9

9

9

9

1

2

9 O~O~O...

2

9

4

2

0

9

3

O1 0

1

=>= 9

3

2

1

9

9

2

2

2

2--2

1

1

1

1 --I

B2 n:

1 --I

BC2n : O : = ~ = O m O ~ O . - . 1 --2

Cn:O~O~O... n(2n+l) 1

,2

0~0~0 3

n-2

n-l--n

1

Cn"

2

2

O=~OmO~O... 1--2

O~O=~=O 2 2--2

0~0~---0,

2

2

2

2

O~I=~==I 2 2"1

2

1

ol C n2:

I ~ O ~ I i, . 1 2

Table 1. Coxeter-Dynkin Diagram of the finite and affine Lie algebras9 THE MATHEMATICAL

INTELLIGENCER VOL. 11, NO. 3, 1989

35

Braunsberg, with a view of the thirteenth-century St. Catherine's Church.

per week in the classroom or tutoring. (Now many mathematicians consider 6 hours a week an intolerable burden!) On the recommendation of Weierstrass, Killing was appointed Professor of Mathematics at the Lyzeum Hosianum in Braunsberg in East Prussia (now Braniewo in the region of Olsztyn in Poland). This was a college founded in 1565 by Bishop Stanislaus Hosius, whose treatise on the Christian faith ran into 39 editions! When Killing arrived the building of the Lyzeum must have looked very much as it appears in the accompanying picture. The main object of the college was the training of Roman Catholic clergy, so Killing had to teach a wide range of topics including the reconciliation of faith and science. Although he was isolated mathematically during his ten years in Braunsberg, this was the most creative period in his mathematical life. Killing produced his brilliant work despite worries about the health of his wife and seven children, demanding administrative duties as rector of the college and as a member and chairman of the City Council, and his active role in the church of St. Catherine. Killing a n n o u n c e d his ideas in the form of Programmschriften [15] from Braunsberg. These dealt with (i) Non-Euclidean geometries in n-dimensions (1883); 36

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

(ii) "The Extension of the Concept of Space" (1884); and (iii) his first tentative thoughts about Lie's transformation groups (1886). Killing's original treatment of Lie algebras first appeared in (ii). It was only after this that he learned of Lie's work, most of which was inaccessible to Killing because it never occurred to the college librarian to subscribe to the Archiv fiir Mathematik of the University of Christiana (now, Oslo) where Lie published. Fortunately Engel played a role with respect to Killing similar to that of Halley with Newton, teasing out of him Z.v.G.I-IV, which appeared in the

Math. Annalen. In 1892 he was called back to his native Westphalia as professor of mathematics at the University of M~inster, where he was quickly submerged in teaching, administration, and charitable activities. He was Rector Magnificus for some period and president of the St. Vincent de Paul charitable society for ten years. Throughout his life Killing evinced a high sense of duty and a deep concern for anyone in physical or spiritual need. He was steeped in what the mathematician Engel characterized as "the rigorous Westphalian Catholicism of the 1850s and 1860s." St. Francis of Assisi was his model, so that at the age of 39 he, together with his wife, entered the Third Order of the Franciscans [24, p. 399]. His students loved and ad-

mired Killing because he gave himself unsparingly of time and energy to them, never being satisfied until they understood the matter at hand in depth [23]. Nor was Killing satisfied for them to become narrow specialists, so he spread his lectures over many topics beyond geometry and groups. Killing was conservative in his political views and vigorously opposed the attempt to reform the examination requirements for graduate students at the University of Mfinster by deleting the compulsory study of philosophy. Engel comments "Killing could not see that for most candidates the test in philosophy was vollstfindig wertlos" (completely worthless). Nor do my sources suggest that he had much of a sense of h u m o u r . He had a p r o f o u n d patriotic love of his country, so that his last years (1918-1923) were deeply pained by the collapse of social cohesion in Germany after the War of 1914-18. Nonetheless, the accompanying photograph of Killing in his old age radiates kindliness and serenity. He was greatly cheered by the award of the Lobachevsky Prize by the Physico-Mathemafical Society of Kazan in 1900 for his work in geometry.

Why was Killing's Work Neglected? Killing was a modest man with high standards; he vastly underrated his own achievement. His interest was geometry and for this he needed all real Lie algebras. To obtain merely the simple Lie algebras over the complex numbers did not appear to him to be very significant. Once Z.v.G.IV had appeared, Killing's research energies w e n t back to Raumformen. I recall that one day in 1940 during the regular tea-coffee ritual in Fine Hall at Princeton, Marston Morse declaimed "A successful mathematician always believes that his current theorem is the most important piece of mathematics the world has ever seen." Few have lived this philosophy with more 61an than Morse! And of course even though I immediately formed a deepseated dislike for Morse, there is something in what he said. If you do not think your stuff is important, w h y will anyone else? But Morse's philosophy is far removed from St. Francis of Assisi! Also Lie was quite negative about Killing's work. This, I suspect, was partly sour grapes, because Lie admitted that he had merely paged through Z.v.G.II. At the top of page 770 of Lie-Engel III [20] we find the following less than generous comment about Killing's 1886 Programmschrift: "with the exception of the preceding unproved t h e o r e m , . . all the theorems that are correct are due to Lie and all the false ones are due to Killing!" According to Engel [9, p. 221/2] there was no love lost between Lie and Killing. This comes through in the nine references to Killing's work in volume III of

Wilhelm Killing, probably about 1889-1891.

[20]. With one exception they are negative and seem to have the purpose of proving that anything of value about transformation groups was first discovered by Lie. Even if this were true, it does not do justice to the fact that there was no possibility of Killing in Braunsberg knowing Lie's results published in Christiana. So if Lie's results are wonderful, Killing's independent discovery of them is equally wonderful! It seems to me that even Hawkins, who has done more than anyone else to rehabilitate Killing, sometimes allows himself to be too greatly influenced by the w i d e s p r e a d negativism surrounding Killing's work. The misunderstanding about the relation of Cartan to Killing would never have occurred if readers of Cartan's thesis had taken the trouble to look up the 63 references to Killing's papers that Cartan supplied.

Conclusion Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing's THE MATHEMATICAL INTELLIGENCER VOL. I1, NO. 3, 1989

37

paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linear associative algebras which culminated in Wedderburn's theorems. Killing's success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl's theory of the representation of semisimple Lie g r o u p s w o u l d have b e e n i m p o s s i b l e without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl's fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper w a y to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac's b o o k [17] are given on p. 21 of Z.v.G.II. (5) It was Killing w h o discovered the exceptional Lie algebra Ea, which apparently is the main hope for saving Super-String T h e o r y - - n o t that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan w a s b a s e d more or less directly on Z.v.G.II. Euclid's Elements and Newton's Principia are more important than Z.v.G.II. But if you can name one paper in the past 200 years of equal significance to the paper which was sent off diffidently to Felix Klein on 2 February 1888 from an isolated outpost of Bismarck's empire, please inform the Editor of the Mathematical Intelligencer.

Acknowledgments M y d e b t to T h o m a s H a w k i n s will be o b v i o u s to anyone w h o has explored his fascinating historical writings. I am also most grateful to I. Kiessling of the University library in Mfinster and to K. Haenel of the library of G6ttingen for invaluable information about Killing's life and the pictures which enliven this article.

References 1. A. Borel in "Hermann Weyl: 1885-1985," ed. by K. Chandrasekharan, Springer-Verlag (1986). 2. I. Z. Bouwer, Standard Representations of Lie Algebras, Can. ft. Math 20 (1968), 344-361. 3. W. Burnside, "Theory of Groups of Finite Order" 2nd Edition. Dover, 1955. Note M p. 503; in note N he draws attention to the "sporadic"groups (1911). 38

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INTELLIGENCER VOL. 11, NO. 3, 1989

4. E. Cartan, Oeuvres Completes, I., Springer-Verlag (1984). 5. C. Chevalley, "Sur la Classification des alg~bres de Lie simples et de leurs representations," Comptes Rendus, Paris 227 (1948), 1136-1138. 6. H. S. M. Coxeter, "Regular Polytopes," 3rd Edition, Dover (1973). 7. H. S. M. Coxeter, "Discrete groups generated by reflections", Annals of Math. (2) 35 (1934), 588-621. 8. L. P. Eisenhart, "Continuous Groups of Transformations", Princeton U.P. (1933). 9. F. Engel, "Killing, Wilhelm," Deutsches Biographisches Jahrbuch, Bd. V for 1923, (1930) 217-224. 10. F. Engel, "Wilhelm Killing," Jahresber. Deut. Math. Ver. 39 (1930), 140-154. 11. W. Feit and J. Thompson, "Solvability of groups of odd order," Pacif. J. Math. 13 (1963), 775-1029. 12. K. G6del, "Ueber formal unentscheidbare S/itze der Principia Mathematica und verwandter System I," Monatshefte fiir Math. u. Physik 38 (1931), 173-198. 13. T. Hawkins, "Hypercomplex Numbers, Lie Groups and the Creation of Group Representation Theory," Archive for Hist. Exact Sc. 8 (1971), 243-287. 14. T. Hawkins, "Non-euclidean Geometry and Weierstrassian Mathematics: The background to Killing's work on Lie Algebras," Historia Mathematica 7 (1980), 289-342. 15. T. Hawkins, "Wilhelm Killing and the Structure of Lie Algebras," Archive for Hist. Exact Sc. 26 (1982), 126-192. 16. V. G. Kac, "Simple irreducible graded Lie algebras of finite growth," Izvestia Akad. Nauk, USSR (ser. mat.) 32 (1968), 1923-1967; English translation: Math. USSR Izvest. 2 (1968), 1271-1311. 17. V. G. Kac, "Infinite dimensional Lie algebras," Cambridge University Press, 2nd Edition (1985). 18. W. Killing, "Die Zusammensetzung der stetigen, endlichen Transformationsgruppen," Mathematische Ann. L 31 (1888-90), 252;//33, 1; III 34, 57; 36, 161. 19. F. W. LeMire, "Weight spaces and irreducible representations of simple Lie algebras," Proc. A.M.S. 22 (1969), 192-197. 20. S. Lie and F. Engel, "Theorie der Transformationsgruppen," Teubner, Leipzig (1888-1893). 21. R. V. Moody and A. Pianzola, "On infinite Root Systems," to appear (1988). 22. R. V. Moody, "A new class of Lie algebras," J. Algebra 10 (1968), 211-230. 23. P. Oellers, O.F.M., "Wilhelm Killing: Ein Modernes Gelehrtenleben mit Christus," Religi6se Quellenschriften, Heft 53, (1929) l~sseldorf. 24. E. Wasmann, S. J., "Ein Universit~itsprofessor im Tertiarenkleide," Stimmen der Zeit, Freiburg im Br.; Bd. (1924) 106-107. 25. H. Weyl "Mathematische Analyse des Raumproblems," Berlin: Springer (1923). 26. H. Weyl, "DarsteUung kontinuierlichen halbeinfachen Gruppen durch lineare Transformationen," Math. Zeit 23 (1925-26), 271-309; 24, 328-376; 24, 377-395; 24, 789-791. 27. H. Weyl, "The structure and representation of continuous groups," Mimeographed notes by Richard Brauer; Appendix by Coxeter (1934-35). 28. E. Witt, "Treue Darstellung Liescher Ringe," Jl. Reine und Angew. M. 177 (1937), 152-160. Department of Mathematics and Statistics Queen's University Kingston, Ontario Canada, K7L 3N6

When You Can't Hear the Shape of a Manifold Carolyn S. Gordon

In his well-known article in the American Mathematical Monthly in 1966, Mark Kac posed the question "Can you hear the shape of a drum?" View the surface of a drum as a plane domain M. We say that K is an eigenvalue of the Laplacian A of M if there is a smooth function f (not identically 0) on M with f = 0 on the boundary such that Af = )~f. The collection of eigenvalues forms the spectrum of M. The eigenvalues determine the frequencies of the sounds produced by M. If you could listen to the drum with a perfect ear and hence hear the spectrum of M, what geometrical or topological information about M would you then obtain? In particular, Kac's question asks: w o u l d you know M up to congruence? The spectrum certainly contains considerable geometrical information a b o u t the domain. Indeed, Hermann Weyl already knew that the spectrum determines the area and perimeter of M. Yet Kac's question remains open. Kac himself said in his autobiography [14] that he thought the answer was probably no. Moreover, recent examples of Hajima Urakawa [17] and of Peter Buser [5] show that you can't always hear the shape of a domain in R n for n i> 3. Kac's question has an analog for closed manifolds. In order to discuss the " s h a p e " or g e o m e t r y of a smooth manifold M, we give M a Riemannian metric. Riemannian metrics on arbitrary smooth manifolds generalize the Euclidean metric on R n. A Riemannian manifold is a smooth manifold together with a choice of Riemannian metric. A.Imost immediately after Riemann introduced Riemannian metrics in his dissertation, Beltrami began studying the analog of the Laplace operator for Riemannian manifolds. For closed (that is, compact without boundary) Riemannian manifolds, the eigenvalues of the Laplacian, repeated ac-

cording to multiplicity, form a discrete set of positive numbers called the spectrum of M. Isometric Riemannian manifolds have the same spectrum. Conversely we can ask: Are isospectral Riemannian manifolds (that is, Riemannian manifolds with the same spectrum) necessarily isometric; that is, can you hear the shape of a Riemannian manifold? If not, what geometric properties can you hear? The sounds produced by certain manifolds became painfully clear to participants at a recent meeting of the American Mathematical Society in Alabama.

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3 9 1989 Springer-Veflag N e w York

39

f

Figure 1. Construction of a flat torus.

Dennis DeTurck used the eigenvalues of the manifolds to simulate their patterns of overtones. The audience enjoyed, or rather suffered through, the strains of Alabama Jubilee played by a two-dimensional flat torus band ("flat" here refers to its geometry, not its sound) and a projective quartet, the latter composed of a real, a complex, a quaternionic, and a Cayley projective space. The audience would perhaps be happy to learn that two-dimensional flat tori and low-dimensional projective spaces are uniquely determined by their spectra. No two of them produce the same terrible sound. Because a Soundsheet is included with this issue of the Mathematical Intelligencer, you too can hear the sound of manifolds, produced by Dennis DeTurck. See the accompanying box for details. In contrast to the elusive case of the plane domains, however, we now have examples of closed manifolds in every dimension greater than or equal to two whose shapes can't be "heard." (The only one-dimensional closed Riemannian manifolds are circles.) John Milnor first discovered a pair of isospectral, non-isometric manifolds in 1964, two years before Kac's article appeared. However, the subsequent fifteen years saw few new examples. In 1980, examples again began to trickle in, and during the last four years we have seen not only an avalanche of examples but also general techniques for p r o d u c i n g examples. You can n o w even construct isospectral manifolds at home with scissors and paste (see [3], [4], [5]). Moreover, in all dimensions greater than or equal to five, we have continuous families of isospectral, non-isometric manifolds. The monograph [1] contains an extensive bibliography. Other basic references are [2] and [6]. 40

THE MATHEMATICAL INTELLIGENCER VOL. II, NO~ 3, 1989

Examples of Isospectral Manifolds We begin with flat tori. A flat torus is a quotient F \ R n of R n by a lattice F of rank n. View elements of F as translations of R ". Beginning with a fundamental parallelepiped of F and identifying those sides that are equivalent under translation by elements of F, we obtain a torus (see Figure 1). The Euclidean metric on R" induces a metric on the torus; the geodesics in the

f

Figure 1. Construction of a flat torus.

Dennis DeTurck used the eigenvalues of the manifolds to simulate their patterns of overtones. The audience enjoyed, or rather suffered through, the strains of Alabama Jubilee played by a two-dimensional flat torus band ("flat" here refers to its geometry, not its sound) and a projective quartet, the latter composed of a real, a complex, a quaternionic, and a Cayley projective space. The audience would perhaps be happy to learn that two-dimensional flat tori and low-dimensional projective spaces are uniquely determined by their spectra. No two of them produce the same terrible sound. Because a Soundsheet is included with this issue of the Mathematical Intelligencer, you too can hear the sound of manifolds, produced by Dennis DeTurck. See the accompanying box for details. In contrast to the elusive case of the plane domains, however, we now have examples of closed manifolds in every dimension greater than or equal to two whose shapes can't be "heard." (The only one-dimensional closed Riemannian manifolds are circles.) John Milnor first discovered a pair of isospectral, non-isometric manifolds in 1964, two years before Kac's article appeared. However, the subsequent fifteen years saw few new examples. In 1980, examples again began to trickle in, and during the last four years we have seen not only an avalanche of examples but also general techniques for p r o d u c i n g examples. You can n o w even construct isospectral manifolds at home with scissors and paste (see [3], [4], [5]). Moreover, in all dimensions greater than or equal to five, we have continuous families of isospectral, non-isometric manifolds. The monograph [1] contains an extensive bibliography. Other basic references are [2] and [6]. 40

THE MATHEMATICAL INTELLIGENCER VOL. II, NO~ 3, 1989

Examples of Isospectral Manifolds We begin with flat tori. A flat torus is a quotient F \ R n of R n by a lattice F of rank n. View elements of F as translations of R ". Beginning with a fundamental parallelepiped of F and identifying those sides that are equivalent under translation by elements of F, we obtain a torus (see Figure 1). The Euclidean metric on R" induces a metric on the torus; the geodesics in the

t o m s are curves that lift to straight lines in R". We say that R" is a Riemannian covering of the toms. If we view functions on the torus as F-periodic functions on R" (f is F-periodic if f(~/ + x) = f(x) V'Y E F), then the Laplacian is just the Euclidean Laplacian = -- ~ C32/0X2. i=l

It's easy to compute the spectrum of a flat toms. Let F* be the dual lattice F* = {'r E R": "r'~/ E ~,V~/ E F}. For "r E F*, the map f, defined by f,(x) = exp(2"rri'r 9 x) is a F-periodic eigenfunction of A with the eigenvalue 4,rr2[r,rll2. (More appropriately, the real and imaginary parts of f, are eigenfunctions.) Using the Stone-WeierTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

41

strass Theorem, we find that all eigenfunctions are linear combinations of the f~'s. Thus the spectrum of the torus is precisely the collection of numbers 4~r2[['rll2 as "r ranges over F*. Define the length spectrum of a lattice F in R" to be the sequence of lengths of all vectors in F. The computation above s h o w s that two flat tori F I \ R n and F2NR n are isospectral if and only if F~ and F 2 have the same length spectrum (or equivalently, by the Poisson Summation Formula, F 1 and F 2 have the same length spectrum). The tori are isometric if and only if F 1 and IF'2 are congruent under an orthogonal linear transformation. Thus, to produce isospectral, non-isometric tori, one needs to produce two lattices that are not congruent under any orthogonal transformation but that have the same length spectrum. Witt did just that in sixteen dimensions in the process of classifying even, positive-definite, integral quadratic forms. Milnor then observed that Witt's lattices were precisely what he needed to obtain isospectral tori. Thus, in an article of only one page [15], he showed us that you can't always hear the shape of a manifold. Before looking at other examples, we view the tori slightly differently. View R" as the subgroup of translations of the group Iso(R n) of Euclidean motions. Two vectors in R n have the same length if and only if they are conjugate in Iso(R"). Thus the tori F I \ R " and F2NR" are isospectral if and only if there is a one-toone correspondence between 1P1 and F2 such that corresponding elements are conjugate in Iso(R"), and they are isometric if and only if F 1 and F 2 a r e conjugate subgroups of Iso(Rn). In all the examples known of isospectral manifolds, the manifolds have a common Riemannian covering M, just as the tori have the common covering R"; i.e., the manifolds are orbit spaces of the form F I \ M and F 2 \ M where F 1 and F 2 a r e discrete subgroups of the group of isometries Iso(M). (Elements of F \ M are equivalence classes of M under the equivalence x - y if y = ~ 9x for some ~/ ~ F.) If F 1 and F 2 are conjugate in Iso(M), then the manifolds are isometric. Unlike the tori, however, the manifolds may occasionally turn out to be isometric even when the groups are not conjugate. The Laplacian of F N M is just the Laplacian of M restricted to F-periodic functions. In 1980 Marie-France Vign6ras [18] constructed pairs of isospectral Riemann surfaces and higher-dimensional hyperbolic manifolds (here M is hyperbolic space). Her higher-dimensional examples have nonisomorphic fundamental groups, showing that you can't hear the fundamental group of a manifold. At about the same time, Ikeda [12] constructed isospectral, non-isometric lens spaces; these are quotients of the sphere S". The construction of continuous families of isospectral manifolds is similar to the torus construction. Instead of the abelian group R", we take M to be a nilpo42 THE MATHEMATICAL [NTELLIGENCER VOL. 1I, NO. 3, 1989

tent Lie group G (say, a subgroup of the group of unipotent matrices in some dimension). The Euclidean metric is a translationJinvariant metric on Rn; we analogously choose a Riemannian metric on G for which the left translations by elements of G are isometries; we call such a metric left-invariant. Let F be a discrete subgroup of G with F N G compact. We view G and hence F as subgroups of Iso(G) by identifying elements of G with left translations. The metric on G descends to a Riemannian metric on the "nilmanifold" F\G. To deform F, let 9 be an almost inner automorphism of G; i.e., for each x E G, ~(x) is conjugate to x, although the conjugating element may depend on x. Then: THEOREM. [11] (See also [8], Part I.) The manifolds F \ G and cI~(F)\ G are isospectral. If @ is an inner automorphism, then F and @(F) are conjugate subgroups of Iso(G) and thus the manifolds are isometric. However, for generic almost inner automorphisms, the manifolds are non-isometric. Many nilpotent groups G admit continuous families of almost inner, non-inner automorphisms, giving rise to isospectral families of manifolds qbt(F)\G. Another way to view the same family of manifolds is to leave F fixed but deform the metric. If d is the distance function for the original metric, then the deformed metric on G has distance function dt(p,q) = d(cI)t(p),r ). In our proof of the theorem, we showed that the space of smooth functions on F \ G decomposes into Laplace-invariant subspaces ~ . For each a, there is an isometry ot E Iso(G) such that for f E H=, f ~ cb = f o I,, where I, denotes conjugation by a; i.e., functions in H~ can't detect the difference between qb and an inner aut o m o r p h i s m . Thus the Laplacians of F \ G a n d @(F)NG, when restricted to H~, behave like the Laplacians of the isometric manifolds F N G and (aFa-1)NG and hence have the same eigenvalues. The formal proof is in the language of representation theory. Example. Let G be the group of 7 x 7 matrices of the

form -I Xx x2 zl 0 0 01 0yl00 00 ly200 0001000 00 0 0 lxlz 00 0 00 ly .00 0 000

0 0 0

2 2 1

where xi, Yi, and z i are real numbers. We denote elements of G by 6-tuples h = (xl, x2, Yl, Y2, zl, z2). An easy computation shows that for h, h' E G, we have

h'h(h') - I = (x 1, x2, Yl, Y2, Zl + x'lYl + x~2 -

xly~ - xzy~, z2 + X'ly~ - Xly~).

Let F be the integer lattice. Set N = F \ G . Define a o n e - p a r a m e t e r family of a u t o m o r p h i s m s of G by (~t(h) ~"

( X l ' X 2 ' Y l ' Y2' ZI' Z2 --

ty2)"

We easily check that ~t is not inner; h o w e v e r , ~t(h) = h'h(h') - 1 w h e r e

h'

J1 (-t,

t y l / y 2, O, O, 0 , 0 )

if Y2 = 0 if y2 ~ 0.

Thus ~t is almost inner. If w e choose any left-invariant metric on G, we obtain an isospectral family of metrics on N. What does N look like? As a manifold, G is simply R 6. Recall that to form the torus in Figure 1, we identified points in R n that differed by a translation b y an element of F. We form N similarly, but n o w the translation of h by ~/is not the ordinary Euclidean translation but rather the left multiplication ~h in the group G; namely, for ~/ = (a 1, a 2, b1, b2, c 1, c2) ff I~ (so a i, b i, a n d c i ~ Z) and h ~ G as above, ~/h = (a 1 + xl, a 2 + x2, bI + Yl, b2 + Y2, c 1 + z I + aly~ + a2y 2,c2 + z 2 + alY2).

Just as the parallelopiped was a f u n d a m e n t a l d o m a i n for the torus, the unit cube in R 6 is a f u n d a m e n t a l domain for N. The faces Yl = 0 and yl = 1 are identified b y t h e o r d i n a r y t r a n s l a t i o n (Xl,X2,0,y2,zI,Z2) (Xl,X2,1,y2,Zl,Z2). (Take "V = (0, 0, 1, 0, 0, 0) in the equation above.) Similar identifications hold for the faces associated with the extreme values of Y2, zl, and z2. H o w e v e r , for i -- 1, 2, the faces xi = 0 and x; = 1 are identified by shear transformations, because, for example, taking ~/ = (1, 0, 0, 0, 0, 0) in the equation above, we obtain ~/ " (0, 9(2,

Yl, Y2, zl,

Z2) -~

(1, X2, Yl, Y2, Zl + Yl, Z2 q- Y2)"

Figure 2. The xlY2Z2 plane covers a submanifold of N obtained by the indicated identificat i o n s . The t o p a n d bottom faces are identified by a translation, as are the front and back faces. The left and right faces are identified by a " s h e a r " transformation. Figure 2 shows the x1Y2Z2 slice of the cube (defined by x2 = Yl = Zl = 0) together with the identifications of its faces. The x l y l z 1 slice is identical. In o r d e r to construct a specific family of isospectral metrics, let's specify that for our initial metric d = d o, the Euclidean unit vectors at the origin are orthonormal. O u r r e q u i r e m e n t that the metric be left-invariant t h e n determines the o r t h o n o r m a l frames (i.e., bases) at other points. Figure 3 illustrates the orthonormal frames in the XlYlZ1 a n d xlY2Z2 slices at points along the x I axis. The effect of (I)t o n the metric is simply to slide the xly2z 2 picture t units to the left while leaving the XlYlZ 1 picture u n c h a n g e d (see Figure 4). The torus and the nilmanifold examples (and also the e x a m p l e s of Vign~ras) s u g g e s t a possible techn i q u e for c o n s t r u c t i n g i s o s p e c t r a l m a n i f o l d s : o n e might seek manifolds of the form F I \ M and F 2 \ M with a one-to-one c o r r e s p o n d e n c e b e t w e e n F 1 and F 2 such that c o r r e s p o n d i n g e l e m e n t s are conjugate in

Figure 3. Orthonormal frames in the x l y l z l and xly2z2-slices of N for the metric do at the points (0, 0, 0) and (a, 0, 0). Note that the metric in the plane x~ = a changes with a, since the group law of G is not the Abelian one on R6. THE MATHEMATICALINTELLIGENCER VOL. 11, NO. 3, 1989 43

Figure 4. Orthonormal frames in the xlY2Z2slice of N for the metric dt. The orthonormal frames in the XlYlZlslice are identical to those of Figure 3.

Iso(M). Is this condition sufficient to prove the manifolds are isospectral? Probably not. However Toshikazu Sunada in 1985 gave an elegant proof of a special case. Sunada was motivated not by the examples discussed above (some of which were not yet known to him) but rather by a result in algebraic number theory. For Sunada's construction we begin with a compact manifold M and a tower of Riemannian coverings

and

(ii) lira K(t,x,y) = 8v(x) where t---~0

is the Dirac measure. K(t, x, y) may be viewed as the amount of heat at the point x at time t if a spark is set off at y at time 0. The solution of the heat equation with initial condition u(O, x) = f(x) is given by

u(t,x) = ;M K(t,x,y)f(y)dy. F 1LM F 2kM N/-/~ M/" In particular, F1, F2 C H C Iso(M). THEOREM. (Sunada [16]) Assume that there exists a bijection between F 1 and F 2 such that corresponding elements are conjugate in H. (Equivalently, if we let [hi denote the conjugacy class in H of h ~ H, then # ( [ h ] N F 1) = #([h]NF 2) for all h.) Then S p e c ( F l \ M ) = Spec(F2\M). Using this theorem, Peter Buser [5] and Robert Brooks and Richard Tse [4] have constructed pairs of isospectral, non-isometric Riemann surfaces of every genus greater than or equal to four. Brooks [3] has written a delightful exposition of Sunada's theorem and these examples. We reproduce in Figure 5 his recipe for constructing isospectral genus-4 surfaces. In this example, F 1 and F 2 have index 7 in H. The drawings show seven copies of the fundamental domain of H together with instructions for identifying the edges to produce F I \ M and F 2 \ M . (The dotted and dashed lines are for later reference.) The proof of Sunada's theorem is simple and elegant. We follow here the exposition given in Brooks's article referred to above. The heat kernel, or fundamental solution, of the heat equation u t + Au = 0 on a closed Riemannian manifold M is the unique smooth function K : R + x M x M ~ R + such that (i) for each y, K(.,.,y) is a solution of the heat equation 44

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

There is a simple formula for K. Let ~'1, h2. . . . be the eigenvalues of the Laplacian and let %, ~2. . . . be the associated normalized eigenfunctions. The eigenvalues h/approach infinity sufficiently rapidly so that the sum Y~e x p ( - hjt)q~j(x)q~j(y) converges, and you can easily check that this sum satisfies (i) and (ii) and hence equals K(t,x,y), since (i) and (ii) characterize K. The trace of K, given by

tr(K)(t) = fM K(t,x,x)dx, satisfies tr(K)(t) = Y, exp(-)~jt). Thus, knowledge of tr(K) is equivalent to knowledge of Spec(M). If F is a finite subgroup of Iso(M) and M covers F \ M , then the map Kr(t,~,~) = s r K(t,~/" x,y) of R + x (FNM) + (FNM) ~ R + (here x and y denote any lifts of ~ and ~ in M) satisfies (i) and (ii) and hence is the heat kernel of F \ M . We can n o w prove Sunada's theorem. Let F = F1 or F2. Let F be a fundamental domain for F in M. Then tr(Kr)(t) = ( Kr(t,~,~)d~ = ~ (K(t,~/" x,x)dx Jr \ M ~EF aV = (1/#F) ~ ( K(t,~/" x,x)dx. 3,EF aM (The last equality uses the fact that the measure is invariant under Iso(M).) If a ~ Iso(M), then K(t,x,y) = K(t,a 9 x,a 9 y). Thus if ~/ ([h], say ~/ = a-lha, then

K(t,~/ 9 x,x) = K(t,a-lha 9 x,x) = K(t, ha 9 x,a 9 x).

Using again the fact that the measure is translation invariant, we obtain

~M K(t,~/ " x,x)dx = fM K(t,h " x,x)dx. Thus, tr(Kr)(t) = (1/#F)

~ [hi~[HI

#([h] n F) I K(t,h" x,x)dx. JM

By Sunada's hypotheses, we have # F 1 = # F 2 and #([h]MF1) = #([h]MF2), and hence the heat kernels of F1NM and F 2 \ M have the same trace. Sunada's theorem follows. A generalization ([8], Part II) of Sunada's Theorem to infinite groups F i (with a more technical conjugacy condition) includes most of the examples known of isospectral manifolds, including the deformations discussed above. In a different vein, Alejandro Uribe has generalized Sunada's theorem to allow the groups F i and H to be connected. He replaces the conjugacy condition with the condition that the right actions of G on L2(FI\G) and L2(F2\G) be equivalent. Unfortunately, no one has as yet found examples of such groups. An example would be very exciting, as it might result jin the construction of isospectral simply-connected manifolds. In all the known examples of isospectral manifolds, the manifolds have a common covering and are therefore locally isometric. There is no consensus among experts as to whether isospectral, non-isometric, simply-connected manifolds are likely to exist. W h a t C a n Y o u Hear?

The examples above amply say that you can't hear the shape of a manifold. What properties can you hear? S. M i n a k s h i s u n d a r a m and A. Pleijel f o u n d an asymptotic expansion for the heat kernel near t = 0. The coefficients of this expansion are spectral invariants. From the expansion we know that the spectrum determines the dimension, the volume, and certain curvature properties of the manifold. For two-dimensional manifolds, the spectrum determines the Euler characteristic. Another geometric property related to the spectrum is the length spectrum, i.e., the collection of lengths of closed geodesics, counted with multiplicities. Note that for tori F N R n, this definition of length spectrum agrees with the one given earlier. If we imagine that a manifold could vibrate like a drum, then the wavefronts would travel along geodesics. Thus we might guess that the Laplace s p e c t r u m - - o r the collection of frequencies of the possible vibrations--determines the length spectrum. At least generically, our guess is correct. (See [7] or [10].) In particular, Sunada's conjugacy condition or its generalization to infinite discrete groups F i implies

that the isospectral manifolds F I \ M and F2NM have the same length spectrum. To understand the idea of the proof, let r be a closed geodesic in F1NM of length 1. Then cr lifts to a geodesic 6 in M such that 6-(0 = ~/1 " 6-(0) for some ~h E F1. By Sunada's assumption, we can find an isometry a E Iso(M) and an element ~/2 of F 2 such that ~/2 = a'Yla-1. Because a is an isometry, a 9 6- is again a geodesic in M and a 9 6-(I) = a~h 9 4(0) = ~Y2.(a 9 ~(0)). Thus a 96 descends to a closed geodesic of length I in F2NM. W h a t C a n ' t Y o u Hear?

One might expect to read off from the examples of isospectral manifolds many geometrical or topological properties that can't be heard. For example, Vign6ras's examples show that one can't hear the fundamental group. Unfortunately, though, in most of the examples, the distinguishing properties of the manifolds are more subtle. Brooks proved that the manifolds in Figure 5 are not isometric as follows: The geodesics indicated by dotted lines in each of the drawings in Figure 5 descend to the shortest closed geodesics in F I \ M and F 2 \ M , respectively. Similarly, the geodesics indicated by dashed lines descend to the next shortest geodesics in the two manifolds. (Note in each case that when the sides of the hyperbolic polygon are glued according to the instructions, the two dashed geodesic segments form a single closed geodesic.) In the first manifold, two of these geodesics cross; in the second one, they do not. Thus the pattern of criss-crossing of closed geodesics is not a spectral invariant. The behavior of the geodesics in Brooks's example again shows up in the isospectral deformation described above. For fixed x 1 = a, the straight lines 11,a and 12,a in R 6 given by ll, a : s ~ (a, O, s, O, as, O) and 12,a : s ~ (a, O, O, s, O,(a + t)s) are geodesics for the metric d t. (See Figure 6. The skewing of the lines in the z 1 or z 2 direction as a changes is due to the effect of left translation in G as in Figure 3.) Note that 11,a is simultaneously a geodesic for all the metrics. The geodesic in N defined by 11,a (respectively, 12,a) is closed precisely when a (respectively, a + t) is an integer. At time t = 0, we thus find that the Yl and Y2 axes (a = 0) project in M to intersecting circles of length 1 which are closed geodesics. For other values of t, the Yl axis and the line 12,_t : s ~ ( - t , O, O, s, 0, 0) project to circles of length 1 w h i c h are disjoint geodesics for the metric d t. This separation of the geodesics as the metric deforms leads to another distinguishing property of the metrics d t. In analogy to the length spectrum, we define the "area spectrum" of a Riemannian manifold. Roughly speaking, we obtain the area spectrum by recording the minimal area of surfaces in each two-dimensional homology class. (The actual definition reTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

45

Figure 5. Construction of a pair of isospectral Riemann surfaces.

Figure 6. Two families of geodesics in N for the metric dr. Recall that the faces of the cubes are identified according to the scheme depicted in Figure 2; in particular, the geodesics indicated in boldface are circles, i.e., closed geodesics. quires the language of currents.) At t = 0, the YlY2 plane covers a torus in M whose meridian and longitude (the Yl and Y2 circles) are orthogonal closed unit length geodesics. This torus has area one and, not surprisingly, has minimal area in its homology class. For the metric dr, the Y2 circle is no longer a geodesic; i n d e e d , no torus h o m o l o g o u s to the YlY2 torus has both a geodesic meridian and a geodesic longitude. As we slide the YlY2 torus to the left along the x 1 axis (see Figure 7), the l e n g t h of the m e r i d i a n steadily increases, while the length of the longitude steadily decreases. The minimal torus is the best compromise (xl = - t / 2 ) and its area is 1 + t2/4. Thus the area spect r u m changes with t (see [9]).

Figure 7. A (Euclidean) unit cube in the xiylz2-slice of G. Each rectangular ylY2-slicedescends to a torus in M with meridian and longitude having the indicated lengths in the metric dr. The slice x1 = -~ descends to the minimum-area torus.

46

THE MATHEMATICAL LNTELLIGENCER VOL. 11, NO_ 3, 1989

We have only a short list of properties that w e k n o w w e can "'hear" and a short list that w e k n o w w e can't "'hear." As the discussion above suggests, we have only a short list of properties that we k n o w we can " h e a r " a n d a short list that we k n o w we can't " h e a r . " The m a n y n e w examples are interesting and fun, but the k n o w n isospectral manifolds have m u c h in c o m m o n geometrically; locally they are indistinguishable since t h e y share a c o m m o n covering. Thus the detective w o r k of deciphering just w h a t geometrical information the spectrum holds has only begun.

References 1. P. B6rard, Spectral geometry: Direct and inverse problems, Springer Lecture Notes 1207 (1986.) 2. M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d'une Varidt~ Riemannienne, Springer Lecture Notes 194 (1971). 3. R. Brooks, Constructing isospectral manifolds, Amer. Math. Monthly 95 (1988), 823-839. 4. R. Brooks and R. Tse, Isospectral surfaces of small "genus, Nagoya Math J. 107 (1987), 13-24. 5. P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier, Grenoble 36 (1986), 167-192. 6. I. Chavel, Eigenvalues in riemannian geometry, New York: Academic Press (1984). 7. Y. Colin de Verdi~re, Spectre du Laplacien et longeur des g6odesiques periodiques II, Comp. Math. 27 (1973), 159-184. 8. D. DeTurck and C. Gordon, Isospectral deformations: Part I, Comm. Pure Appl. Math. 40 (1987), 367-387; Part II, to appear in Comm. Pure Appl. Math. 9. D. DeTurck, H. Gluck, C. Gordon, and D. Webb, You can't hear the size of a homology class, to appear in Comm. Math. Helv. 10. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math. 29 (1975), 39-79.

11. C. Gordon and E. Wilson, Isospectral deformations of compact solvmanifolds, I. Diff. Geom. 19 (1984), 241-256. 12. A. Ikeda, On lens spaces which are isospectral but not isometric, Ann. Sci. Ec. Norm Sup. 13 (1980), 303-315. 13. M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23. 14. , Enigmas of chance, an autobiography, New York: Harper and Row (1985). 15. J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci., U.S.A. 51 (1964), 542. 16. T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), 169-186. 17. H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ec. Norm. Sup. 15 (1982), 441-456. 18. M. Vign6ras, Vari6t6s riemanniennes isospectrales et non isom6triques, Ann. of Math. (2) 112 (1980), 21-32. Department of Mathematics Washington University St. Louis, MO 63130 USA STATEMENT OF OWNERSHIP, MANAGEMENT, AND CIRCULATION (Required by 39 U.S.C. 3685). (1) Title of publication: The Mathematical Intelligancer. A. Publication No.: 03436993. (2) Date of filing: 10/1/88. (3) Frequency of issue: quarterly. A. No. of issues published annually, 4. B. Annual subscription price, $28.00. (4) Location of known office of publication: 175 Fifth Avenue, New York, NY 10010. (5) Location of the headquarters of general business offices of the publishers: 175 Fifth Avenue, New York, NY 10010. (6) Names and addresses of publisher, editor, and managing editor:. Publisher. Springer-Verlag New York Inc., 175 Fifth Avenue, New York, NY 10010. Editor. Sheldon Axler, Dept. of Mathematics, Michigan State University, East Laming, MI 48824. Managing Editor. Springar-Verlag New York Inc., 175 Fifth Avenue, New York, NY 10010. (7) Owner. Springer Export GmbI~I, Tiergartenstr 17, 6900 Heidelberg, West Germany, and Springer-Verlag Berlin, Heidelberger Platz 3, D-1000 Berlin 33, West Germany. (8) Known bondholders, mortgagees, and other security holders owning or holding 1 percent or more of total of bonds, mortgages or other securities: none. (9) The purpose, function, and nonprofit status of this organization and the exempt status for Federal income tax purposes: has not changed during preceding 12 months. (10) Extent and nature of circulation. A. Total no. copies printed (net press run): Average no. copies each issue during the preceding 12 months, 6700; no. copies single issue nearest filing date, 8050. B. Paid circulation: 1. Sales through dealers and carriers, street vendors, and counter sales: Average no. copies each issue during preceding 12 months, 2085; no. copies single issue nearest to filing date, 2085. 2. Mail subscriptions: average no. copies each issue during preceding 12 months, I937; no. copies single issue nearest to filing date, 2023. C. Total paid circulation: average no. copies each issue during preceding 12 months, 4022; no. copies single issue nearest to filing date, 4108. D. Free distn'bution by mail, carrier, or other means. Samples, complimentary, and other free copies: average no. copies each issue during preceding 12 months, 558; no. copies single issue nearest to filing date, 558. E. Total distribution: average no. copies each issue during the preceding 12 months, 4580; no. copies single issue nearest to filin/g date, 4666. F. Copies not distributed. 1. Office use, left-over, unaccounted, spoiled after printing average no. copies each issue during the preceding 12 months, 2102; no. copies single issue nearest to filing date, 3374. 2. Return from news agents: average no. copies each issue during the preceding 12 months, 18; no. copies single issue nearest to filing date, 10. G. Total: average no. copies each issue dttring preceding 12 months, 6700; no. copies single issue nearest to filing date, 8050. I certify that the statements made by me above are correct and complete.

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Ute Bujard Vice-President, Production

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989

47

The Gleichschaltungof Mathematical Societies in Nazi Germany* H. Mehrtens

Translated by Victoria M. Kingsbury

During the 1933 business meeting of the Deutsche Mathematiker-Vereinigung (DMV) [German Mathematicians Union[, only the allocation of 300 Reichsmarks "for national w o r k " s h o w e d that the " n e w time" had begun [1]. Other signs were apparent, however, in the report on the meeting of the Mathematischer Reichsverband (MR) [Reich Mathematical Association], which took place at the same place and on the same date. The Fi~hrer (leadership) principle was accepted and the former chair and new Fi~hrer of the group, Georg Hamel, made the following statement:

The difference in the behavior of the three groups is understandable; the GAMM and DMV were scientific specialty organizations, whereas the MR was a political interest organization. The GAMM and DMV were not spared the pressure of Gleichschaltung, however. I wish to report here about the history of these three organizations in those years. The contemporary term Gleichschaltung 1 does not adequately define the process of fitting into the system of

We want to cooperate sincerely and loyally in accordance with the total state. Like all Germans, we place ourselves unconditionally and happily in the service of the National Socialist movement, behind its F/ihrer, our Chancellor Adolf Hitler. [2] At the meeting of the Gesellschaft fiir angewandte Mathematik und Mechanik (GAMM) [Society for Applied Mathematics and Mechanics], which also met in the same location, two Jewish members of the executive board resigned; other than that, no political statements were made.

* This article is a translation of "Die 'GIeichschaltung' der mathematischen Gesellschaften im nationalsozialistischen Deutschland," Jahrbuch UberblickeMathematik 18 (1985), 83-103. t This work developed in the context of a research project, supported by the Stiftung Volkswagenwerk, on mathematics during the National Socialist period. A preliminary version was presented at the annual convention of the DMV in K61n in 1983. I would like to thank Professor M. Kneser and the Max-Planck-Institut ffir StrOmungsforschung, who made available the most important sources for this project. 1 Translator's footnote: Gleichschaltung is difficult to translate into English. When applied to events during the Third Reich, Gleichschaltung means the process of bringing organizations into line with Nazi ideology. 48 THEMATHEMATICALINTELLIGENCERVOL.11, NO. 3 9 1989Springer-VerlagNew York

the National Socialist state. It was not simply a matter of being geschaltet. In all cases it was a much more complex process, one in which the political and professional interests of those affected interacted with the interests of various state and party authorities who were sometimes even antagonistic toward each other. 2 The atmosphere was laden with the pressure and uncertainty developing from the rise of radical Nazis and the persecution and expulsion of "racial" and political enemies of the system. In addition, the basically conservative stance of the German academicians led them to accommodate themselves all too willingly to the s u p p o s e d l y legal measures that were taken; for the most part they placed their hopes on the new, strong state in the belief that the barbaric abuses of Nazi domination were just a passing phenomenon. Historians have often spoken of Selbst-Gleichschaltung'" (self-Gleichschaltung) in the context of science and universities (e.g., [6]). Actually there was indeed a quick accommodation to the new system, with no perceptible opposition. Already by 1932 the professors in Heidelberg had willingly bent under the pressure of N a t i o n a l Socialist s t u d e n t s and r e m o v e d Julius Gumbel from his position as associate professor of mathematical statistics, because of his political and pacifist activities [7]. Gumbel was the only mathematician on the Nazis' first "withdrawal of citizenship" list, which included Albert Einstein and other prominent figures. To depict fully the Gleichschaltung of the mathematical institutes, one would have to discuss the expulsion of mathematicians from their positions in universities as well as the history of the institutes, journals, publishing companies, and local organizations. That will not be done here because only a detailed discussion of individual cases w o u l d make them understandable. To begin, I just want to touch quickly on the topic of expulsion. 3 The formal basis of persecution was the "Law for the Restoration of the Civil Service" of 7 April 1933, according to which officials could be fired or forced to retire for political or racial reasons, as well as for "simplification of the administration." Although there were exceptions for "old officials," or "front-line fighters" of the First World War, increased pressure was placed on members of these two groups so that they "voluntarily" went on leave or retired, as did Richard Courant and Edmund Landau. As in the case

2 O n the structure a n d d e v e l o p m e n t of the National Socialist system, cf. Franz Neumannls "Behemoth" [3]; on the subject of universities and science, cf. above all Heinemann I4] and Mehrtens [5]. 3 On the subject of persecution and expulsion of mathematicians, cf. Pinl and Furtm(iller [8],[9], and Schaper [10], as well as a detailed case study of the G6ttinger Institut by Schappacher 111].

Richard Courant

E d m u n d Landau

of Landau, the pressure was intensified by actions of the student body. Otto Blumenthal in Aachen was denounced by the general student council, placed into "protective custody" for fifteen days, and then fired according to the Law for the Restoration of the Civil Service [Q1]. In some places, like Bonn, things remained quiet, and the "exceptions" could remain in their positions. But in 1935, according to the "Nuremberg laws," they too had to leave, as did Felix Hausdorff and Otto Toeplitz in Bonn, for example. In 1937, the " N e w Official's Law" affected even those w h o were "related to Jews by marriage," like Erich Kamke or Emil Artin. Pinl and Furtm(iller [8],[9] mention 127 persecuted mathematicians, of whom 101 were "racially" persecuted. The numbers are certainly too low, and one should not forget the foreign victims such as the Polish mathematicians. In view of the weight of each individual fate, quantitative precision is meaningless. Nevertheless, to give an idea of the extent to which expulsions took place, a few further figures will be mentioned. 4 In 1931, 227 mathematics instructors taught in German universities. By 1938 more than one-fourth of them had been expelled. Of 95 full professors, 26 were expelled; of 45 associate professors, 13 were expelled; and of 55 university lecturers, 29. In addition, m a n y assistants and advanced students were also expelled. The losses could barely be replaced. A striking deficit of younger scholars was unavoidable, made worse

4 The following figures, as well as those given later in this article for filled positions, rely on a combination of data from Ferber [12] and Pinl and Furtm~iller [9], with a few corrections and additions. The expulsion of mathematicians in Austria and other annexed regions is not taken into account here. The presentation of complete data remains s o m e w h a t uncertain, since there are always borderline cases. THE MATHEMATICALINTELLIGENCER VOL. 11, NO. 3, 1989 49

Otto Toeplitz

Deutsche Mathematiker-Vereinigung (German Mathematicians Union) The currents of the time are against rationality, in the trouble and confusion of the day the passions of youth turn to other gods and idols. [14]

Emil Artin

by the drastic fall in the number of students, which was more severe in the fields of physics and mathematics than in other subjects. In 1932, 7139 students still studied mathematics, actuarial theory, or physics. In 1939, still before the beginning of the war, there were only 1270. In G6ttingen during the same period the number of mathematics students decreased from 432 to 37 [13]. As a consequence, m a n y positions remained unfilled or had to have continual replacements. The loss of filled positions is approximately as high as the number of expulsions. In addition, through cancellations and demotions, approximately ten percent of the positions were completely lost. Those who remained were confronted with massive professional and political challenges. The pressure to engage in politics in the interest of the profession, the institutes, and the younger generation of mathematicians was intense. There was tremendous pressure to conform to the politics of the regime, because the power lay on the side of the state and the party. Unlike the party, and often in contradiction to it, the administrative bureaucracy, with its aura of legal and orderly conduct, provided a feeling of confidence, especially given the widespread bureaucratic mentality of the German university faculty and staff. For this reason the legality of expulsion and m u c h else was accepted. Those wanting to achieve anything for themselves or their professions had to conform to an unsympathetic system and renounce a n y s y m p a t h i e s that could have negative consequences. The anxiety and uncertainty engendered by the expulsion of colleagues contributed to this feeling. 50

THE MATHEMATICAL INTELLIGENCER VOL. n , NO. 3, 1989

These words were part of the congratulations the DMV sent to David Hilbert on the occasion of his 70th b i r t h d a y in 1932. It is signed by the four board members: H e r m a n n Weyl, who retired by rotation from the board of the DMV and in 1933 emigrated to the U.S.; Ludwig Bieberbach, the secretary, who became the Nazi ideologist of mathematics; H e l m u t Hasse, the treasurer, who took over the mathematical institute at G6ttingen; and Otto Blumenthal, publisher of the DMV's journal, the Jahresbericht, with Bieberbach and Hasse, who was dismissed in Aachen and died in 1944 in the Theresienstadt concentration camp. The positions of secretary, treasurer, and publisher of the Jahresbericht were practically permanent positions. Therefore it was very difficult for the DMV, when it was politically challenged by its secretary Bieberbach in 1934, to distance itself clearly from him. Blumenthal retired as publisher of the Jahresbericht in 1933, and Konrad Knopp succeeded him; the reasons for this are u n k n o w n to me. Blumenthal stayed as head publisher of the Mathematische Annalen until 1938 [15]. Oskar Perron, who certainly did not sympathize with the Nazis, was elected to the board of the DMV in September 1933. On 8 April 1934 an article entitled "Neue Mathematik" [New Mathematics] about a lecture by Bieberbach a p p e a r e d in the n e w s p a p e r Deutsche Zukunft [German Future] [16]. According to this article, his lecture appeared to show that the lessons of blood and race also apply to mathematics and place the most abstract of all sciences under the total state. Bieberbach's speech may be seen in this connection to have historical significance for the Third Reich. The newspaper report goes into detail about the justification of the student boycott against Landau: "Bieberbach's rejection of Landau is of biting sharpness, a rejection in the G e o r g e s c h e n style of d a m n i n g . " "Jewish versus Aryan" mathematics, "mental arrogance," "devilish cleverness," "juggling with concepts," and the "cunningness" of Jewish mathematicians like Jacobi were some of the expressions used. Reports of the lecture were quickly circulated and aroused international indignation. Courant wrote to Weyl: Have you read the newspaper articles about Bieberbach's scandalous lecture on Jewish and Aryan mathematics? l have arranged to have a copy of the newspaper sent to you. For his robust attitude Bieberbach was rewarded with the position of substitute vice-president of the University of Berlin. [Q2], No. 59

Harald Bohr, a Dane, published a newspaper article about Bieberbach's new mathematics [17], which he s e n t to a few G e r m a n m a t h e m a t i c i a n s . O s w a l d Veblen, an American, circulated the indignant letter he had written to Bieberbach [Q3], and G. H. Hardy in England wrote a letter to the editor of Nature [18], after a summary of the first lecture was published by Bieberbach himself. Bieberbach had spoken about "the structure of personality and mathematical creativity" [19],[20]. He began with a justification of the G6ttingen student boycott against E d m u n d Landau, divided the "styles of mathematical creativity" into a "positive German Jtype" and a "negative S-type," and professed his faith in the "German type" in mathematics. The anti-Semitism in Bieberbach's lecture was hidden by a psychological theory of mental types. The newspaper article was in this respect somewhat more sharply formulated. Even if Bieberbach used the term Jewish mathematics carefully, it was quite clear at that time what the juxtaposition of the terms inherent mathematics and the opposite type meant, s The DMV reached a crisis point when Bieberbach, against the will of the other two editors, published an "open letter to Harald Bohr" [22] in the Jahresbericht, in which he defended his lecture and called the newspaper article a "malicious caricature." In his justification, however, Bieberbach retracted none of his statements and called Bohr a "well-poisoner" and "damager of all international cooperative work," one who "was blinded by hatred against the new Germany, an impenetrable hatred, which emanated from every sentence in his article." With the argument between Bohr and Bieberbach, the conflict within the DMV began; at the same time, the DMV saw itself as being endangered from the outside. If it became identified with Bieberbach's statements, its international reputation w o u l d be lost. Masses of foreigners, who formed a significant part of the 1100 members at that time, might resign from the organization. In Germany, Carl Ludwig Siegel, although himself not a member of the DMV, reacted as a matter of principle: "Let your resignation from the DMV be a symbol for everyone," he wrote on 7 July 1934 to Hasse [Q4]. Since it seemed best that the DMV distance itself clearly from Bieberbach, so the question of leadership cropped up, a question that had to be faced at the annual meeting in September. The result was ambiguous; how it affected emigrees' and foreigners' opinion of the DMV is illustrated in a letter from George P61ya to Weyl on 2 February 1935~

5 We cannotgo into the contentand motivesof Bieberbach'stheories on race and style here (cf. [12],[44]).

Oskar Perron

The report on the meeting in Pyrmont and the entire volume of the Jahresbericht (without Bieberbach on the title page) unequivocally suggests the dilemma: either to resign, because of the principal mistake of censuring Bohr, or not to resign because of, all things considered, the actual triumph over Bieberbach. I am deciding on a third option: to wait until the situation becomes clearer, e.g., until the "triumph" over Bieberbach-- which is not yet finalized--develops into a triumph by Bieberbach and a complete "Gleichschaltung," and then to resign in corpore, causing a scandal. [Q2], No. 420 What had happened? Both a censureship of Bohr and Bieberbach's defeat? A detailed protocol of the business meeting in Pyrmont on 13 September 1934 was published [23] and provides quite a bit of information. It does not record, however, that Bieberbach had appeared with a following of students, some of them in uniform, including Fritz Kubach, the math department's leader of the German student body; nor does it state that Perron, the chairman of the board, bravely and as a matter of principle, had halted this demonstration at the beginning of the meeting by excluding the n o n m e m b e r s according to the bylaws [Q5], [11]. Perron's further leadership of the proceedings seems not to have been so successful. Erich Hecke wrote in a letter to Hermann Weyl [Q2], No. 56, that the first decision against Harald Bohr was "unavoidable in the situation in which we f o u n d ourselves, after the matter had been pushed onto this track because of the ineffectual chairman." The "matter" was the question that was asked as the first point of order of the day in the "Secretary's Report": whether Mr. Bieberbach had the right to publish . . . his open letter, a n d . . , to decide on its insertion, despite the fact that both other editors were against its insertion. [23], p. 86 THE MATHEMATICAL INTELLIGENCER VOL. I1, NO. 3, 1989

51

The "track" onto which the matter was pushed was the general political question of whether Bohr's statem e n t was an attack on Germany and National Socialism. An attempt to keep the conflict at the internal level of the DMV was apparently not successful. The discussion led to a "national" avowal by the DMV. After a lengthy discussion Mr. Mi~ller from Hannover put the following motion to a vote: "The membership meeting decries the attack of Mr. Bohr against Mr. Bieberbach, inasmuch as it represents an attack on the German state and on National Socialism. It regrets the form of Mr. Bieberbach's open letter and his act of publishing it, which was done against the will of both editors and without the knowledge of the chairman." Mr. Sch6nhardt from T~ibingen put the following additional motion to a vote: "The members present recognize that, in the matter of Bohr, Mr. Bieberbach was concerned with protecting the interests of the Third Reich." The secret written vote on these two motions had the following result: 50 yeas and 10 nays with 2 abstentions for M~iller's motion and 31 yeas, 20 nays, and 10 abstentions for Sch6nhardt's motion. ([23], p. 87) In addition, Bieberbach made known that he had retorted to Bohr only because of the " h i d d e n but infamous attack" on the Third Reich. The professional behavior of the members of the DMV was logical, and they were hardly timid regarding internal organizational matters. Under the circumstances, however, they did not have the courage to avoid taking a political position, and they dared not reject M~iller's motion. In this area they were scared. So Bohr was sharply criticized, whereas Bieberbach's behavior was only "regretted," and was, as a matter of general politics, even accepted.

In 1931, 227 mathematics instructors taught in German universities. By 1938 more than one-fourth of them had been expelled.

This suggestion provided for a weak Fidhrerprinciple and was at the same time a trick, apparently devised by Hasse and Hecke. With this change in the bylaws, the newly elected F~ihrercould dismiss Bieberbach in short order from his position as secretary [Q12]. The " G e r m a n " mathematicians were outvoted: Bieberbach's resolution received 40 nays, 11 yeas, and 3 blank pieces of paper; Tornier's got 38 nays, 10 yeas, and 5 abstentions. In contrast, Hecke's resolution was accepted with the required three-fourths majority: 38 yeas, 8 nays, and 4 abstentions. Blaschke was approved as chairman in accordance with the change in the bylaws, and as secretary Bieberbach had the task of entering the change in the bylaws into the official record. The result of the meeting was ambiguous. On one hand, without explicitly having to bend to the will of the party, a weak Ff~hrerprinciple had been instigated, to the DMV's advantage, and with.it was the chance to distance Bieberbach from the DMV. But first the change in the bylaws had to become effective, and the ministry's reaction could not be predicted. On the other hand, in the Bohr affair the DMV had allied itself too closely with Bieberbach. The DMV still needed to distance itself clearly from Bieberbach to affect foreign opinion positively. Bieberbach apparently found it difficult to complete his own demise and delayed registering the change in the bylaws, although he seemed to have already given up his hope of getting control of the DMV. As the first step toward the creation of a new society of national mathematicians, four weeks after the meeting of the DMV, he requested financial help from the German Research Association for the founding of a Deutsche Zeitschrift fiir Mathematik [German Journal of Mathematics] and commented that the DMV had shown "that it at present did not want to be a proponent of the national interest" [Q6], 14 October 1934. Four days later Blaschke asked Bieberbach h o w things s t o o d with registering the change in the bylaws. 6 The answer was evasive, and repeated questioning brought no new result. Then Blaschke's tone sharpened. On 6 November 1934 he wrote to Bieberbach.

The elections followed; Wilhelm Blaschke was elected chairman for the following year. Then came the "motions regarding the transition to the Fiihrer principle." Bieberbach proposed voting for a F~ihrer who would have the job of revising the bylaws in accordance with the ministry, and suggested Erhard For the third time I would like to ask you the following Tornier as F~ihrer. Tornier in turn nominated Bieberquestion. Have you filed the change in the bylaws for the DMV at the district court in Leipzig or not? bach. The F~hrer principle was another general political issue. To deny the motions would have created Bieberbach answered that he had requested an exthe suspicion that the DMV was taking a position pert legal opinion because of legal considerations and against one of the basic elements of National Socialist he wanted to spare the DMV the disgrace of a refusal ideology. If the DMV did not want to endanger itself, or challenge. Hasse and Knopp then wrote to Bieberit had to find a compromise solution. The suggestion for such a compromise came from Hecke. The chairman will be elected for two years by the assembled members by acclamation or by secret written vote with relative voting majority. He will appoint and dismiss the board members. ([23], p. 87) 52 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989

6 The following depiction of t h e occurrences until the c o m m i t t e e m e e t i n g in J a n u a r y 1935 relies o n t h e e x c h a n g e of letters b e t w e e n K n e s e r a n d Bieberbach, a n d K n e s e r a n d H a m e l [Q7].

b a c h o n 18 N o v e m b e r , s u m m a r i z e d the letter exc h a n g e b e t w e e n Bieberbach a n d Blaschke, and closed w i t h the following words: We are forced to regard your behavior and actions as insincere and uncollegial. Therefore, cooperation with you is no longer possible for us. Since we two form a twothirds majority of the board of the DMV, we have decided to restore, and therewith take into our hands, the statutory changes, which have been endangered because w~u hold both the secretary and editor positions. Blaschkehas sanctioned our decision. We have sent a corresponding letter to the Teubner company. The b o a r d believed it n o w had leverage against Bieberbach. Blaschke, H a s s e , a n d K n o p p a s s e m b l e d quotations f r o m the entire exchange of letters, w h i c h was t h e n sent as a letter to Bieberbach a n d as a circular "to m e m b e r s of the D M V . " It e n d e d with the conclusion " t h a t collegial c o o p e r a t i o n with y o u in the leadership of the D M V is no longer possible for us. We request that y o u realize the c o n s e q u e n c e s . " Bieberbach h a d a right to be indignant, h o w e v e r , b e c a u s e in the m e a n t i m e he h a d g o t t e n an e x p e r t o p i n i o n on the c h a n g e in the bylaws. The expert h a d d e t e r m i n e d that b e c a u s e of a few insignificant mistakes in form, the c h a n g e in the b y l a w s could not be recorded. The letter i n f o r m i n g the b o a r d of this had arrived w h e n the a b o v e - m e n t i o n e d circular was sent out. Bieberbach a n s w e r e d : Basically I maintain that raising an attack against me through a circular, without a complete presentation of the facts, is to be sharply criticized and contradicts all sense ot fairness. It is especially unfair that the position of fine registry j u d g e - - t h e main point that justifies my actions was intentionally kept back from the recipients of the circular. The other three did not back d o w n . A further circular followed, one in w h i c h the formal m i s t a k e s of the c h a n g e in the b y l a w s w e r e m e n t i o n e d . It stated that the accusation of uncollegial b e h a v i o r applied to o c c u r r e n c e s in c o n n e c t i o n w i t h the Bohr letter a n d Bieberbach's b e h a v i o r before he h a d g a t h e r e d all the information. N o t h i n g w a s t a k e n back. This circular is d a t e d C h r i s t m a s 1934. Bieberbach h a d already b r o u g h t a n e w a n d s u p p o s edly decisive factor into all this. H e h a d r e q u e s t e d the a d d r e s s list of the first circular but did not receive it until a f t e r C h r i s t m a s . O n 4 J a n u a r y 1935 Bieberbach sent a r o u n d his o w n circular. Of the 130 recipients, 36 are foreigners--not counting emi g r a n t s - f r o m all countries, including America and Japan. As a consequence I feel compelled to explain that the distribution abroad of a circular directed against the integrity of a German colleague--especially in the present time of organized agitation against everything G e r m a n - - i s an action which speaks of a deplorable lack of national feeling and national pride. With that letter the m a t t e r w a s again p u s h e d onto the track that h a d a l r e a d y led to the " j u d g m e n t " on THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989 5 3

Bohr at the Pyrmont meeting. Bieberbach had created a general political question out of a question of collegiality and correctness of behavior within the DMV. Here the members of the board of the DMV were in a quandary. On one hand, the DMV was an organization with an international membership; it needed to make clear to the foreign members that the DMV was warding off any Nazification of mathematics in Bieberbach's style. If it wanted to preserve its reputation, the DMV had to make the conflicts internationally public. In "the national state" of the Third Reich, however, that was a political act, especially when emigrants were involved. With his new tactic, Bieberbach tried to swing the a r g u m e n t in his favor. He c o m m u n i c a t e d w i t h Theodor Vahlen, the mathematician in the Reich's M i n i s t r y of E d u c a t i o n (cf. [24]), and w i t h Georg Hamel, in order to overturn the board of the DMV. On 19 January an ultimatum was sent to the board of the DMV.

Hamel was thus elected the new chair, and Emanuel Sperner was made the secretary, both sufficiently nationally minded men. The "attributes of the leader" [F~ihrereigenschaft] were laid d o w n as follows in the succinct published protocol [25]: It was agreed that the two remaining members of the board, Misters Hasse and Knopp, are to secure Mr. Hamel's agreement on all important decisions.

Bieberbach was in this way shut out, the DMV had avoided the instigation of an explicit Ffihrer principle and a radical Gleichschaltung had been avoided. The result must be seen as a compromise with the ministry. The autonomy of the DMV was not disturbed, and in exchange it was made certain that the leadership of the DMV remained loyal to the state. This compromise came about because Bieberbach hardly found undivided support. Erhard Schmidt and Hellmuth Kneser, who were involved in the conflict on Bieberbach's side, did not really stand behind him, but had a vested interest in the autonomy of the DMV and Undersecretary Director Vahlen has instructed me to communicate the following to the board: it must be taken into in its reputation abroad. Even more important is the account that the Reich's Ministry of Education will pro- behavior of the ministry, which could have actually hibit its officials from becoming members of the German exerted pressure, as it had in the ultimatum of 19 JanMathematicians Union if the following two conditions are uary. Obviously, however, neither Vahlen nor Minnot met: 1) Mr. Blaschke must step down from his posi- ister Rust was interested in raising Bieberbach to the tion on the board, since he is responsible for the defamation of a colleague in respect to foreigners, and 2) a change throne of leadership. Here, as in other cases, the ministry, which itself in the bylaws must be enacted in which it is stated that the responsible leadership of the German Mathematicians disagreed with party ideologues like Goebbels and RoUnion demands the trust of the leadership of the land, senberg, had retained in its own area of competency a i.e., confirmation through the Reich's Ministry of Educa- remnant of bureaucratic rationality, which was dition. rected t o w a r d a real functioning science and with Bieberbach further suggested Hamel as the new which it could bind scientific and educational instituchairman and commissioned a notary to work out new tions to the state. Bieberbach's hope for a n e w " G e r m a n " mathebylaws. At the end of the letter he says, "With the instigation of the n e w bylaws my position in the matics society also soon disappeared. In 1936 his German Mathematicians Union expires." This honor- journal Deutsche Mathematik was started, with a high able decision was, however, not completely voluntary. circulation and massive financial support. In the folHamel had insisted that Bieberbach step down as a lowing year, after the Nazi physicist Johannes Stark condition that he would play along. Also, Bieberbach was toppled from the presidential seat of the German apparently withdrew his request for the resignation of Research Association, donations were reduced. A the whole board of the DMV only after being pres- m e m o r a n d u m stated somewhat cynically that, "in accordance with its purpose, the journal should be supsured. On 25 January a committee of the DMV, a type of ported entirely by the political will and spirit of sacriexpanded board, met in Berlin. No one insisted any fice of the participants" [Q6], 3 March 1937. The circulonger on the resignation of Hasse and Knopp, after lation d e c l i n e d steadily, a n d in 1942, w h e n the Blaschke had declared that he alone was responsible publishing company wanted to pulp the numerous old for sending a r o u n d the circular. Vahlen even re- issues, it wrote: n o u n c e d the change in the by-laws. Hamel wrote It is hardly to be assumed that Professor Bieberbach has about this to Hellmuth Kneser. plans for greater propaganda use for the journal after the war, for which sample copies would be required. ([Q6], 21 Even Vahlen himself had not desired a change in the March 1942) by-laws; it was enough that the chairman to be voted on needed to have the approval of the ministry. The imThe DMV elected Wilhelm Sfiss as its chair in 1937 mense formal difficulties of a change in the by-laws--for and enacted a change in the by-laws that would allow each small change a three-fourths majority is required-his reelection until the end of the war [26], p. 20. With were for the present disregarded, and the attributes of a leader were ascribed to me only as a matter of protocol. that action the hidden Ffihrerprinzip in the spirit of the 54

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compromise of 1935 was completed. Sfiss proved to be an extraordinarily skillful and successful professional politician, who stood on good terms with Minister Rust and cooperated closely with other research politicians during the war. With a view to the situation after the war, he understood how to preserve the interests of mathematics, which clearly was part of the compromise of the DMV with the National Socialist regime.

Mathematischer Reichsverband (Reich Mathematical Association) The Reichsverband deutscher mathematischer Gesellschaften und Vereine [Reich Association of German Mathematical Societies and Organizations] (the MR's full name), as mentioned at the beginning of this article, is the example of "joyful" setf-Gleichschaltung in mathematics. The reason for this lies in the nature of the MR and in the person of Georg Hamel, who was the chair from its founding in 1921 until the end of the MR during the war. At the beginning of the Weimar Republic the curriculum allotment for mathematics in the schools was in danger of being decreased, so the MR was formed out Jof the DMV "for the effective representation of common interests" [27], p. 44. In Berlin the MR had a work group, and an advisory board took care of the connection with all parts of the country. The MR was closely tied to the DMV and always met during the DMV's annual convention. Its activities were directed above all to the curricula of the higher institutions of learning and to university instruction. The group was especially concerned with promoting applied mathematics. 7 Close relations existed between the MR and the Deutscher Verein zur FOrderung des mathematischen und naturwissenschaftlichen Unterrichts [German Group for the A d v a n c e m e n t of M a t h e m a t i c a l a n d N a t u r a l Science Instruction], called FOrderverein (advancement group) for short. It is known that the teachers and their organizations turned quickly to the Nazi party [30]. The FOrderverein associated blatantly with the new state in the spring of 1933, offered its services, aligned itself with the National Socialist Teachers' Union, and assimilated the Fiihrer principle and the "Aryan Paragraph" into its by-laws (cf. [31], [32]). If the MR wanted to continue to be active with any prospect of success, it could not distance itself from the F6rderverein and had to subjugate itself to the new leaders. Political or moral resistance was hardly to be expected from the members of the work group and

7 O n t h e subject of t h e MR, cf. t h e short orthodox-Marxist interpretation [28], as well as [29], pp. 86ff. Since no collection of H a m e l ' s files exists, t h e source situation is very bad. I h a v e not yet f o u n d a n y larger collection of t h e circulars that the MR regularly s e n t a r o u n d .

especially not from Hamel, as the tone of his vicechancellor's speech of 1928 [33] shows. At the meeting of the MR in May, the stance of the F6rderverein was affirmed and its resolution quoted in full in the circular of 29 May 1933 [Q8]. In June, Hamel sent a memorandum to the Ministry of Culture, in which the necessity of m a t h e m a t i c s " f o r the e d u c a t i o n of the German people" in line with National Socialist ideology was confirmed [Q9]. At the annual meeting in September the avowal cited at the beginning of this article was formulated and the Fiihrer principle confirmed. In October, at a demonstration of the F6rderverein in Berlin, Hamel spoke on "Mathematics in the Third Reich." He talked about "the practical," "blood and race," and "the heroic." At the conclusion he stated: Mathematics as a teaching of spirit, of spirit as action, belongs next to the teachings of blood and soil as an integral part of the entire educational process. The unity of body, mind, and spirit in the human parallels the unity of body hygiene, mother tongue, and teachings of blood, soil, and active creative spirit in education. Mathematics is the central core of the latter. ([34], p. 309) One does not have to believe that Hamel actually felt very strongly about blood and soil, bodily hygiene, and the mother tongue. He was concerned only about mathematics. The actions of the FOrderverein and the MR were obviously aimed at securing a safe place for mathematics in the National Socialist school curriculum.

Hamel sent a memorandum to the Ministry of Culture, in which the necessity of mathematics "'for the education of the German people" in line with National Socialist ideology was confirmed. In 1934 the MR proclaimed: "The work of the future should above all be practical, and we should be prepared to create concrete assignments in areas important for the education of the people" [35], p. 3. The result was a "Handbook for Teachers" with the title "Mathematics in the Service of National Socialist Education" [36]. Two examples from it will suffice. Under the titles "Colonies" and "Race and Kin" are found the following assignments: Assignment 85. In the German Reich the population density of one square km is 140 inhabitants, in Poland 80, in southeast Europe 48, in northeast Europe 32, in Soviet Russia 8. Depict this in five squares, with each square having a shaded portion to reflect the population density." (p. 40) Assignment 95. The building of an insane asylum required 6 million Reichsmark. How many housing settlements at 15,000 Reichsmark each could have been built with that amount? (p. 42) THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989 55

Here Nazi ideology is packaged in objective, mathematical terms. The editor writes: "This handbook methodically strives to hammer into the people the basic facts that determine the policy of the government" (p. 34). In the foreword, Hamel himself makes it again clear why the handbook has been created: It should be clear that mathematics is indispensable for the basic understanding of the science of people and state [Volkswissenschaft] and National Socialist reconstruction work. Mathematics will thus retain the number of instructional hours needed for furthering the goals of the fatherland. (p. 1) How much or how little National Socialist conviction stood behind this statement appears immaterial. Hamel and others played a role as representatives of the professional interests of m a t h e m a t i c i a n s and teachers. Where only school instruction was involved, t h e i r politics w e r e a c c o m m o d a t i n g a n d w i t h o u t scruples. The MR functioned as a buffer for the professional scientific societies, especially the closely allied DMV: because the MR conformed so radically, the DMV could defend its autonomy. I am not aware of a n y protests a g a i n s t the MR by the DMV or the GAMM. The only protest came from Siegel, who had also distanced himself from the DMV. His Frankfurt seminar group withdrew from the MR in 1934 [35], p. 3. By 1936/1937, service to the fatherland had replaced "conformation to ideology" as the basis of National Socialist scientific and university politics (cf. [5]). The MR, with its traditional lobbying for applied mathematics, could easily adapt itself, in cooperation with the DMV. One result of this cooperation was the establishment of a new degree for mathematicians qualifying for jobs outside the teaching profession (cf. [43]). During the war Hamel worked together with Wilhelm Sfiss, but because of the wartime conditions and the dominance of S~iss, the MR lost its relevance and no longer existed by the end of the war.

Gesellschaft ffir angewandte Mathematik und Mechanik (Society for Applied Mathematics and Mechanics) Different from the MR and even stronger than the DMV, the GAMM was an organization with primarily intra-scientific interests. In 1921 the Verein Deutscher Ingenieure [Organization of German Engineers[ (VDI) founded the Zeitschrift fiir angewandte Mathematik and Mechanik (ZAMM) [Journal for Applied Mathematics and Mechanics]. One year later the representatives of these fields tried to separate themselves from their traditional disciplines (mathematics, the various engineering sciences, and physics) to create their own professional organization. How this new discipline was to be defined was not clear. Richard von Mises, editor of the ZAMM, stressed above all "practical necessities," 56 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989

Carl Ludwig Siegel while the a e r o d y n a m i c e n g i n e e r L u d w i g Prandtl would have preferred to have used only the words technical mechanics in the name of the society. A few applied mathematicians totally opposed the founding of the society, in order to avoid a splitting of mathematics, s Despite such non-unanimity, the founding of the GAMM in 1922 was a success. In Germany the GAMM had close ties with the VDI. Internationally, since 1924, it had taken part in the organization of the International Congresses for Mechanics despite the boycott of German science [40]. In 1933 the GAMM had 444 members, of which more than a third were foreigners. Since 1922 the board had been composed of Ludwig Prandtl as chairman, Hans Reit~ner as his deputy, and von Mises as secretary. Reil~ner and von Mises were of Jewish background. On 22 June 1933 Prandtl wrote to Erich Trefftz:9 Von Mises and Reit~ner have pointed out to me that at the next main meeting, the Society for Applied Mathematics and Mechanics must be "gleichgeschaltet," and that they wish to resign at this occasion. Prandtl suggested the resignation of the entire board and asked Trefftz to take over the chairmanship. In a further letter on 26 June he clarified that von Mises should remain as editor of the ZAMM and that the resignation meant that both of them no longer

8 Unless otherwise noted, the depiction of the history of the GAMM follows Gericke [37]; regarding the founding of GAMM pp. 5-10 (cf. also [38]); regarding Prandtl [39]. On applied mathematics in general, cf. [43]. 9 The following exchange of letters between Prandtl and Trefftz is found in [Q10a].

wanted to be "publicly" active for the GAMM because that could bring difficulties. A Gleichschaltung with a National Socialist at the top was not appropriate for a scientific organization. Trefftz answered on 7 February that changing the entire board would endanger the GAMM and asked Prandtl to remain as the chair. Instead of changing the board, "it would be better to dissolve the society, which I would in any case prefer to see, if we should be forced to exclude Jewish members." Trefftz repeated this idea in the same letter: "If we must exclude Jewish members, I would consider dissolution to be the most honorable course of action." Prandtl answered him very clearly on 7 June. The society was founded to secure the proper influence at meetings for our field, which had found no appropriate place at the annual meetings of either the mathematicians or the physicists. I believe that the necessity for organizing the mechanics scientists exists today even more so than at the time of the society's founding, and that we must not allow ourselves to be influenced by political moods; therefore, even if the exclusion of Jewish members should be requested, the GAMM must continue to be maintained. According to my feelings, considerations about what would be the most honorable course of action have nothing to do with this, since it is simply a matter of the necessity of our profession. The argument about the self-preservation of the profession, in which politics must not play a role, is used to counter Trefftz's political/moral considerations. Prandtl is clearly referring to the competition from the mathematicians and physicists. One may assume that he had the development of German flight technology in mind. His attitude was successful. Without showing himself to be subordinate to the National Socialists, he became one of the most important men of German aviation research (he became one of the four members of the Luftwaffe's "Leadership of Research" group) and his institute expanded considerably [39], pp. 17-20. At the main meeting of the GAMM, Trefftz and Constantin Weber replaced von Mises and Reif~ner on the board. Prandtl remained as chair. Von Mises had emigrated in 1933 to Turkey. Reit~ner was not dismissed until 1936 and left soon after for the United States. Until his death in 1937, Trefftz was the editor of the ZAMM, on the title page of which the names of the other coworkers were struck out in order to avoid difficulties. Prandtl and his colleagues on the board of the GAMM consequently led a persistent fight, which repeatedly revolved around the "non-Aryan question." To circumvent this problem, the GAMM avoided the connection with the "Reichsarbeitsgemeinschaft Techn i s c h - w i s s e n s c h a f t l i c h e r V e r e i n e " [the Reich's Working Organization of Technical-Scientific Groups], especially the VDI. Prandtl argued that the GAMM was not a professional organization, but was only re-

sponsible for holding conferences. Its members were all members of other organizations "so that there is no interest in including the members of our particular society in a politically aligned organization" (21 April 1937 to the VDI, [Qll]). In 1938 negotiations were renewed concerning the connection with the VDI, with the result that the VDI requested the GAMM to exclude "non-Aryan" members. Prandtl decided at first to do nothing further, and the VDI renounced the connection [Qll]. According to Prandtl's exchange of letters with the VDI a n d his colleagues on the board, he was not averse to a connection with the VDI under the condition that the GAMM retain its autonomy. He had renounced the connection at first because of the plan to invite the International Congress for Mechanics to Germany in 1942. The negotiations with the VDI took place before the International Congress of 1938, and any measures taken regarding the "non-Aryan question" would have endangered the acceptance of the German invitation.

The Reich's Ministry of Education, which was responsible for such matters, communicated to Prandtl on 28 August 1938 that "'Jews of foreign citizenship" who took part in the congress would not be regarded "'as Jews here," but there would be no place for "'non-Aryans of German citizenship.'" It is clear from the source materials about the International Congress [Q10b] that already in 1934 there were plans to invite the congress to Germany for 1938. Before the 1938 congress, which took place in the USA, Prandtl had explored the question of whether a German invitation had a chance. He had the complete support of the Ministry for Aviation in Germany, and the mechanics scientists appeared not to be averse to coming to G e r m a n y - - a s s u m i n g that no distinction would be made between Jews and non-Jews. Despite intensive efforts, Prandtl could not guarantee that condition. The Reich's Ministry of Education, which was responsible for such matters, communicated to Prandtl on 28 August 1938 that "Jews of foreign citizenship" who took part in the congress would not be regarded "as Jews here," but there would be no place for "non-Aryans of German citizenship." This decision destroyed the hope for a Congress in Germany in 1938. The GAMM was unable to do anything about the "non-Aryan question" no matter how much it argued that international communication was important for the development of flight t e c h n o l o g y - - a n d with it, the Luftwaffe. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

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By 1940 the GAMM itself was "Jew-free," because of a paragraph of the by-laws that allowed members who had been in arrears with their subscriptions for two years to be struck from the membership role. Despite that rule, five years elapsed before 37 names were struck, among them von Mises. Soon, after two years of arrears, Courant and von K~irman's names were struck, among others (Weber to Prandtl, 18 May 1938 and 14 December 1938 [Qll]). In 1940 it was ascert a i n e d that t h e r e w e r e o n l y t w o " n o n - A r y a n " m e m b e r s to be struck, one of w h o m was Reit~ner (Weber to Prandtl, 1 November 1940 [Q10c]).

Conclusions The atmosphere in which mathematicians in the Nazi state had to work is described by Karl-Dietrich Bracher as a "double track on which the enterprise of National Socialism and the Third Reich were run: radical politicizing and nonpoliticized specialization, reactionary political romanticism and glorification of modern technological progress at the same time" [41], p. 546. In the early years of the regime, the mathematical societies had to cope with radical political changes. Later they could find their place as representing unpolitical specialization. German fascism only appeared to q u e s t i o n the n e c e s s i t y of mathematics, w h i c h is unrenounceable in a modern industrial state, especially when a war is being prepared for and carried out. In that respect mathematics was not seriously endangered. Early on, the existing groups and organizations clearly could have recognized and fought against the criminal expansionism and racism; it was necessary, however, that they be integrated into the "enterprise of the Third Reich." I have depicted h o w that happened for the mathematical societies. In conclusion, I want to attempt an analysis of these various histories.

the DMV was bent straight into a functioning organization of scientific specialists w h o , although not enthralled by the Nazi state, were sufficiently loyal to it.

2. Schools as fields of legitimation A science like mathematics is dependent for its existence on governmental help and public recognition. It repeatedly has to legitimize its importance. For a science to survive in the institutes of higher education, it must be taught in primary and secondary schools. The MR was founded for that reason, when the curriculum allotment for mathematics seemed to be endangered. It was also for that reason that the MR took pains to point out the National Socialist value of mathematics education. There seems to have been no opposition from either the DMV or the GAMM. The ideologization of school instruction in no way endangered scientific work, b u t it c o n t r i b u t e d t o w a r d making clear to the Nazis that one could not do without a sufficient number of university mathematics teachers.

3. Applications as legimitation The mathematical professional organizations used possible technical, industrial, and military applications of mathematics to legitimize their existence: Prandtl and the GAMM in their associaton with aviation research and their seeking support from the Luftwaffe; the DMV and the MR in their concerns about the promotion of applied mathematics and the education of "industrial mathematicians," in the preparation of the "Diploma Examination Regulation," and in the programs Siiss organized on basic mathematical research important for war. The National Socialist leadership, w h i c h w a s p r e p a r i n g for and c o n d u c t i n g a war, always had an open ear for such ideas.

1. The squeeze The conservative powers saved the governmental and societal order for National Socialism by protecting them from National Socialism ([42], p. 106) This paradoxical formulation applies to the history of the DMV: the board of the DMV saved the association and German mathematics from National Socialist "German" mathematics. The conflict with the radical Bieberbach pushed the DMV into the arms of the ministry, which was as National Socialist as Bieberbach, although less radical. In 1935 and afterward, the ministry no longer had much interest in Bieberbach and his following. On the contrary, to keep the "German" mathematics from taking over was quite in the interest of the ministry. In the squeeze between the radicals on one hand and the governmental power on the other, 58

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4. "Unpolitical" mathematics Prandtl described himself after the war as an "unpolitical German" [39], p. 65ff. This argument, that one had been completely "unpolitical" and had been concerned only with "good science," can be read and heard often. To be "unpolitical" is a prerequisite of successful scientific research. For that reason the DMV defended itself so clearly against Bieberbach's political attacks. As far as methods and criteria for quality are concerned, mathematics could defend its autonomy, which was the prerequisite for producing useful mathematics and educating c o m p e t e n t specialists. The leadership of the ministries of Education and Aviation understood that too much political interference would

cization and the technical, bureaucratic, and military rationality of the system, they accommodated themselves as "unpolitical specialists." The only political potential for opposition was the international openness of science. Perhaps this at least can be a positive lesson. Acknowledgments

The translator wishes to thank Michael Eric Bennett, Martin Fuchs, and Bill Gear for their assistance with this translation. References A.

This photograph is from a report on the Mathematisches Arbeitslager Kronenberg, 1936. Lecturer E. A. Weiss (1900--1942) of Bonn University is at the far right.

hurt mathematics. In every system that depends on industry and the military, the "unpolitical" character of science and mathematics is necessary. Of course, the Nazis made it easy to give in to this illusion, because politics was so very much defined as ideology. Some mathematicians held themselves consciously apart from the applications of mathematics and really tried to be unpolitical. The mathematical societies did not try that.

5. I n t e r n a t i o n a l i s m

The histories of the GAMM and the DMV have shown that the international connection was as stubbornly defended as the autonomy of scientific work. These organizations could not persevere against the chauvinism and racism of the Nazis, especially not under wartime conditions. International communication is also an "unpolitical" condition for successful science, but this professional value stood counter to the regime. The balance is rather depressing. The mathematical organizations conformed to the new time by a division of labor. While the MR bowed deeply before the National Socialist ideology, the GAMM and the DMV defended professional values as a condition of scientific work and tried to legitimize themselves through practical usefulness. In the squeeze between radical politi-

Unpublished sources

Q1. Durchf/,ihrung des Berufsbeamtengesetzes [Implementation of the Civil Service Law], TH Aachen, Nordrhein-Westfalisches Hauptstaatsarchiv, D~isseldorf, NW 5, 831. Q2. Hermann Weyl Papers, Wissenschaftshistorische Sammlungen, ETH Z/irich, Hs 91. Q3. Letter from O. Veblen to L. Bieberbach, 19 May 1934, Wissenschaftshistorische Sammlungen, ETH Z(irich, Ms 653:1. Q4. Helmut Hasse Papers, Handschriftenabteilung der Nieders/~chsischen Staats- und Universit/itsbibliothek, G6ttingen, according to a communication from Martin Kneser (the nonscientific part of the memoirs is not yet accessible). Q5. Oral communication by B. L. van der Waerden, Ziirich, 21 March 1984. Q6. Deutsche Forschungsgemeinschaft: Zeitschrift Deutsche Mathematik, Bundesarchiv Koblenz R73, 15934. Q7. Hellmuth Kneser Correspondence, Private collection of Martin Kneser, G6ttingen. Q8. Richard von Mises Papers, Harvard University, Cambridge, MA, HUG 4574.5, Box II. Q9. NSLB: Mathematik, Bundesarchiv Koblenz, NS12.806. Q10. Ludwig Prandtl Papers, Archiv des Max-Planck-Instituts f~ir Str6mungsforschung, G6ttingen. (a) Prandtl's personal correspondence, (b) folder "V Internationaler Kongref~ f/ir technische Mechanik 1936-1938," (c) folder "GAMM/ZAMM, Nr. 2." Qll. GAMM-Archiv, Sammlung Gericke, Archly des MaxPlanck-Instituts f~ir Str6mungsforschung, G6ttingen. Q12. Letter from E. Noether to O. Veblen, O. Veblen Papers, General Correspondence, Manuscript Division, Library of Congress, Washington, DC. B.

Publications 1. Gesch/iftssitzung der D.M.V. am 20.9.1933 in W/~rzburg. Jber. DMV 43 (1934), 2. Abt., 81. 2. Jahresversammlung des Mathematischen Reichsverbandes in W/irzburg am 20. September 1933. Jber. DMV 43 (1934), 2. Abt., 81-82. 3. Neumann, F., Behemoth--Struktur und Praxis des Nationalsozialismus 1933-1944. Europaische Verlagsanstalt, K61n 1977. English original: Behemoth: The Structure and Practice of National Socialism. Oxford University Press, New York (1944). 4. Heinemann, M. (ed.), Erziehung und Schulung im Dritten THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989 5 9

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Reich, Teil 2: Hochschule, Erwachsenenbildung. Klett-Cotta, Stuttgart (1980). Mehrtens, H., Das "Dritte Reich" in der Naturwissenschaftsgeschichte: Literaturbericht und Problemskizze. In: H. Mehrtens, and S. Richter (eds.), Naturwissenschaft, Technik und NS-Ideologie. Suhrkamp, Frankfurt (1980), 15-87. Faust, A., Die Selbstgleichschaltung der deutschen Hochschulen. Z u m politischen Verhalten der Professoren und Studenten 1930-1933. In: S. Harbordt (ed.), Wissenschaft und Nationalsozialismus. Technische Universit/it, Berlin (1983). Benz, W., Emil J. G u m b e l - - D i e Karriere eines deutschen Pazifisten. In: U. Walberer (ed.), 10. Mai 1933-Biicherverbrennungen und die Folgen. Fischer Taschenbuch Verlag, Frankfurt (1982), 160-198. Pinl, M., Kollegen in dunkler Zeit. Jber. d. DMV 71 (1969), 167-228; 72 (1971), 165-189; 73 (1972), 153-208; 75 (1973/74), 166-208. Pinl, M., and L. Furtm~iller, Mathematicians under Hitler. Year Book, Leo Baeck Institute 18 (1973), pp. 129-182. Schaper, R., Mathematiker im Exil, manuscript. In: E. B6hne and W. Motzkau-Valeton (eds.), Die Kidnste und Wissenschaften im Exil. Beitr/ige zur Woche der verbrannten B~icher 1983 in Osnabriick, Vol. 1, Lambert Schneider, Heidelberg (1984). Schappacher, N., Das Mathematische Institut der Universit/it G6ttingen In: H. Becker et al. (eds.): 1929-1950. Die Universitdt G6ttingen unter dem Nationalsozialismus. K. G. Saur, M~inchen (1987), 345-373. Ferber, C. von, Die Entwicklung des Lehrk6rpers der deutschen Universitf~ten und Hochschulen 1864-1954. Vandenhoeck und Ruprecht, G6ttingen (1956). Lorenz, C., Zehnjahresstatistik des Hochschulbesuchs und der AbschluJ3pr~ifungen. Ed. by the Reichsminister f~ir Wissenschaft, Erziehung und Volksbildung, Verlag f(ir Sozialpolitik, Wirtschaft und Statistik, Berlin (1943). David Hilbert zum siebzigsten Geburtstag am 23. Januar 1932. Jber. DMV 42 (1933), 2. Abt., 67-68. Sommerfeld, A., and F. Kraul~, Otto Blumenthal zum Ged/ichtnis. Jahrbuch der Rheinisch-WestfaIischen Technischen Hochschule Aachen 4 (1951), 21-25. Neue Mathematik--Ein Vortrag von Ludwig Bieberbach. Deutsche Zukunft, 8 (April 1934), 15. Bohr, H., "Ny Matematik" i Tyskland. Berlingske Aften (1 May 1934). Hardy, G. H., The J-type and the S-type among Mathematicians. Nature 134 (1934), 250. Bieberbach, L., Pers6nlichkeitsstruktur und mathematisches Schaffen. Forschungen und Fortschritte 10 (1934), 235-237. Bieberbach, L., Pers6nlichkeitsstruktur und mathematisches Schaffen. Unterrichtsbldtter f. Math. u. Naturwiss. 40 (1934), 236-243. Lindner, Helmut, "Deutsche" und "gegentypische" M a t h e m a t i k - - Z u r Begr/~ndung einer "arteigenen Mathematik" im "Dritten Reich" durch Ludwig Bieberbach. In: H. Mehrtens and S. Richter (eds.), Naturwissenschaft, Technik und NS-Ideologie, Suhrkamp, Frankfurt (1980), 88-115. Bieberbach, L., Die Kunst des Zitierens--Ein oftener Brief an Herrn Harald Bohr in KCbenhavn, Jber. DMV 44 (1934), 2. Abt., 1-3. Mitgliederversammlung der Deutschen MathematikerVereinigung 11. bis 13. September 1934 in Bad Pyrmont. Jber. DMV 44 (1934), 2. Abt., 86-88. Siegmund-Schulze, R., Theodor Vahlen--zum Schuld-

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5.

6.

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8. 9. 10.

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14. 15. 16. 17. 18. 19. 20. 21.

22. 23.

25. 26. 27. 28.

29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

39.

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44.

anteil eines deutschen Mathematikers am faschistischen Mit~brauch der Wissenschaft. NTM-Schriftenr. Gesch. Naturwiss., Technik, Med. 21 (1984), No. 1, 17-32. Sitzung des Ausschusses am 25. Januar 1935. Jber. DMV 45 (1935), 2. Abt., 1. Gericke, H., Aus der Chronik der Deutschen Mathematiker-Vereinigung, lber. DMV 68 (1966), 46-74. Angelegenheiten der Deutschen Mathematiker-Vereinigung. Jber. DMV 29 (1920), 2. Abt., 41-53. Tobies, R., Zur Einflu~nahme des Reichsverbandes deutscher mathematischer Gesellschaften und Vereine auf die Verstarkung der angewandten Mathematik in der Ausbildung. In: Naturwissenschaften, Mathematik, Technikwissenschaften und Bildung in Geschichte und Gegenwart (Philosophie und Naturwissenschaften in Vergangenheit und Gegenwart, Heft 19), Humboldt Universit/it zu Berlin, Berlin (1980), 66-71. Lietzmann, W., Aus meinen Lebenserinnerungen. Vandenhoeck & Ruprecht, G6ttingen (1960). Erger, J., Lehrer und Nationalsozialismus. Von den traditionellen Lehrerverb/inden zum Nationalsozialistischen Lehrerbund (NSLB). In: [4], pp. 206-231. Lorey, W., Der deutsche Verein zur F6rderung des mathematischen und naturwissenschaftlichen Unterrichts e. V. 1891-1938. Verlag Otto Salle, Frankfurt (1938). Br/imer, R., and A. Kremer, Physikunterricht im Dritten Reich (Soznat Sonderband 1), AG Soznat, Marburg (1980). Hamel, G., Ueber die philosophische Stellung der Mathematik. Verlag Studentenhaus, Charlottenburg (1928). Hamel, G., Die Mathematik im Dritten Reich. Unterrichtsbldtter f. Math. u. Naturwiss. 39 (1933), 306-309. Bericht ~iber die Veranstaltungen des Mathematischen Reichsverbandes in Bad Pyrmont und in Hannover 1934. Jber. DMV 45 (1935), 2. Abt., 2-4. Dorner, A., Mathematik im Dienste der nationalpolitischen Erziehung. Diesterweg, Frankfurt (1936). Gericke, H., 50 Jahre GAMM. Ingenieur-Archiv 41 (1972), Beiheft. Tobies, R., Die "Gesellschaft fi~r angewandte Mathematik und Mechanik" im Geftige imperialistischer Wissenschaftsorganisation. NTM-Schriftenr. Gesch. Naturwiss., Technik, Med. 19 (1982), No. 1, 16-26. Tollmien, C., Das Kaiser-Wilhelm-Institut f~ir Str6mungsforschung verbunden mit der Aerodynamischen Versuchsanstalt. In: H. Becker et al. (eds.): Die Universit~t G6ttingen im Nationalsozialismus. K. G. Saur, Miinchen (1987), 464-488. Schr6der-Gudehus, B., Deutsche Wissenschaft und internationale Zusammenarbeit 1914-1928. Universit6 de Gen~ve, Genf (1966). Bracher, K.-D., Die deutsche Diktatur. Entstehung, Struktur, Folgen des Nationalsozialismus. Kiepenheuer & Witsch, K61n, 5th ed. (1976). Mason, T. W., Sozialpolitik im Dritten Reich, Westdeutscher Verlag, Opladen, 2d ed. (1978). Mehrtens, H., Angewandte Mathematik und Anwend u n g e n der Mathematik im nationalsozialistischen Deutschland. Geschichte und Gesellschaft 12 (1986), 317-347. Mehrtens, H., Ludwig Bieberbach and "Deutsche Mathematik." In: E. Phillips (ed.), Studies in the History of Mathematics. Mathematical Association of America, Providence (1987).

Jenaerstr. 6 D-IO00 Berlin 31 Federal Republic of Germany

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61

The Truth and Nothing But the Truth David Gale

In this essay I have written about a particular aspect of mathematics. The exposition, however, is deliberately informal and non-rigorous. Readers who are so inclined will find plenty of nits to pick, but if I were to proceed throughout with complete rigor, it would impede the flow of ideas, so I hope the reader will not begrudge me some limited use of poetic license. When I started writing this article, my idea was to give a rough exposition, mainly via examples, of a controversial area in the philosophy of mathematics w i t h o u t taking sides myself, but in the course of

writing I found myself reaching positions on at least some of the issues. I decided to present my conclusions in the final paragraphs. I realize that others have thought about these matters over the years, probably in much greater depth than I, and if some of them disagree with m y conclusions I know that I and perhaps other readers of this journal would enjoy reading their rejoinders.

The Meaning of Mathematical Results Mathematics is distinguished among academic disciplines by the property that it alone establishes facts with absolute certainty. Once a theorem has been correctly proved, no "on the other h a n d " is possible. Thus, for example, every positive integer is the sum of at most four squares, period. This non-controversial nature of the results of mathematics extends, it seems to me, to other aspects of mathematical culture. Mathematicians generally agree, for example, as to which results in the subject are important, e.g., the fundamental theorem of algebra, Cauchy's Theorem, the c e n t r a l limit t h e o r e m , G 6 d e l ' s i n c o m p l e t e n e s s theorem (you can probably make up your own list of the top ten theorems of all time). There is also a fair consensus within the mathematical community as to which contemporary mathematicians are doing the most significant and exciting work. It is therefore almost refreshing to find in this sea of harmonious concord one facet of the subject on which mathematicians sharply disagree. The dispute is not about the validity or profundity but rather about the meaning of mathematical results and, more specifically, how they relate to reality.

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THE MATHEMATICAL INTELLIGENCER VOL. I1, NO. 3 9 1989 Springer-Verlag New York

Let me begin with some illustrations. We know that it is impossible to tour the city of K6nigsberg in such a way as to cross each of its seven bridges exactly once, and the way we arrive at this knowledge is by reasoning that is surely mathematical. We also know that it is impossible to trisect an arbitrary angle using only ruler and compass, and the reasoning here is considerably more profound and subtle. We know that in a right triangle a 2 + b2 = c2 (this is on my top-ten list), but we don't know whether the analog of this equation has integral solutions w h e n the e x p o n e n t is greater than two. Most of us believe, however, that this is a matter of our ignorance and in reality there either do or do not exist such integral solutions. On the other h a n d , some mathematical results clearly have no counterpart in reality. It is illuminating to contrast a pair of w e l l - k n o w n d e c o m p o s i t i o n theorems. The first is the Bolyai-Gerwin Theorem, which says that if two polygons P and P' have the same area, then we can cut up P into a finite number of pieces (triangles in fact) and reassemble them to get P'. This theorem certainly has a real counterpart, say, in the construction of jigsaw puzzles. I have a computer program that allows the user to place a more or less arbitrary triangle on the face of the monitor. The machine then chops it up and reassembles the pieces to form a square of the same area. You can't get much more real than that. By contrast, the Banach-Tarski Theorem gives the same conclusion as Bolyai-Gerwin for any two balls B and B' of different v o l u m e s in 3space. I think most would agree that this theorem has no counterpart in the real world (much less an implementing computer program). What I hope to do in this essay is to explore the "reality gap" between results like Bolyai-Gerwin on the one hand and Banach-Tarski on the other, because it turns out, as I have stated, that there is wide difference of opinion on these matters. I will begin by quoting two of them. The subject, as one might have guessed, is set theory. Raymond Smullyan is lecturing on the continuum hypothesis to a group of non-experts. 1 Having explained that both the hypothesis and its negation are now known to be consistent with the standard axioms of set theory, he says, "Now, some very formalistic mathematicians say this means that the continuum hypothesis is neither true nor false. Well, I don't buy that and neither would m o s t - - m a n y - - m a t h e m a t i cians. We want to know whether the continuum hypothesis is true or not" [my italics]. His position is that our present tools, say, Zermelo-Frankel set theory, are insufficient to let us know whether or not there are cardinals between aleph null and the continuum, but he is in no doubt that either there are or there aren't.

1 From the Public Broadcasting System NOVA program, "Mathemat-

ical Mystery Tour."

Saunders Mac Lane has quite a different point of view. Writing in this journal [2] he says, "the belief of logicians in Zermelo-Frankel set theory is apparently based on a sort of Platonic imagination that there is out there in the real world some universe of sets to which their axioms will apply . . . . I submit that the Platonic notion of a real universe of sets is absolute balderdash. ,,2 Without taking sides on the issue of the continuum hypothesis (yet), I believe Smullyan is wrong on one point. It is not true that most, or even many, mathematicians believe the continuum hypothesis is either true or false. Indeed, I have yet to find a mathematician, aside from those working in set theory, who espouses this view. As one person put it, we know that we can have both Euclidean and non-Euclidean geometry. Now we see that there are different kinds of set t h e o r y - - a n d w h a t ' s w r o n g with that? On the other hand, some, though not all, of the set theorists I have talked to take the Platonic view, In my limited sample two of them believed the hypothesis was probably false, as did GOdel. In fact, toward the end of his life, G6del had come to the view that the cardinality of the continuum was probably aleph two, and there were even some writings which attempted to formulate a system that would have this as a consequence. In recent times, I am told, some of the younger set theorists believe the CH is true; there are a fair n u m b e r of undecideds, and also, even a m o n g set theorists, many who consider the question meaningless.

True, False, or Meaningless We are dealing of course with the famous law of the excluded middle. For clarity let us say that a statement is d e t e r m i n a t e if it satisfies the law of the excluded middle and indeterminate if it does not. An example of a statement I think most people would consider determinate is, "Cleopatra had blood type A." Although we will surely never know whether the statement is true or false, it is clearly one or the other. On the other hand, consider the statement, "If Gauss had never been born, someone else would have proved the fundamental theorem of algebra before the end of the nineteenth century." While it is clear what is being asserted here, it is not appropriate to ask whether the statement is true or false. The statement is rather a manner of speaking. Shall we then consider all such conditional statements to be indeterminate? What about, "If I hadn't sent you that letter you would not have received it?"

2 Absolute balderdash is presumably a strong form of balderdash by analogy with absolute and ordinary convergence. THE MATHEMATICALINTELLIGENCERVOL.11, NO. 3, 1989 63

My point is that even in ordinary language it is not always clear whether a particular statement is determinate or indeterminate. As a final illustration, consider, "The sun will continue to exist for at least 10 billion years after all life on earth has disappeared." I'm not sure I know where I stand on that one myself. Physics is such a crazy subject that perhaps the very notion of time, that is, years, is indeterminate! Getting back to mathematics and mathematicians, there seems to be a broad spectrum of beliefs on the excluded-middle issue. On the extreme right are those who reject using the law of the excluded middle in any argument, while at the left we have the Platonists, like the set theorists referred to earlier, who believe the law holds in great generality. For us "middle"-of-the-

Toward the end of his life, G6del had come to the v i e w t h a t the c a r d i n a l i t y of the continuum was probably aleph two.

roaders, things like Fermat's last theorem are either true or false. Not so for the right-wingers who feel it may well be indeterminate. On the other hand, the mainstream people believe the continuum hypothesis is indeterminate. Why? Let us examine this question in a specific context a n d start from a specific set, namely, the natural numbers N. Mainstream people believe there is such a set which has m a n y interesting properties, one of which is that in N Fermat's last theorem either is or is not true. Next we go on to the set variously denoted by p(N) or 2N, which is the set consisting of all subsets of N. The continuum hypothesis asserts that every nondenumerable subset of p(N) is equinumerous with p(N). So the question is, why does the "mathematician on the street" consider this statement to be indeterminate while Fermat's theorem is determinate? By chance I h a p p e n e d to be present on the street and overheard the following dialogue. Plato. W h y do y o u believe the c o n t i n u u m hypothesis is neither true nor false? Mathematician on the Street. Well, G6del showed that it was consistent and Cohen showed it was independent of the rest of set theory. Plato. What do you mean, "the rest of set theory?" MoS. Well, t h e r e ' s s o m e t h i n g called ZermeloFrankel set theory. It's a collection of axioms, I believe. Plato. Do you know what these axioms are? MoS. Well, no, not really. Plato. Have you worked through the proofs of G6del and Cohen? 64

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

MoS. No. That's not my area. I work in number theory. I'm trying to prove Fermat's last theorem. Plato. Oh, number theory! So you work with the natural numbers. MoS. All the time. Plato. And do you sometimes consider sequences of natural numbers? MoS. Quite often. Plato. Do you ever think about all sequences of natural numbers? MoS. Once in a while. I usually give it the product topology and denote it by Z. Plato. I see. And I take it then that this Z is quite a w e l l - d e f i n e d object in y o u r m i n d . And do y o u sometimes look at subsets of Z? MoS. Oh yes. Plato. And would it be fair to say that you feel quite comfortable working with such subsets? MoS. Sure. Some of them are open, some are closed, some are measurable, some are not. It's very interesting. Plato. And w h e n you work with these various kinds of subsets, are you always working in the framework of the Zermelo-Frankel axioms? MoS. Oh no, certainly not. As I've already said, I don't even know precisely what these axioms are, and in any case I don't think such formal axioms play a n y role in my work or that of most other working mathematicians. Plato. Well, now you have me really confused. You tell me you feel comfortable working with general

On the extreme right are those who reject using the law of the excluded middle in any argument, while at the left we have the Platonists, like the set theorists referred to earlier, who believe the law holds in great generality.

subsets of Z, yet when I ask you a straightforward question about the cardinality of such subsets, you reply that the answer is indeterminate, neither true nor false. And when I ask w h y you feel that way you tell me that you've heard via the grapevine that a couple of famous mathematicians have shown that the hypothesis can be neither proved nor disproved from a particular set of axioms--axioms that as far as you're concerned " d o n ' t play any role" in the work you are doing anyway. Isn't it possible that this is just a shortcoming of those axioms and not an indeterminacy of the facts? MoS. I seem to be getting a headache. Plato. I have no further questions.

Having heard all this and being a street person myself, I too felt some confusion. To exclude or not to exclude the middle, that was the question. I began to think about the right-wing non-excluders. Surely even t h e y m u s t agree t h a t some s t a t e m e n t s are determinate. For example, what about the statement The 1001~176decimal digit of ~r is a 7.

(1)

Now there is an algorithm, actually many, for settling this question (and in fact some of them are being actively executed at the present time; see [3]). Nevertheless it seems certain we will never know the answer to (1) because the time required to get it probably exceeds the expected duration of life on earth (if not, add on a few more zeros in the exponent). But in principle the calculation could be carried out, and I don't believe anyone has proposed a theory that maintains that whether a statement is determinate or not depends on the length of time required to verify it. Next I thought of the following statement: The decimal expansion of ~r contains a string of 100 l~176 consecutive 7's.

(2)

At this point the right-wingers might claim that the statement may be indeterminate, for while there is a finite (though lengthy) procedure that would in principle verify the statement if it is true, there is no such procedure in case the statement is false. This situation, with signs reversed, is the same as that of the Fermat problem. If Fermat's theorem is false a finite proced u r e will c o n f i r m t h i s - - n a m e l y , f i n d s u i t a b l e numbers a, b, c, and n and perform the required arithmetic. If the theorem is true, no such finite procedure exists. Then I went one step further and considered The decimal expansion of ~r contains an infinite number of 7's.

(3)

At first glance this would seem like a more tractable statement than (2) (there is very strong "empirical evidence" that the statement is true; again, see [3]). But from another point of view, statement (3) is a better candidate for indeterminacy than (2). In this case neither the truth nor the falsehood of the proposition can be verified by finite calculation, as a moment's reflection will show. (The analogous situation among the classical unsolved problems is the twin prime conjecture.) Although we will almost certainly never know if (1) is true, we may conceivably someday know the answer to (2) and (3). Indeed, there is the well-known conjecture that ~ is a normal number, meaning that all its digits to base 10 or any other base b are uniformly distributed and, more generally, every consecutive se-

The first 1,000 digits of ~r

quence of digits of length n occurs with asymptotic density b-n. If someone should prove this, then statements (2) and (3) would be obvious consequences. So let the next statement be is a normal number.

(4)

Now, I would like to speculate that (4) will never be either proved or disproved. In fact, let me suggest that perhaps there is no proof or disproof of (4) in exactly the same sense that there is no proof or disproof of the continuum hypothesis; namely (excuse the repetition), it is not provable from standard Zermelo-Frankel set theory. This may strike some people as farfetched. It's one thing, they might say, to show that a nebulous proposition about the continuum is unprovable, but when it comes to concrete questions about the density of subsequences in a well-defined sequence, one would expect that such questions ought to be resolvable by standard though possibly very involved arguments. In response to this I would point out that some THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989 6 5

statements much more " d o w n to earth" than (4) are to be u n p r o v a b l e from Z e r m e l o - F r a n k e l . Perhaps the best example is a blproduct of the work on Hilbert's tenth problem. A specific polynomial P with integer coefficients and, I believe, nine variables has the property that there is no proof or disproof (from Zermelo-Frankel) that this polynomial has a root in the natural numbers, i.e., a 9-tuple of positive integers n = (n 1. . . . . n 9 ) such that P ( n ) = 0. This problem is, in one sense at least, simpler than the Fermat problem; the Fermat equation a n + b n - c ~ = 0 is more complicated than a polynomial because variables occur as exponents. So n o w my question is this: Suppose someone should actually prove that (4) could not be proved or disproved from Zermelo-Frankel. 3 Would you then conclude that it was neither true nor false? If your answer is yes, then let me ask the same question about (3). Surely, I thought, my friend MoS believes that the expansion of "~ contains either a finite or an infinite number of 7's. Would he change his mind if someone showed that the statement was unprovable? I decided to ask him this when I ran into him on the street a few days later. I could see he was feeling well again, h a v i n g recovered from his earlier encounter with Plato. I put the above question to him and he responded right away. " O f course, there are either a finite or infinite n u m b e r of 7's in the expansion of w regardless of whether or not someone proves it to be unprovable. This is quite a different story from the continuum hypothesis. As what's-his-name said, God gave us the natural numbers. All the rest, like sets and cardinal numbers, are the creation of people like, in this case, Cantor and Zermelo." (When I mentioned that "~ was not a natural number he pointed out rightly that this was beside the point. What we were really talking about were algorithms that for each natural n u m b e r n produced a digit. The fact that this particular algorithm came from exp a n d i n g ~ was clearly irrelevant.) I persisted and asked whether the fact that (3) could not be proved from the Zermelo-Frankel axioms would raise any questions in his mind. He replied, with some impatience, "Axioms, schmaxioms! We know there are lots of things that can't be proved from those bloody axioms. Plato was right about that." I wasn't going to let him off the hook yet, and so I asked " A n d what if the normality of w was shown to be undecidable in Zermelo-Frankel? That involves s t a t e m e n t s about a s y m p t o t i c densities of subseknown

/

.

.

3 This possibility is admittedly highly hypothetical. The k n o w n techniques for proving undecidability do not seem appropriate for statem e n t s like (3) and (4), but one can imagine that some as yet unk n o w n m e t h o d s w o u l d give the result. 66 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989

quences and so on. Did God give us that, too, or was it human invention?" He looked at me for a moment and then said, "I'll have to get back to you on that." Unfortunately, I haven't run into him since. Anyway, here was more food for thought. Where should one draw the line? At what point does God leave off and man take over? Where is the boundary between reality and balderdash?

Reality I started experiencing some head symptoms myself at this point and decided to let the matter rest for a while, but some days later I happened to be reading Donald K n u t h ' s article, " W h a t Is a R a n d o m Sequence?" [1], and the question came up all over again. I thought b a c k t o statement (4) again. It appears that satisfies conditions much stronger than merely being normal. For example, if one looks at the digits dl,d 4. . . . . dn2 . . . . they too seem to be uniformly distributed. No doubt the same is true for the subsequence (dn3). We have a lot of data by now (at last count we knew more than 16 million digits). This suggests that we should strengthen the definition and talk about supernormal numbers as those that are not only normal but such that all "d~scribable" subsequences should also be normal. (Obviously we cannot require that all subsequences be normal because then given any number one could choose the subsequence to be those terms where the digit was 7, and there would be no supernormal numbers.) Fortunately, the word describable, as probably most mathematicians and all computer scientists know by now, is capable of being given a precise definition. The correct technical term is recursive. One should think of such sequences as those for which there is an algorithm for calculating each term. The crucial fact is that there are only countably many such sequences because there are only countably many algorithms. Let us then call a number supernormal if all of its discribable subsequences are normal. One thing is immediately clear. The number Tr cannot possibly be supernormal because obviously the places where the expansion of ~r has the digit 7 is a describable subsequence (I just described it) and the corresponding number, .7 7 7 . . . . is highly abnormal. Intuitively, we think of a supernormal number in base 2 as one obtained by repeated tosses of a fair coin. It would surely be surprising, would it not, if in 16 million tosses the coin fell heads whenever the corresponding decimal digit of "rr was a 7? The point is that supernormal numbers, if they exist, must be "indescribable." But they do exist. In fact, the set of n o n - s u p e r n o r m a l numbers is a set of m e a s u r e zero! F u r t h e r , the set of s u p e r n o r m a l numbers is a nice set, an F~a to be exact. The proof of

this result, which is essentially due to E. Borel, is not difficult and involves nothing more profound than a form of the law of large numbers. I find this a very strange result, not because of the mathematics, but because of what it seems to mean. If we think of flipping the true coin repeatedly, the sequence we get will almost surely correspond to a supernormal number and hence, by definition, a sequence that is impossible to describe. The mathematics is clear but the question is, in the words of Mac Lane, do such sequences "exist in the real world" or are they just figments of our mathematical imagination? I don't know what Plato would say at this point, but I have reached some conclusions of my own. I believe the continuum hypothesis is neither true nor false, but not because of the results of G6del and Cohen, though these results strongly reinforce my

Up to the present the set of all subsets of a set has not led to any contradictions, but my thesis is that it marks the point at which mathematics loses contact with reality. conviction and I would not have arrived at it without them. To explain my position, let us go back to the specific context of the natural numbers N and the set p(N) of all subsets of N--repeat, all subsets of N. That "all" is the joker. In "reality" none of us has ever seen an infinite set. We've seen plenty of finite sets, and we certainly know what we mean by all subsets of such a set. Most mathematicians also think they know what they mean by the set of all subsets of any set, finite or infinite. Even my friend MoS is among them with his set Z. But wait! When we see the word all in connection with set theory, a red light should go on in our minds. Cantor believed it was acceptable to think of the set of all sets, and we know the kind of difficulty this led to. Up to the present the set of all subsets of a set has not led to any contradictions, but my thesis is that it marks the point at which mathematics loses contact with reality. The left-wing set theorists, and this included the great G6del, believe there exists in reality a specific set that is the set of all subsets of the natural numbers (and some of them believe a lot more). Well, then, either they know something I don't know or they're kidding themselves, extrapolating from something that is clear and verifiable in the finite case to the infinite case where, in my opir~ion, the concept is probably meaningless, just as meaningless as asking what would have happened if Gauss had never been born. This is not to say that the concept of p(S) does not play an important part in many branches of mathematics. I'm sure I've used it many times. All I'm saying is that

at that point we're talking mathematics, not reality. So for me the reason the continuum hypothesis is indeterminate is because the sets involved are indeterminate. As to supernormal numbers, again we must start with the set of all real numbers (or all sequences of digits), throw away a bunch of them, and consider those that are left. Because the set of all real numbers makes me nervous, I certainly don't place much stock in the existence of this weird subset. Looking at it the other way, we are talking about the heads-tails sequences we w o u l d get almost surely if s o m e o n e flipped a fair coin an infinite number of times; in other words, we are talking about w h a t would happen if something that can't happen happened. I find this even less acceptable than talking about what would happen if Gauss had not been born. Unfortunately, this is not the end of the story. I still have to face the problem of the infinite number of 7's in the expansion of ~. It comes down to this: What are we doing w h e n we write d o w n expressions like, "three point one four one five nine dot dot dot"? I feel quite comfortable with three, one, four, five, nine, and point. It's the "dot dot dot" that bothers me. I think I know what I mean by i t - - b u t do I? (And where did I put that aspirin bottle?) So where does all this place me in the spectrum of beliefs about the law of the excluded middle? Somewhat right of center I would say. Perhaps I should call myself a fuzzy-minded conservative. Anyway, that's the way I feel right now. Of course, all this could change by next week, especially if I should happen to run into Plato in the interim.

References [1] D. E. Knuth, The Art of Computer Programming, MA: Reading, Addison-Wesley (1969), Vol. 2. [2] S. Mac Lane, "Are we all just specialists?" Mathematical Intelligencer 8, 4 (1986), 74-75. [3] S. Wagon, "Is 7r normal?" Mathematical Intelligencer 7, 3 (1985), 65-67.

Department of Mathematics University of California Berkeley, California 94720

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

67

Significant New Titles for Mathematicians and Students Polynomials By E.J. Barbeau, Univ. of Toronto, Ontario, Canada 9 Extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis and complex variable theory; 9 Introduces many techniques and topics in the theory of equations, such as evaluation and factorization of polynomials, solution of equations, interpolation, approximation and congruences; 9 Illustrates the theory of equations through examples; 9 Tests understanding, ingenuity and skill with over 300 problems drawn from journals, contests, and examinations; 9 Includes answers to many of the exercises and solutions to all of the problems. This is a wonderful introduction to the fascinating study of polynomials! 1989/441 pp., 36 illus./Hardcover $59.00 ISBN 0-387-96919-5 Problem Books in Mathematics

Classical Fourier Transforms By K. Chandrasekharan, EidgenOssische Technische Hochschule Z~rich, Switzerland 9 Gives a thorough introduction to classical Fourier transforms in a clear and compact form. Chapter h Devoted to the L~-Theory: basic properties are proved as well as the Poisson summation formula, the central limit theorem and Wiener's general tauberian theorem. As an illustration of a Fourier transformation of a function not belonging to L~ ( ~,~) an integral due to Ramanujan is given. Chapter Ih Devoted to the L2-theory, including Plancherers theorem, Heisenberg's inequality, the Paley-Wiener theorem, Hardy's interpolation formula and two inequalities due to Bernstein. Chapter IIh Deals with Fourier-Stieltjes transforms. After the basic properties are explained, distribution functions, positive-definite functions and the uniqueness theorem of Oftord are treated. 9 Intended for undergraduate students with basic knowledge in real and complex analysis. 1989/172 pp./Softcover $29.50/ISBN 0-387-50248-3 Universitext

Continua with Microstructure By G. Capriz, Universita Pisa, Italy Proposes a new general setting for theories of bodies with microstructure when they are described within the scheme of the continuum: then, besides the usual fields of classical thermomechanics (displacement, stress, temperature, etc.) some new fields enter the picture (order parameters, microstress, etc.) Continua with Microstructure can be used in a semester course for students who have some background on the classical theory of continua as an introduction to special topics (for example: materials with voids, liquid crystals, meromorphic continua). Research students studying continuum theories of new materials will find helpful the book's appropriate framework for new developments and a link between apparently disparate themes such as the topological theory of defects, phase transitions and boojums. 1989/92 pp./Hardcover $49.00/ISBN 0-387-96886-5 Springer Tracts in Natural Philosophy, Vol. 35

Percolation By G. Grimmett, Univ. of Bristol, England 9 Presents a fresh new look at the mathematical theory of percolation; 9 Contains a definitive and coherent account of the subject in an orderly manner unrestricted to the nonspecialist; 9 Includes the shortest and neatest proofs currently known; 9 Describes the subcritical and supercritical phases in considerable detail; 9 Uses the recent proofs of the uniqueness of critical points and the infinite open cluster extensively. 1989/approx. 320 pp., 77 illus./Hardcover $49.80 ISBN 0-387-96843-1

Continuity, Integration and Fourier Theory By A.C. Zaanen, Univ. of Leiden, The Netherlands The first part of this thorough textbook is devoted to continuity properties, culminating in the theorems of Korovkin and Stone-Weierstrass. The last part consists of extensions and applications of the Fourier theory, for example, the Wilbraham-Gibbs phenomenon, the Hausdorff-Young theorem, the Poisson sum formula and the heat and wave equations. Since the Lebesque integral is indispensible for obtaining familiarity with Fourier series and Fourier transforms on a somewhat higher level, the book also contains a brief survey with complete proofs of abstract integration theory. 1989/264 pp./Softcover $39.00/ISBN 0-387-50017-0 Universitext

Modular Forms T. Miyake, Hokkaido University, Sapporo, Japan Translated from the Japanese by Y. Maeda 9 Provides the reader with the basic knowledge of elliptic modular forms necessary to understand the recent developments in number theory; 9 Gives the general theory of modular groups, modular forms and Hecke operators, with emphasis on the Hecke-Weil theory of the relation between modular forms and Dirichlet series; 9 Contains a section on the unit groups of quaternion algebras, which are seldom dealt with in books; 9 Includes the so-called Eichler-Selberg trace formula of Hecke operators and gives the explicit computable formula; 9 Discusses the Eisenstein series with parameter following the recent work of Shimura. i989/336 pp, 11 illus./Hardcover $73.00 ISBN 0-387-50268-8

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Digital Geometry* Azriel Rosenfeld and Robert A. Melter

digitization. We shall define digitization here only for

1. Images and their digitizations Digital geometry is the study of geometric properties of sets of lattice points produced by digitizing regions or curves in the plane. In particular, natural concepts of paths, arcs and curves, arcwise connectedness, and distance can be defined for lattice points. One can also characterize sets of lattice points that are the digitizations of planar regions or curves having given (conventional) properties, such as convexity or straightness. This paper introduces the basic concepts of digital geometry, surveys the work done on the subject over the past 20 years, and discusses open questions and generalizations (e.g., to 3-space). Computers are used extensively to process and analyze images. In order to do this, the image must first be converted into an array of numbers suitable for computer input. The process of conversion is called

"black-and-white" images; digitization can also be defined for images having varying shades of gray, or even for color images. Formally, a black-and-white image is a function defined over a rectangular region R of the plane, say 0 x ~ m, 0 ~ y ~ n, and taking on the values 0 and 1 only. Thus a black-and-white image is the characteristic function of a subset S of R. The region R contains the lattice points {(i,j)10 ~ i m, 0 ~ j ~ n}, where i,j are integers. We digitize S by associating with it a set (S/of these lattice points. This must be done carefully if we want (S) to be, in some

* T h e s u p p o r t of t h e N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t DCR-86-03723 is gratefully a c k n o w l e d g e d , as is t h e help of Barbara Bull in p r e p a r i n g this paper.

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3 9 1989Springer-Verlag New York 69

natural sense, an "approximation" to S (so that we can obtain useful information about S by computer analysis of (S)). To illustrate the need for care in defining (S), we consider three poor definitions before giving a good one. a) A bad definition. It might seem natural to simply define (S) as the set of lattice points contained in S. Then even a "large" S (e.g., all of R except the lattice points!) could have an empty digitization, so that (S) would in no sense "approximate" S. b) A sloppy definition. A better idea is to define (S) as the set of lattice points that lie at distance n o t greater than 5 from S. (Recall that the distance of a point from S is the infimum of its Euclidean distances from the points in S.) For completeness, we define ((~) = (~. This is closer to what we want, but doesn't quite do the trick. In particular, let S consist of a single point P; then (S) m a y consist of 0, 1, or 2 lattice points, d e p e n d i n g on the coordinates of P. 1 I For example, if P is (i,5), then (P) is empty; if P is (0,5), then (P) contains both (0,0) and (0,1). c) A less sloppy definition. We can do s o m e w h a t better if we use " c h e s s b o a r d distance" instead of Euclidean distance. The chessboard distance between two points (a,b), (c,d) is max ([a - crib - d[). If we define (S) as the set of lattice points t h a t lie at chessboard distance not greater than 5 from S, then (S) can never be e m p t y if S is n o n e m p t y . Indeed let P ~ S have coordinates (i + c~, j + [3), where 0 ~ i < m, 0 ~ j < n (i,j integers) and 0 ~ o~ < 1, 0 ~ ~ < 1. Then (S) D (P) contains at least one of the four lattice points (i,j), (i + 1,j), (i,j + 1), (i + 1,j + 1), which ones d e p e n d s on the sizes of oLand ~ relative to 5" Note that (P) still consists of two lattice points if oL = 5 or [3 = 5 (but not both), and can even consist of four lattice points if oL = [3 - 5" d) An acceptable definition. To fix the problem with definition (c), we n e e d only introduce some m e t h o d of handling the cases where one or both of the coordinates of P is 5" In fact, all we need do is define (P) -= ((i + oL,j + ~)) to be the lattice point whose coordinates are obtained by rounding the n u m b e r s i + a n d j + ~ in some standard way. From n o w on, w h e n oL = 5 or [3 = 5, we shall round d o w n w a r d . Evidently, this insures that for a n y point P, (P) consists of exactly one lattice point. We can n o w define the digitization of any S as (S) -= UpEs (P), where (P) -= ((i + oL, j + [3)) is the lattice point obtained by r o u n d i n g i + o~ and j + [3 to the nearest integers, or r o u n d i n g t h e m d o w n w a r d y f ~ = 5 or Digitization clearly commutes with union: ( U i S i ) =

Ui(Si). On the other h a n d , for intersection we have only (AiSi) C Ni(S i) (in fact, disjoint sets can have the same digitization); a n d there is no simple relationship b e t w e e n the digitization of a set and that of its complement. 70

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

2. "Topological" properties of digital images The " n a t u r a l " topology on the lattice points is the discrete topology in which every lattice point is an open set; in fact, this is the only nontrivial topology if we impose the natural requirement that open sets go into open sets u n d e r (integer) translation. This topology, unfortunately, is not very interesting; for example, the concept of connectedness becomes trivial in this topology, because a set containing more than one lattice point cannot be connected. In this section we show that it is in fact possible to define nontrivial "topological" concepts for sets of lattice points by introducing a natural digital concept of a path. This will allow us to define pathwise connectedness for sets of lattice points and to s h o w that this concept has a natural relationship to arcwise connectedness in the plane.

2.1. Neighbors and paths A lattice point (i,j) has four horizontal and vertical neighbors, n a m e l y (i _+ 1,j) and (i,j +_ 1). We call t h e m the 4-neighbors of (i,j). In addition, (i,j) has four diagonal neighbors, namely (i + 1,j ___ 1 ) a n d (i - 1,j + 1); t h e 4 - n e i g h b o r s , t o g e t h e r w i t h the d i a g o n a l neighbors, are called the 8-neighbors of (i,j). Let P,Q be lattice points. A path from P to Q is a sequence of lattice points P ~ P0, P1 . . . . . Pr = Q such that Pi is a neighbor of Pi-1, 1 ~ i ~ r. Note that this is t w o d e f i n i t i o n s in o n e , d e p e n d i n g o n w h e t h e r " n e i g h b o r " m e a n s "4-neighbor" or "8-neighbor"; we call these two types of path 4-path and 8-path.

2.2. Pathwise connectedness Let L be a n o n e m p t y set of lattice points. We say that lattice points P, Q are (pathwise) connected in L if there exists a p a t h from P to Q consisting entirely of points of L. Again, this is two definitions in one 4-connected a n d 8-connected. Evidently, " c o n n e c t e d in L" is an equivalence relation. The equivalence classes defined by this relation are called the (connected) components of L. If L has only one component, it is called connected. Clearly, a n y L can be the digitization of a non-connected subset of R. On the other hand, it is not true that an arbitrary L can be the digitization of a connected subset of R; for example, {(i,j),(i + 2,j)} cannot be. (Why?) In fact, it is not hard to see that if S is arcwise connected, then (S) is 8-connected. This result establishes a relationship between arcwise connectedness in the plane a n d 8-connectedness for sets of lattice points. We will soon see w h y we also introduced the concept of 4-connectedness.

2.3. Holes Let L be a n o n - e m p t y set of lattice points that does not meet the border of our rectangle R, i.e., such that 0 ~ i

< m a n d 0 < j < n for all(i,j) EL. Thus the set of lattice points on the b o r d e r of R (~ the lattice points (i,j) such that i = 0 or m,j = 0 or n) is contained in the c o m p l e m e n t R\L of L (i.e., the set of lattice points of R not belonging to L). Evidently this set of b o r d e r lattice points is 4-connected; h e n c e it is contained in a 4-comp o n e n t of R\ L. This 4 - c o m p o n e n t is called the background of L. O t h e r 4 - c o m p o n e n t s of R\L, if any, are called holes in L. If L is 8-connected and has no holes, it is called simply 8-connected. W h y do we use 8-connectedness for L and 4-conn e c t e d n e s s for R\ L? Consider the four lattice points P1 Q1 Q2

P2

w h e r e the P's are in L and the Q's in R\ L. Intuitively, if the P's are r e g a r d e d as connected, the Q's should not be. In fact, using opposite types of c o n n e c t e d n e s s for L and R\L allows us to prove various intuitively reasonable results that w o u l d otherwise be false, as w e shall see in the r e m a i n d e r of this section.

2.4. Arcs L is called a (simple) arc if it is connected, and all but two of its points (the " e n d p o i n t s " ) have exactly two neighbors in L, while those two have exactly one. It is easily seen that a simple arc can be regarded as a path t h a t n e i t h e r crosses n o r " t o u c h e s " itself; in o t h e r w o r d s , the points of an arc can be arranged in a seq u e n c e Q1 QF, beginning at one e n d p o i n t and e n d i n g at the other, such that Qi is a neighbor of Qj iff i = j _+ 1. To rule out d e g e n e r a t e cases, we a s s u m e that an arc always has at least two points. Up to n o w we have not specified w h e t h e r the connectedness, neighbors, etc., in the definition of an arc are of the 4- or 8-type. We can m a k e either choice, p r o v i d e d we do it consistently, but we must be careful to use opposite definitions for L and its c o m p l e m e n t . For e x a m p l e , it is n o t h a r d to s h o w that an arc is s i m p l y c o n n e c t e d , i.e., t h a t an 8-arc has n o holes w h e n we use 4-connectedness for R\ L, and a 4-arc has no holes w h e n we use 8-connectedness for R\ L. If we try to use 4-connectedness for both L and R\ L, on the o t h e r h a n d , we can easily construct examples of arcs that have holes, e.g., .

.

.

.

P

P P

P c a n n o t be the digitization of an arc. (Why?) It is easily seen that a n y digital arc is the digitization of an arc, but note that the digitization of an arc n e e d not be a digital arc, because an arc can double back on itself and give rise to a "thick" digitization. It is more interesting to characterize the sets of lattice points that can be digitizations of straight line segments, but w e will not give the details here.

2.5. Curves L is called a (simple closed) curve if it is connected, and each of its points has exactly two neighbors in L. The points of such an L can be n u m b e r e d Q1 . . . . . Qr in such a w a y that Qi is a neighbor of Qj iff i ~- j _+ 1 (modulo r). To avoid degeneracies, we assume that an 8-curve always has at least four points, and a 4-curve at least eight points. If we use opposite types of c o n n e c t e d n e s s for L and R\ L, it can be p r o v e d that a curve has exactly one hole; this is the J o r d a n C u r v e T h e o r e m for digital curves. O n the other hand, if w e try to use the same type of c o n n e c t e d n e s s for both L and R\ L we get into trouble. For example, the 4-curve

PPP P PP PP P PPP

.

has two holes if we use 4-connectedness for R\ L, and the 8ocurve P P

P P

has no holes if w e use 8-connectedness for R\ L.

3. Digital convexity A n y set L of lattice points can be the digitization of a non-convex subset S of the plane, but it is not true that an arbitrary L can be the digitization of a convex S; for example,

PPP P P

PPP P P PP

cannot. If L is the digitization of a convex set, we call it is a 4-arc but has a hole. It is not true that an arbitrary set L of lattice points can be the digitization of an arc. By Section 2.2, any such L must be 8-connected. Moreover, if we delete any lattice point from L, w h a t remains must have at most two 8-components; for example,

digitally convex. It can be s h o w n that if L is digitally convex, it has the following two properties (which are equivalent to each other): (x) If P, Q, R are collinear lattice points with Q bet w e e n P a n d R, and P, R are in L, t h e n Q is in L. THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989 71

~) The convex hull of L contains no lattice point of R\L. The converse is not true; for example, P PP PPPP PP P has properties (oL,~), but it cannot be the digitization of a convex set. (Why?)

4. Extensions This article has introduced some of the basic ideas of digital geometry in the plane; many other topics could also have been discussed. In the following paragraphs we briefly touch on a few of the major omitted topics. A set of lattice points L can be "thinned" by deleting points from it in such a way that the components of L become smaller, and those of R\ L become larger, but components are never destroyed, split, or merged. Thinning is used in digital image processing to simplify digital objects. The trick is to give "local" characterizations of the points that can safely be deleted, so they can be quickly identified by examining their neighborhoods. It is of interest to characterize sets of lattice points that are digitizations of sets S having shapes of specific t y p e s - - e . g . , disks, triangles . . . . - - a n d to infer properties of S by analyzing (S). For example, if L is the digitization of an arc, what can we say about the length of the arc? If L is the digitization of a straight line segment, what can we say about the slope of the segment? Digital geometry becomes significantly more complicated when we move from two dimensions to three. In recent years m a n y computational methods have been developed for reconstructing solid volumes from sets of two-dimensional data, e.g., from x-rays. The result of such a reconstruction is a three-dimensional digital image. This provides a motivation for studying the geometric properties of sets of lattice points in 3space. A 3D lattice point P has three types of neighbors, one, two, or all three of whose coordinates differ by 1 from those of P. This gives rise to three types of connectedness, and it is not immediately clear which types are safe to use for a set of lattice points and its complement. The topology of 3D sets is also quite a bit more complicated than that of 2D sets; in addition to the components of L and of its complement ("cavities" in L), we must also take into account "holes" in L in the sense that a torus has a hole. It is not obvious how to define simple surfaces and simple closed surfaces for 3D sets of lattice points; they are much harder to characterize than simple arcs and closed curves. Dig72

THE MATHEMATICAL INTELLIGENCER VOL. n , NO. 3, 1989

ital space arcs and curves are easy to define; but no one has attempted to study digital knots! The concept of digital convexity generalizes more or less straightforwardly from 2D to 3D. One can also characterize sets of 3D lattice points that are the digitizations of arcs, surfaces, straight line s e g m e n t s , planes, or shapes of specific types.

5. What's up: a brief guide to the literature The topics introduced in this paper first appeared in the computer science and pattern recognition literature during the late 1960s and early 1970s. Some early references on digital topology are [1-3]; on metrics, [4]; on convexity, [5-7]; and on straightness, [8]. Tutorial introductions to much of this material can be found in [9-10]. The 3D case was occasionally considered during this period, but it began to receive serious attention about the beginning of the 1980s. A survey of digital topology, containing over 80 references and treating both the 2D and 3D cases, can be found in [11]. Currently some 10 to 20 papers a year appear on digital topology, distance, convexity, straightness, and related topics, most of them appearing in the pattern-recognition, image-processing, and computer vision literature. A bibliography covering much of this literature appears annually in the journal Computer Vision, Graphics, and Image Processing.

References 1. A. Rosenfeld, Connectivity in digital pictures, J. Assoc. Comp. Mach. 17 (1970), 146-160. 2. A. Rosenfeld, Arcs and curves in digital pictures, J. Assoc. Comp. Mach. 20 (1973), 81-87. 3. A. Rosenfeld, Adjacency in digital pictures, Information Control 26 (1974), 24-33. 4. A. Rosenfeld and J. L. Pfaltz, Distance functions on digital pictures, Pattern Recognition 1 (1968), 33-61. 5. J. Sklansky, Recognition of convex blobs, Pattern Recognition 2 (1970), 3-10. 6. L. Hodes, Discrete approximation of continuous convex blobs, SIAM J. Appl. Math. 19 (1970), 477-485. 7. J. Sklansky, Measuring concavity on a rectangular mosaic, IEEE Trans. Computers 21 (1972), 1355-1364. 8. A. Rosenfeld, Digital straight line segments, IEEE Trans. Computers 23 (1974), 1264-1269. 9. A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), 621-630. 10. C. E. Kim and A. Rosenfeld, How a digital computer can tell whether a line is straight, Amer. Math. Monthly 89 (1982), 230-238. 11. T. Y. Kong and A. Rosenfeld, Digital topology: introduction and survey, Computer Vision, Graphics, and Image Processing, submitted. Azriel Rosenfeld Center for Automation Research University of Maryland College Park, MD 20742 USA

Robert A. Melter Department of Mathematics Long Island University Southampton, NY 11968 USA

Chandler Davis*

"'A

very large slice of pi"**

Pi and the A G M . A study in analytic number theory and computational complexity by Jonathan M. Borwein and Peter B. Borwein Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons. N e w York, 1987. R e v i e w e d by John Todd

1. The authors This is an old English book by two young Canadians, both born in Scotland, w h o got their first degrees at the University of Western Ontario and who presently have appointments at Dalhousie University in Nova Scotia. They are erudite, ingenious, and industrious. Their father, to w h o m the book is dedicated, got his doctorate from L. S. Bosanquet in London. The ideal reviewer for this book would have been G. N. Watson [1886-1965]9

On the other hand, the Arithmetic Geometric Mean (AGM), discussed in the first chapter, has been with us only since Legendre, Lagrange, and Gauss. It should be in the repertoire of every mathematician and teacher of mathematics. AGM. If 0 <~ bo <<-ao are given and if a, + 1 = 1/2(an + then the sequences b,), b, + 1 = X/(a, b,), n = 0, 1, 2 . . . . . {a,}, {b,} converge quadratically to a common limit M = M(a o, bo), the A G M of ao, bo. [At each stage the positive square root is to be taken, otherwise the story is quite different: see e.g., Gauss [10], H. Geppert [11], D. A. Cox [8], and the reviewer [19].] An obvious question was to find a formula for M(a0, b0). Gauss as a teenager had done much computation, e.g., of M(1, X/2), and noted in his diary (30 May 1799) the coincidence to eleven decimals of M(1, X/2) and (2/,tr) .f~dt/X/(1 - ta) and that the proof of the identity of these quantities would surely open a new field of analysis. On 23 December 1799 he established the equality, and events to this day are confirming his prophecy. Several proofs are available. One method uses "invariant integrals." In fact, for n = 0, 1, 2 . . . . m_1

2. The titles

0=/2(a2cos20 + ~sin20) ~d0

Pi has been long with us. The last chapter contains a history of the computation of "rr and discusses the transcendence of ~r and the rate of approximation of -rr by rationals.

F~I2

= J0

9

1

(a2+lCOS2q~ + b~n+lsln2q~)-~dq)'

(1)

so that, passing to the limit,

* Column editor's address: Mathematics Department, University of Toronto, Toronto, Ontario M5S 1A1 Canada

M = "rr[2f0"v/2(a2cos20 + ~sin20)-~d0] -z.

**This is the title of the third leader in the Los Angeles Times of 16 February 1986, referring to the computation to 29,360,000 decimal places by David H. Bailey [1].

Transformations due to Landen or to Gauss established (1); for instance:

THEMATHEMATICALINTELLIGENCERVOL.11, NO. 3 9 1989Springer-VerlagNew York 73

sin 0 = 2 ao/[(a o + bo)csc q~ + (a0 - b0)sin q~]. This m e t h o d has been exploited by B. C. Carlson [6] in the case of other means. A second proof uses the O constants to parametrize the AGM algorithm. If a 0 = 02(q), b 0 = 02(q), then a 1 = O2(q2), b 1 = O2(q2). (2)

Here 03(q) = 1 + 2q + 2q4 + 2q9 + . . . . ~4(q) = 1 - 2q + 2q4 - 2q9 + . . . . where q = e ~'i" with-r having a positive imaginary part. Clearly q2, ~ 0 and M(O2(q), 0](q)) = 1. The basic result (2) d e p e n d s on identities that are combinatorial in character. To complete the proof, given a 0, b0, we m u s t establish the existence of a q satisfying (ao/bo) = [02(q)/O2(q)]. For a more leisurely account of the first proof with computer programs a n d numerical examples we refer to Harley Flanders [9]. For later use we record the expressions for the complete elliptic integrals K, E. -~/2

K = K(k) = Jo (1 - k2sin20)-89

where

= M(1,k'),

k '2 = 1 - k2.

(3)

/-~/2

E = E(k) = Jo (1 - k2sin20)89

It is of interest to have a brief account* of this paper in E n g l i s h ; we f o l l o w H a l p h e n ' s t r e a t m e n t . The problem is one in a t t r a c t i o n s - - a subject which seems to have disappeared from mathematical programs, alt h o u g h an excellent source of applications of the calculus. [I h a d an u n d e r g r a d u a t e course in the subject in Belfast in 1931 but I do not think this work was included.] The conjecture referred to was that the secular perturbation of Venus on Mercury is essentially as if the mass of Venus were smeared over its orbit so that the mass on an element ds of the orbit becomes proportional to the time spent describing it, i.e., to ds/(ds/dt) = dt; however, according to Kepler the areal velocity do./dt is constant and so dt is proportional to do.. Thus the attraction on Mercury is i n d e p e n d e n t of the position of Venus, and accepting this, we have to integrate to obtain it. Let the planes ~, v of the orbits of Mercury a n d Venus intersect in a line, X. Take rectangular axes at M, a point on the orbit of Mercury, let V(x, y, z) be a point on the orbit of Venus distant p = X/(x 2 + y2 + z 2) from M, a n d let V' (x + dx, y + dy, z + dz) be a neighboring point. Let the coordinates of the Sun be (x0, Y0, z0) a n d let h denote the perpendicular distance from M to the plane v. The v o l u m e of the tetrahedron M V V' S can be obtained from a determinant formula or from the formula: v o l u m e = 1/3 (area of base) x h. Equating these gives, for the area of the base, do-:

oo

= (1 - ~-~2"-1c2)0 K,

w h e r e c 2 = a2 - ~ , a 0 = 1, b0 = k'.

(4)

The only actual publication by Gauss on the AGM was in 1818 a n d dealt with secular (i.e., non-periodic) perturbations of planets, in particular that of Venus on Mercury. In some respect the work was i n c o m p l e t e w earlier, according to F. Klein [15] a n d K a u f m a n n Bfihler [5], Gauss h a d written in pencil on his work s h e e t "'Der Tod ist mir lieber als ein solches Leben" ( " D e a t h is preferable to me t h a n such a life"). His w o r k d e p e n d s on w h a t he called an elegant theorem, w h i c h he said, if it has not already been completely s t a t e d , could easily be p r o v e d by e l e m e n t a r y ast r o n o m y . A formal proof was given in 1882 by G. W. Hill [14], w h o also p u r s u e d the problem numerically, carrying out the numerical integration that was left u n d o n e by Gauss. Gauss's original paper was in Latin: an account of it, a n d an alternative treatment, was given in French by H a l p h e n [12] in 1888, a n d a translation into German by Geppert [11] was published in 1927. Actually, Geppert [11, p. 194] gives a brief account of Bessel's proof of the conjecture in his 1821 review of Gauss's paper. 74 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989

do. = (1/2h) [xo(ydz - zdy) + yo(ZdX - x & ) + z0 ( x @ - y & ) ] .

The c o m p o n e n t s of the attraction of the element V V' of the ring are proportional to x/p, y/p, z/p times do./p 2. To integrate we choose the axes so that the equation of the cone s u b t e n d e d at M by the orbit of Venus is (in Gauss's notation)

- ( x 2 / G ') - (y2/G") + (z2/G) = 0; t h i s c o n e c a n be r e p r e s e n t e d , p a r a m e t r i c a l l y as x/X/G ' cos T = y/X/G" sin T = z / V G , z arbitrary. Take e.g., the y-component. Using s y m m e t r y it reduces to a multiple of fo~' - sin 2 TdT (G + G' cos 2 T + G" sin 2 T)~ ' which is also a multiple of

* A n y o n e to w h o m this material a p p e a l s will e n j o y J. E. Littlew o o d ' s e s s a y o n " T h e discovery of N e p t u n e " a n d others in his Miscellany [16].

f0Ir/2

-- sin 2 TdT

(1 - k2 sin 2 T)~

for a certain k. But this integral is just 2 d K / d ( k 2) = (E - K(1 - k2))/k2(1 - k2) in the standard notations. We have already n o t e d ((3), (4)) that E, K can be comp u t e d rapidly by the AGM and we are therefore in a position to integrate the equations of motion of M numerically.

3. T h e s u b t i t l e s

H a v i n g introduced "rr and the AGM we n o w discuss their conjunction. Up to 1976 computations of ~ were carried out essentially by arctangent relations of the f o r m 1/4 ~r = 4 a r c t a n (1/5) - a r c t a n (1/239) a n d Gregory's series for arctan x. However, breakthroughs were then m a d e by Brent [4] and Salamin [18]. We can begin with a result of Euler. AB=

1

where A =

So

(1 - t4)-Wdt, B =

So

t2(1 - t 4 ) - ~ d t

(5)

are the lemniscate constants. This can be proved using F-functions or in a m a n n e r similar to the e l e m e n t a r y derivation of Wallis's formula. Taking a 0 = X/2, b0 = 1 in the A G M , we find A = (2/~r) M ( X / 2 , 1): the other lemniscate constant B can also be c o m p u t e d rapidly a n d the results combined to give:

Jonathan M. Borwein, David Borwein, Peter B. Borwein (left to right).

As a b y p r o d u c t of his work on ~r, Brent [ 4] s h o w e d h o w to c o m p u t e various elementary functions rapidly. An essential ingredient of his recipes is the quadratic convergence of the N e w t o n process, especially for the inversion of a function. We have introduced the b - f u n c t i o n s - - i t has been suggested that they were introduced to analyse the A G M - - a n d from them it is natural to digress to analytic n u m b e r t h e o r y a n d to discuss d r a m a t i c happ e n i n g s a n d various aspects of R a m a n u j a n ' s work, e.g., the Rogers-Ramanujan identities (cf. H a r d y [13]). In addition it is natural to s t u d y m e a n s in general: are there m e a n s other t h a n the AGM with exciting properties? 4. T h e i n t e r i o r

9r =(M(X/2,1)) 2 [1 - E2"-l(a 2 - b2)] -1,

(6)

which enabled existing records to be easily b r o k e n - about 106 decimals were t h e n obtained. In [3] the authors m e n t i o n their quartically convergent algorithm: Let Y0 = X/2 - 1 and s 0 = 2 y02. For n = 0, 1, 2 . . . . compute Y,+I = [1 - ~,(1 - y.4)]/[1 + ~/(1 - y4)] and oL,+1 = [(1 + y,+l) 4 c~,] - 22"+3 Y,+I (1 + Yn+l + Y.+12) 9 Then c~15-1 agrees with ~r for more t h a n 2 x 109 decimals. At present this is only Gedankenrechnen. In actual fact Bailey [1] has carried out (and checked) such a computation to more t h a n 29 x 106 decimals a n d reports that Kanada has got more than 134 x 106 decimals. The subject of efficient computation, i.e., computational complexity, in the context of automatic computers can be traced back to the work of Ostrowski on H o m e r ' s Method in 1951, to Strassen (1969) a n d Winograd a n d Pan for matrix multiplication, to various authors on the fast Fourier Transforms, a n d to Ninomiya [17] for optimal m e t h o d s for square roots.

W h a t we have discussed so far is the basis of the book. It is not feasible to reveal the treasures of the authors a n d others which fill its interior, but it is possible to e n u m e r a t e some of the concepts that occur. Even this requires some notation. From the m o d u l u s k (usually a s s u m e d to satisfy 0 ~ k ~ 1) a n d its complement k' (where k '2 = 1 - k2) we get the quarter periods (of the Jacobian elliptic functions) (cf.(3)) K(k) a n d K' = K(k'). The notations q = e -~K'/K and ,r = i K ' / K are also used (in the restricted case 0 < q < 1 a n d ,r is a pure imaginary). As usual j('r) denotes the elliptic modular function. Since K ' / K steadily decreases from ~ to 0 as k increases from 0 to 1, corresponding to a n y k and to any positive integer n there is a unique e = s for which the corresponding L, L' satisfy: K'/K = nL'/L

(7)

The moduli k, (~ are connected by a m o d u l a r equation of d e g r e e n. For instance the basic relation (1) gives, with a change in notation. THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3, 1989 75

L = 1/2(1 + k')K, L' = (1 + k')K', n = 2,

2' = 2X/k'/(1 + k'), e = (1 - k')/(1 + k').

(8)

In the classical (hand) computations of m o d u l a r equations of various types extremely large n u m b e r s turned u p - - n o w a d a y s c o m p u t e r s e n a b l e p r o g r e s s to be made. The transformation (7) above applies for any k. Particularly interesting are the cases w h e n e = (~(k) = k' so that K'/K = L/L' = X/n. Such k are called singular moduli. W h e n n = 1, the lemniscate case, we have: k = k' = 1/V'2, e = 1/X/2, K' = K =

(r(1))2/(4~89

"r = i,j('r) = 123 . W h e n n = 2, the L a n d e n case, we have: k = X/2 - 1, f = V(2(V2 - 1)), K' = X/2 K, K = (X/2 + 1)}

can be integrated to the form A(z, w) = 0, A algebraic. The authors have not neglected applications. For ins t a n c e , s u m s over lattice p o i n t s arise in crystallography a n d elliptic transformations have been u s e d at least since P. P. Ewald's work in 1921.

5. The

review

The book is clearly printed a n d I have noticed very few typographical slips. The reviewer in the Monthly [94 (1987), 908] writes "Suitable for sharp undergradu a t e s . " The a u t h o r s say that is it accessible to a n y graduate student; although this should be true a n d m u c h n e c e s s a r y b a c k g r o u n d material is s k e t c h e d , those having some familiarity with, e.g., Whittaker & Watson a n d H a r d y & Wright (to keep to English references) will h a v e an easier time. A s e m i n a r built a r o u n d this book could educate real mathematicians. A n y w a y , go get a copy, play with the exercises, pass on material to y o u r classes, a n d hope that the authors will write a sequel.

= %/2i,j(T) = 203. References

W h e n n = 3, the Legendre case, we have: k = 88

- 1), e = 88

K' = V 3 K, K =

+ 1),

3v'[r(89

T = %/3 i,j('r) = 16" 153. These matters are related to what is called the complex multiplication of elliptic functions. Whereas rep e a t i n g the a d d i t i o n f o r m u l a for elliptic f u n c t i o n s always leads to real multiplication of the a r g u m e n t (as in the multiple angle formula of trigonometry) %(nz) = A(%(z))(A algebraic), in certain e x c e p t i o n a l cases ( w h e n I~ is a quadratic surd) we have complex multiplication %(t~z) = A(%z). According to Olga Taussky, Hilbert said, in introducing Fueters's Presidential Address to the 1932 International Mathematical Congress at Ziirich, that complex multiplication was not only the m o s t b e a u t i f u l p a r t of m a t h e m a t i c s b u t of all science. For an introductory account see H a r v e y C o h n

[7]. Naturally the approximation of "rr attracted Ramanujan a n d one of his earlier papers s h o w e d h o w singular moduli could be u s e d for this purpose. The authors develop this theme. See also the writing of B. C. Berndt [2] and his collaborators. This subject is also related to the discussion of the circumstances in which a differential equation of the form. dz [(1 - z2)(1 - k2z2)]~ 76

M(k,e)dw [(I

-

wa)(l

-

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

e2w2)lW

1. David H. Bailey, The computation of ~ to 29,360,000 decimal places using Borweins' quartically convergent algorithm. Math. Comp. 50 (1988), 283-296. 2. B. C. Berndt, Ramanujan's Notebook, 3 vols. Part I. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag (1985). 3. Jonathan M. Borwein and Peter B. Borwein, Ramanujan and Pi, Scientific American, February 1988, 112-117. 4. R. P. Brent, Fast multiple-precision evaluation of elementary functions, J. ACM 23 (1976), 242-251. 5. W. Kaufmann-Biihler, Gauss, A Biographical Study, Berlin, Heidelberg, New York: Springer-Verlag (1981). 6. B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), 496-505. 7. Harvey Cohn, Introductory remarks on complex multiplication, Internat. J. Math. & Math. Sci. 5 (1982), 675690. 8. D. A. Cox, The arithmetic geometric means of Gauss, L'Enseign. Math. (2) 30 (1984), 275-330. 9. Harley Flanders, Algorithm of the bi-month: Computing ~t, College Math. J. 18 (1987), 230-235. 10. C. F. Gauss, Werke, 12 vols. 1863-1933, Reprint, Hildesheim: Olms (1981). 11. H. Geppert, ed. Ostwalds Klassiker, #225, C. F. Gauss, Anziehung eines elliptischen Ringes, Leipzig: Akad. Verlag, (1927). 12. G. H. Halphen, Trait6 des fonctions elliptiques et de leurs applications, 3 vols. Paris: Gauthier-Villars (188691). 13. G. H. Hardy, Ramanujan, Cambridge, (1940). 14. G. W. Hill, On Gauss's method of computing secular perturbations, with an application to the action of Venus on Mercury, 1882. CollectedMathematical Papers, Memoir 37, vol. 2, 1-46. 15. F. Klein, Vorlesungen iiber die Entwicklung der Mathematik im 19. Jahrhundert, 2 vols., Berlin: Springer-Verlag, (1926). 16. J. E. Littlewood, Littlewood's Miscellany, ed. B. Bollob~is, Cambridge, (1986). 17. I. Ninomiya, Best rational starting approximations and

improved Newton iteration for square roots, Math. Comp. 24 (1970), 391-404. 18. E. Salamin, Computation of ~ using Arithmetic-T-Geometric Mean, Math. Comp. 30 (1976), 565-570. 19. John Todd, The many limits of mixed means, I, II, ISNM#41 (1978), 5-22 and Numer. Math. to appear.

Department of Mathematics Caltech Pasadena, CA 91125, USA

Das mittelalterliche Zahlenkampfspiel by Arno Borst Heidelberg: Carl Winter, 553 pp., 1986

Rhythmomachia by Detlef Illmer Munich: Hugendubel, 96 pp., 1987

Reviewed by Benno Artmann A r o u n d the y e a r 1408 the f o l l o w i n g v e r s e s w e r e written by John Lydgate, Benedictine m o n k a n d a pupil of Chaucer (I quote from Borst, p. 239): The play he kan of Ryghtmathye, Which dulle wittis doth encombre, For thys play stant al by noumbre, And hath al his conclusions Chefly in proporsions By so sotil ordynaunce, As hyt ys put in remembraunce By thise Philosophurs olde. The "Battle of N u m b e r s , " or "Rithmomachia" was a fashionable game in the English intellectual circles of Lydgate's time. After 1500 it was more and more forgotten, until some historians of the nineteenth century sporadically rediscovered a n d described it. In 1911 David Eugene Smith and Clara C. Eaton presented the game in the American Mathematical Monthly in the following words: When the subject of number games shall be adequately treated, and the long and interesting story is told of how the world has learned to handle the smaller numbers quite as much through play as through commerce, the climax will probably be found in the chapter relating to the Battle of Numbers, the Rithmomachia of the Middle Ages. For here was a tournament worthy of intellectual foes, a play that outranked chess as much as chess surpasses mere dicing, and a game that was by its very nature closed to all save selected minds that had been trained in the Boethian arithmetic, the Latinized Nicomachus, the last great effort in the Pythagorean philosophy of numbers. But when this story comes to be told the one who relates it will have no easy task, and the object of this paper is rather to set forth the problem than to solve it. The goal set by Smith and Eaton has finally been attained by Arno Borst, Professor of History at the University of Konstanz, in the first book u n d e r review. In the second half of his book Borst gives a crit-

ical edition of the relevant sources for the formation of the game in the century 1030-1130 (all in Latin, naturally). The first part of the book contains a detailed a n d t h o r o u g h c o m m e n t a r y , p u t t i n g the game in its proper context of the intellectual, religious, and political life of the monasteries and clerical schools where it was taught and played. Detlef Illmer learned to play the game from Borst and decided to write a popular description, giving a set of simplified rules and practical hints for the actual game, complete with instructions about h o w to manufacture y o u r o w n set. He also includes some information about the number-theoretic b a c k g r o u n d of the game. Let us first have a look at the board, the pieces, and the rules. The game is played on a board that consists of two boards for chess or checkers p u t together. Each of the two players c o m m a n d s a " t e a m of n u m b e r s , " the "even t e a m " (white pieces) a n d the " o d d team." The names of the teams come from the generation of the members according to the following formula (Borst p. 63): From the even numbers k = 2, 4, 6, 8 we get the rest of the even team by calculating k2, k (k + 1), (k + 1)2, (k + 1)(2k + 1), (2k + 1)2, a n d one additional number, the so called " p y r a m i d " 91 = 62 + 52 + 42 + 32 + 22 + 12. The odd team is calculated similarly, starting from k = 3, 5, 7, 9 a n d the pyramid 190 = 82 + 72 + 62 + 52 + 42. Hence the "even team" contains some o d d n u m b e r s and vice versa. The pieces m o v e according to their shape, somew h a t similar to the pieces in chess; the detailed rules are given by Illmer a n d by Smith a n d Eaton. The players move their pieces in turn as in chess. The capture of pieces is effected in this way: If a smaller number, multiplied by the n u m b e r of vacant spaces between itself and a larger one, equals the larger one, it m a y take it. Corresponding rules use division, addition, and subtraction. (If you actually w a n t to play the game, y o u h a d better k n o w y o u r multiplication table!) Victory is not achieved by taking pieces, but by establishing an arithmetical, geometrical, or harmonic sequence of three pieces in the opponent's half of the board. T h u s the e v e n t e a m wins, for example, by placing 16, 20, 25 (in an evenly spaced row!) in the " o d d " territory. An arithmetic sequence leads to a "minimal victory," a geometric sequence to a " m e d i a n victory," a n d a harmonic sequence like 9, 15, 45 (or a, b, c with c:a = (c - b):(b - a)) to a "maximal victory" (Borst, p. 451). Fixed rules are w h a t everybody expects of a game today. Not so in the Middle Ages: The manuscripts are mostly vague and no two of t h e m agree in their outlines of the game. The inventor of the game seems to have been a cleric around 1030 n a m e d Asilo of Wfirzburg. The next person to write about the game is Herm a n n the Lame around 1040 from the Abbey of Reichenau at Lake Konstanz. He in turn is followed by an THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3, 1989

77

a n o n y m o u s writer from L/ittich (Liege in Belgium), which was "the Athens of the North." (The mathematicians from Lfittich were the first ones to actually calculate with arabic numerals; Borst, p. 99.) Fortolf of Bamberg in 1130 gave a comprehensive version of the rules and even composed some tunes based on the musical harmonies (= proportions) of the Rithmomachia (Borst, pp. 466-469). The basic motive of Asilo seems to have been the intention to familiarize his contemporaries with the Arithmetica of Boethius by way of the game, and this had been accomplished surprisingly well by the time of Fortolf (Borst, p. 208/209). With Fortolf the early phase of the development is concluded and from his treatise originates the game's success over all of Western Europe. The Rithmomachia gained a popularity parallel with chess, which came from the Orient and was played by the knights (who could not read or write in these times, let alone do arithmetic), w h e r e a s the i n t e l l e c t u a l s - - t h a t is, clerics and m o n k s - - p l a y e d Rithmomachia. The general picture drawn by Borst shows a rapid development of intellectual life. New ideas travelled swiftly from monastery to monastery, communication was good and ideas from antiquity were rediscovered and assimilated. One of the antique treasures was the basis of Rithmomachia: The Arithmetica by Boethius, the so called "last Roman," written about 500 A.D., h a d been s t u d i e d from the time of C h a r l e m a g n e (about 800). The generation of numbers in each team, for which I gave the formulas above, follows easily from Boethius's classification of different types of proportions. Boethius's Arithmetica was a translation (into Latin) and minor modification of Nicomachus's Arithmetica, written in Greek around 100 A.D., which in turn is based on Pythagorean ideas. The arithmetical foundations are presented by both Borst and Illmer in detail, and both of them are led to the natural question: Is the Rithmomachia a truly medieval game or did it come somehow from Greek sources? Circumstantial evidence points very much in the Greek direction, as already observed by Smith and Eaton: The name, which appears in a variety of forms, points to a Greek origin, the more so because Greek was little known at the time when the game first appears in literature. Based as it is upon the Greek theory of numbers, appearing as it does with a Greek name, necessarily a game known to but few and one that would naturally pass from the elite to the elite, attracting no attention from the populace, it is easy to feel that the origin of the game is to be sought in the Greek civilization, and perhaps in the later schools of Byzantium or Alexandria. In spite of extended searches by Borst, no direct evidence for the game in antique writings or other sources showed up. Apart from the general feeling, there are only hypotheses. Illmer elaborates similarities between the game and Roman (and Greek) military tactics. Some resemblances can be gathered from 78

THE MATHEMATICAL

INTELLIGENCER

V O L . I 1 , N O . 3, 1 9 8 9

',ma

" I!

Jim'v vlVl i

.VVi@ @O VV__ O0O@ L,

|

,

n

,,~

I i I

,

| i

i

i

i

A sixteenth-century Rithmomachia board (see [6]).

Euclid and Plato, which I will discuss below. The numbers in each team are generated from k, k + 1, and 2k + 1, using certain products. Here k(k + 1) is the old Pythagorean "heteromekes" (special rectangular arrangement of dots), and 2k + 1 is a "duplex super-particular" of Boethius and Nicomachus, which plays a prominent role in a quite different context: The standard proportion for the columns around a classical Greek temple in Doric style is (2n + 1):n, most frequently 13 columns on the long side and 6 on the small one (Gruben, p. 43, with numerous examples). The whole design of the Parthenon in Athens is governed by the ratio 9:4, which is 32:22, but also (2 • 4 + 1):4. The second form seems to have been more im-

portant to the architect: The columns are 17:8, not 18:8 = 9:4. Let us n o w have a look into Euclid's arithmetical books. These, presenting proper mathematics in the sense of today, are quite different from the writings of Nicomachus, w h o describes this and that sort of number, lists many kinds of proportions, but gives no proofs at all. What might be relevant from Euclid is his interest in the following question: What pairs a,c of (natural) numbers have a (natural number as) geometric mean? He first gives the easy case r2:rs = rs:s2 (in VIII, 11) and proceeds to the complete solution in VIII 18/20 in terms of "similar rectangular numbers," i.e., again products represented by rectangular arrangements of dots. This question is mathematically interesting enough so that it needs no outside motivation, especially w h e n seen in contrast to the same problem in geometry, but still, its answer would give a complete list of winning configurations for the "median victory" of Rithmomachia. (In the Middle Ages the players had lists and needed no theorem; the same lists are presented by Illmer.) In the philosophical dialogues of Plato we find numerous allusions to a certain boardgame, which he links with arithmetics. On several occasions he lists it as a preliminary stage of arithmetics, e.g., Statesman

Did you know - that the number of abstracts and reviews published each year by the Zentralblatt fiir Mathematik is greater than 50.000? -

that the Zentralhlatt fiir Mathematik has more than 6.000 distin-

299e: "The art of the boardgame and the whole art of arithmetics with pure n u m b e r s . . . " (similarly Gorgias 450d, Phaedros 174d). It is not very different from mathematics (Laws 820c). It is difficult to learn (State 333b, 374c) and only a few in a thousand can play it well (Statesman 292e). The players put their pieces on suitable positions on the board (Laws 903d). This all sounds very much like a description of the Rithmomachia. On two occasions, however, Plato mentions special features of his game which are different from what we know about Rithmomachia: The pieces in the boardgame move away from "the holy line" (Laws 739a), and the better player wins by enclosing the pieces of his opponent, so that he does not know h o w to move any more (State 487bc). Hence the situation with respect to the origins of the Rithmomachia remains inconclusive, and Smith and Eaton's second question is still open. Arno Borst believes that a cooperation between mathematicians and classical scholars might uncover some surprises, but for the moment we have the typical case of a missing link between Plato of Athens and Asilo of Wi~rzburg. References

1. Boethius, Boethian Number Theory (A translation of 'De Institutione Arithmetica' by Michael Masi), Amsterdam: Rodopi (1983). 2. Euclid, The Elements (translated by Sir Thomas L. Heath), New York: Dover Reprint (1956). 3. Gottfried Gruben, Die Tempel der Griechen Darmstadt: Wissenschaftliche Buchgesellschaft (1984). 4. Nicomachus of Gerasa, Introduction to Arithmetic (translated by M. L. D'Ooge), (F. E. Robbins and L. C. Karpinski, ed.), New York: (1926). 5. Plato, Opera (J. Burnet, ed.), Oxford: (1900-1907). 6. David Eugene Smith and Clara C. Eaton, Rithmomachia, the great medieval number game, The American Mathematical Monthly 18 (April 1911), 73-80. Fachbereich Mathematik SchloJ3gartenstraJ3e 7 D-6100 Darmstadt, FederalRepublic of Germany

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l

THE MATHEMATICAL

INTELLIGENCER

V O L . 11, N O . 3, 1989

79

by Robin Wilson*

Mathematics in the Low Countries Simon Stevin (1548-1620) was a highly versatile Flemish inventor, engineer, a n d m a t h e m a t i c i a n . As q u a r t e r m a s t e r general of t h e D u t c h a r m i e s , well k n o w n for his w o r k s on military engineering, he inv e n t e d a s y s t e m of dike sluices for d e f e n s e purp o s e s . He w r o t e on the duties of a citizen and invented a land yacht for carrying twenty-eight people along the seashore. He also helped to refute Aristotle's ideas that heavy bodies fall faster than light ones by dropping two lead spheres of different weights from a height and timing them. The first to advocate decimal weights and measures, his De Thiende (The Tenth) was influential in establishing the use of decimal fractions for everyday mathematics. His two important treatises on statics and hydrostatics contained the first explicit use of the triangle of forces. In pure mathematics, he showed how to find greatest common divisors, gave the modern definition of a polynomial, and translated works of Diophantus. This Belgian stamp was issued for the Anti-Tuberculosis Fund in 1942. Johan de Witt (1625-1672) was a talented mathematician and political leader. He studied at the University of Leiden with Frans van Schooten the Younger, whose commentary on Descartes' La gdomdtrie was to have such a profound influence on Isaac Newton. Although he had little time to devote to mathematics, he

produced the important Elementa curvarum linearum, one of the earliest systematic treatments of analytic geometry, which appeared in Schooten's commentary. Appointed grand pensionary of Holland in 1653, he became one of the foremost European statesmen of the seventeenth century, guiding the affairs of the United Provinces through a critical period of Dutch history. He was m u r d e r e d by a hostile mob. This Dutch stamp was issued for the Cultural and Social Relief Fund in 1947.

Christiaan Huygens (16291695) w a s the f o r e m o s t Dutch mathematician, astronomer, and physicist of the s e v e n t e e n t h century. B o r n in The H a g u e , he s t u d i e d in L e i d e n a n d Breda. He lived in Paris from 1665-1681 and was a f o u n d i n g m e m b e r of the French Acad6mie des Sciences. His contributions to the sciences were many and varied. He expounded the wave theory of light, invented the pendulum clock and the spiral watch spring, and discovered the shape of the rings of Saturn. In mathematics, he made several original contributions to classical mechanics, developed the theory of evolutes of curves such as the cycloid and parabola, and helped to formalize the theory of continued fractions. He also wrote the first formal treatise on probability, introducing the concept of expectation. This Dutch stamp was issued for Child Welfare in 1928.

* Column editor's address: Faculty of M a t h e m a t i c s , The O p e n University, Milton K e y n e s MK7 6AA E n g l a n d

80 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 3 9 1989Springer-VerlagNew York