No title

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.

~.The Apurba Chandra Datta Collection Reading about William Marshall Bullit and his amazing mathematical collection (Mathematical InteUigencer, vol. 11, no. 4 [1989]), I am encouraged to write briefly about the Apurba Chandra Datta Collection in the Dhaka University Library, Dhaka, Bangladesh. The many valuable (and quite a few invaluable) titles in this remarkable collection, which includes collected works and tables, were donated to Dhaka (former spelling: Dacca) University (founded 1920) in 1962 by the heirs of Apurba Chandra Datta. As a member of the Indian Education Service, Mr. Datta had served as Professor of Mathematics at several colleges in British India in the early years of this century. From entries made by Mr. Datta on these books we know that he was a resident student of Emmanuel College, Cambridge, England, while reading for the Mathematical Tripos in 1891-92. He became a wrangler, that is, obtained a first class. Most of the items in the collection were acquired during his stay in Cambridge. Many of these are of great historical interest, and some are quite rare. Evidently, astronomy was his favourite subject, for books on astronomy are the pride of the collection. Probably the most valuable and rare item in it are the Tabulae Rudolphinae, compiled by Kepler on the basis of earlier tables prepared by Tycho Brahe, published in 1627. Other titles of interest include The Elements of Physical and Geometrical Astronomy by David Gregory ("to which is appended Dr. Halley's Synopsis of the Astronomy of Comets"; 2 volumes, 1726); Opere di Galileo Galilei divise in quattro tomi (Padua, 1744); Samuel Horsley's edition of works of Newton (Isaaci Newtoni Opera Quae Exstant Omnia, 5 v o l u m e s , 1779-85); M. Bailley's Histoire de L'Astronomie Ancienne, depuis son origine jusqu'a l'6tablissement de l'6cole d'Alexandrie (1771), Histoire de l'Astronomie Moderne, depuis la fondation de l'6cole d'Alexandrie, jusqu'a l'6poque de M.D.:CC.XXX (3 volumes, 1785), Histoire de l'Astronomie indienne et orientale, ouvrage qui peut servir de suite d l'Histoire de l'Astronomie ancienne (1787); Gauss's Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (1809); Bessel's Tabulae Regiomontanae (1830); M. G. de Pontecoulant's Theorie Analytique du Syst~me du Monde (3 volumes, 4

1829-34); U-J. LeVerrier's D~veloppements sur plusieurs points de la Thdorie des Perturbations des Plan~tes (1841), Recherches sur les Mouvements de la Plan~te Herschel (1846), Recherches Astronomiques (6 volumes, 1855-61). The collection also contains the abridgement, in 18 volumes (by Charles Hutton et al.), with notes and biographic illustrations, of The Philosophical Transactions of the Royal Society of London, from their comm e n c e m e n t in 1665 to the year 1800, published in 1809. A complete list of the titles comprising the collections is available from the undersigned.

M. R. Chowdhury Department of Mathematics Dhaka University Dhaka-l O00, Bangladesh

9First Statistical Laboratory. Wacfaw Szymar~ski states that Jerzy Neyman founded the first statistical laboratory in the United States (Who Was Otto Nikodym?, Mathematical Intelligencer, vol. 12, no. 2 [Spring, 1990] pp. 27-31). From my (admittedly limited) readings in the history of statistics this appears to be in error. Neyman joined the faculty at the University of California, Berkeley in August 1938 and set up the statistical laboratory there in late 1938 or early 1939. Initially it was staffed by only Neyman and one other person, Sarah Hallam, who worked as a part-time research assistant and secretary (see Constance Reid's Neymanfrom Life, pp. 160-166). Yet R. L. Anderson writes that after his first year at Iowa State he held an assistantship in the Statistical Laboratory, and Anderson entered Iowa State as a graduate student in 1936 (see The Making of Statisticians, ed. J. Gani, pp. 131-132). This fits in with what I have picked up from talking to statisticians, namely that the Iowa State Statistical Laboratory predates the one Neyman started in California by several years. Henry Heatherly Department of Mathematics The University of Southwestern Louisiana Lafayette, LA 70504 USA

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4 9 1990 Springer-Vefiag New York

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.

Calculus Reform Murray H. Protter Calculus reform is in the air. The U.S. government is pouring millions of dollars into it. There have been calculus conferences with attendant reports, and a few universities have initiated experimental programs. Most mathematicians involved in teaching calculus have long recognized the great difficulties in making significant changes in the way calculus is taught. Conferences on the teaching of high school and beginning college courses in mathematics have been going on for years, but they have had little if any impact on what actually gets taught. A recent conference in Berkeley, supported by the National Science Foundation, came up with two empty recommendations: "Recommendation 1: It is essential to encourage and support both (1) sustained and serious investigation, and (2) pluralism and diversity, regarding the role of technology in calculus reform." The second recommendation was similar. Obviously we are all in favor of motherhood and apple pie, but mere platitudes will not help. One of the most important problems in teaching calculus is the need to develop a new set of textbooks that would be used on a wide scale throughout the country. The new texts must be concerned with the way students study the subject.

text and work a certain number of problems at the end of the section. That evening the student sits down to do his mathematics homework. What does he do? He goes immediately to the problems at the end of the section and tries to do the first one. On the basis of what he has heard in the lecture and his notes, he manages to do the problem. He goes on to the next one, which perhaps he can also do. However, the solution to the third one eludes him. Then he leafs through the pages of the assigned reading until he finds an illustrative example that is the same as the problem except it has different numbers. He can handle that. And so it goes. Finally, however, he

How a Student Studies Calculus The overwhelming majority of students taking calculus do not become mathematics majors. After many years of involvement in teaching calculus, I believe that the average calculus student learns the subject by the following formula: On a typical day the student attends a lecture, pays attention to what the instructor is saying and may even take notes. At the end of the hour he* gets an assignment to read a few pages in the

* I use the generic "he" to mean "'she or he." 6

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comes to a problem that really stumps him. At this point he has several options: (a) He can ask his roommate or the math major down the hall how to do it; (b) Since most of the major calculus texts have solution manuals, he can look it up there; (c) He can wait until the next meeting of class and ask the instructor or teaching assistant (TA) how to do it; (d) As a last resort he can read the material in the text and try to do the problem on the basis of the principles developed there. Is there anything we can do to alter the way a student studies so that he will digest the content of the subject before he tries to work the problems? Sometimes telling the student that he has to know definitions and theorems will get him to read those--especially the ones in color (more on that later). However, with the typical system of lectures by faculty members a n d section m e e t i n g s with TA's, I d o n ' t see any changes forthcoming. As a matter of fact, many students find that they can simply skip the lectures altogether and get all their work done by attending TA meetings only. I am sure that many calculus teachers notice the exponential decay in attendance at lectures during the term. One way to break this cycle would be to have certain sections of the calculus book discuss theory only - - w i t h no problems at the end of the section. The student would be required to learn the substance and would be quizzed on it at a later date. I have never dared to do this in any of my calculus books, since any textbook is doomed to failure if it is substantially different from the vast majority of texts. For example, Courant's two-volume calculus book was never widely used because, among other reasons, integration is presented before differentiation. Many people believe that it is easier and sounder pedagogically to teach integration first, but such beliefs have no import, since all calculus books must be similar if the authors expect wide adoptions, and all the other calculus books cover differentiation first.

Growth in the Size of Textbooks Fifty years ago the typically well-prepared freshman started his mathematics program with a semester of analytic geometry; he then took one semester of differential calculus followed by a semester of integral calculus. Those students who continued rounded out their lower division mathematics schedules with a semester of either advanced calculus or differential equations. It is interesting to note the size of textbooks at that time. A customary analytic geometry text was about 150 pages long with a page size of 6" x 9". The length of calculus books varied between 250 and 300 pages. In the 1930s, D. Griffin of Reed College was the first, or one of the first, to write a combined analytic geometry and calculus book. It totalled 260 pages.

That was the genesis of the three-semester calculus and analytic geometry course; since that time, threes e m e s t e r texts have proliferated. They have also grown enormously in size. For example, the book by George Thomas published in 1951 has 590 pages with a 6" x 9" page size, but the most popular books today have 1200 or even 1300 pages with an 8" x 10" page size. Moreover, these books are accompanied by exp l a n a t o r y m a n u a l s for the i n s t r u c t o r , s o l u t i o n manuals, computer-generated test problems for the instructor's use, and so on. What caused this expansion in size? One explanation involves the general perception that entering students are now less well prepared than they formerly were. Therefore, authors provide many more illustrative examples, with more diagrams and fuller explanations of these examples. This fits in neatly with my description above of how a typical student uses his textbook. With many more illustrative examples the student finds it easier to do the homework without reading the text. Such books are popular. Another feature of modern texts is the huge number of exercises provided. Most recent books have over six thousand problems, with additional sets of review exercises at the end of each chapter. Because it is difficult to assign more than 900 or 1000 problems over three semesters, we have a problem of overkill. Some faculty like to change the assignments from year to year, but in such a case two or three thousand problems should be more than enough. The problem is not new. Many years ago some calculus books came in two versions, with the only difference between them being that the exercises in the two books were entirely different. In any case, books could be shortened considerably if the number of exercises was cut in half. Currently the competition among textbook writers is such that each new author tries to outdo his predecessors. We see it in the ads, which boast about the enormous numbers of problems in the newest edition. A second reason for the increase in the size of texts is the fear of competition a m o n g authors. I have k n o w n of instances in which textbook committees went through several texts considered for adoption merely by seeing which texts might have omitted a topic the committee considered important, such as radius of curvature or moment of inertia. The book with the most topics wins. Potential authors learn quickly (usually from their publishers) that the more topics covered, the better. For example, in the most recent edition of my calculus book I included about one h u n d r e d pages on differential equations. I don't believe this material should be in a calculus book, but I was told that if I omitted it the book had no chance of adoption at most institutions. Differential equations deserves a course of its own. The same is true for Green's and Stokes's theorems and related topics. A person in some field other than mathematics can use THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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these theorems successfully only if he has a more sophisticated knowledge of the topic than is presented in an ordinary calculus course. Such subjects should be taught in advanced calculus. However, I also had to include Green's and Stokes's theorems because of the competition from other calculus books. We now have calculus textbooks containing many more subjects than can be covered in three semesters. Moreover, with experimental programs that use computers as an aid in teaching calculus, it is likely that the texts will become even larger. I believe we should divide calculus into a first-year program and a separate second-year course. One result that I hope will emerge from the reform movement is a careful, restricted selection of topics that we can all expect a calculus student to learn in the first year. Because most students don't need and don't want more than one year of calculus, the subject matter can easily be organized so that all of the essential topics are covered in the first two semesters. A second-year program with many alternatives should be recommended for those students who wish to continue.

Publishers M o s t mathematicians don't realize h o w significant the role of the publisher is in both the adoption and the content of calculus texts. When I wrote my first calculus book with Charles Morrey in the early 1960s, Addison-Wesley, my first publisher, paid no attention to the content of the book. However, Addison-Wesley had an excellent sales force and with their experience at getting Thomas's calculus book adopted widely, they were able to do the same with our book. In my view, a major factor in getting a text adopted widely is signing a contract with a publisher that has a large field organization. While a few adoptions may occur on the basis of a faculty member having studied a number of books carefully and then deciding that one in particular is most suitable, most adoptions are the result of close personal contact b e t w e e n the publisher's representative and the person on the faculty scheduled to teach calculus next fall. There are instances in which publishers with no field people have mailed out thousands of complimentary copies to faculty members, and yet few adoptions resulted. Then there is a glut of second-hand copies on the market as faculty m e m b e r s unload their copies to agents for used-book dealers. My experience is that a book may do extremely well with a publisher that has a large sales force and is likely to do poorly with a publisher that has no sales force, and which restricts its advertising to one or perhaps two print ads. Such books are frequently doomed to oblivion. Sometimes editors at publishing houses have good ideas about books. Some years ago one of the editors 8

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at Addison-Wesley had the idea of publishing bilingual editions of several of my books. The books were translated into Spanish and appeared with Spanish on one page and the same text in English on the facing page. Inexpensive paperbacks, they were put up for sale throughout Central and South America. Many thousands of copies were sold. Some years later a number of South American mathematics graduate students w h o came to study at Berkeley sought me out to tell me h o w much they appreciated the opportunity I gave them to learn mathematics and English at the same time. Of course, I had had absolutely nothing to do with i t - - t h e plan was thought up and executed entirely by one of Addison-Wesley's editors.

I recall a remark Peter Lax is purported to have made about his book on calculus and c o m p u t e r s (paraphrased): " I k n e w I w a s writing a book that w o u l d n ' t be popular, but I didn't realize I w o u l d be so successful at it.'" Publishers can also influence the content of calculus books. Several years ago I discussed a revision in my basic calculus b o o k with the editor of A d d i s o n Wesley. He offered to make a survey of all the popular books and suggest what changes I might make. It was at that time I learned that I should include differential equations and other topics if I wished to be competitive. But more than that, he urged me to change the order of the material. As an illustration, the editor told me that from his study of the most popular books, he deduced that the calculus of trigonometric functions should be introduced much earlier than it was in my book. I thought about it for a while and concluded that there was no earthly reason to make the change. On the other hand, I felt that even though it would be a lot of work to make all the required changes, there wouldn't be much, if any, harm done. So I did it. All told, the Addison-Wesley editor had twelve recommendations of this type. However, I was able to slip in an elementary section on the relation of probability to integration that none of the Addison-Wesley editors noticed. None of the competing texts had such a section. From the publisher's point of view color is very important. No potential calculus author can expect to succeed with a mainstream calculus book unless it is printed in several colors. (Several years ago AddisonWesley published a calculus book in black and white just at the time when color was becoming popular. It disappeared without a trace.) It is interesting that no books more advanced than calculus have ever been printed in other than black and white (at least to my knowledge). It is important, apparently, for a beginner

to read a theorem printed in blue or red but not so once he goes on to the next course. Color, of course, raises the price of books with, as far as I can see, no corresponding benefit. I think authors of books printed in several colors should petition their publishers to produce alternate editions in black and w h i t e - - a t a much reduced price, of course. It would be interesting to see whether students prefer a book in color or the same book in black and white at a much lower price.

Applications Most applications in calculus books are artificial. There is the ladder sliding down a wall, the person rowing across a stream with a uniform current, the lengthening shadow of a person walking away from a street light, and so on. We know them all. I think it is important to provide problems that students can find believable or at least useful. Such problems are hard to find and even after they make their way into textbooks, the difficulties do not end. For example, will the author provide the necessary background in the text so that the student can make the connection of the problem with an appropriate principle in calculus? Moreover, there is the TA problem with respect to applications. About a dozen years ago we started a course at Berkeley for TA's to learn how to run section meetings. This course turned out to be most useful, especially for those foreign graduate students who had little or no experience with the U.S. system of undergraduate education. Each time I taught the course I polled the students on their educational backgrounds. It turned out that more than 50% of our TA's had never had a college course in physics! Thus the entire burden of teaching applications falls on the faculty members (many of w h o m were once TA's without applied backgrounds) and on the textbooks. If the typical study habits of students are as I described them earlier, the books will provide the illustrative examples and the students will work those problems that match the examples. Nothing is gained in the attempt to teach the applications of calculus.

recall a remark Peter Lax is purported to have made about his book on calculus and computers (paraphrased): "I knew I was writing a book that wouldn't be popular, but I didn't realize I would be so successful at it."

Conclusions The calculus sequence is a juggernaut and there are strong indications that we are heading for the 1500page text. If enough mathematicians are willing to devote their energies to writing one-year texts that include standard topics decided on by a national committee and if major publishers with a large fixed focus can be found to promote such texts, there is a chance of reversing the trend toward mammoth books. However, it is important to avoid a "consensus text"; we need several competing books, each with a stamp of its author or authors. Negating the influence of publishers will be difficult, and I have no idea how to bring it about. The calculus market is enormous, and the commercial firms active in it will continue to attempt to gain an advantage any way they can. It would be helpful if all TA's were required to have at least one course in physics so that topics with applications could be discussed with authority. For example, one of the most beautiful applications of integration is that of computing areas and volumes of irr e g u l a r l y s h a p e d objects. Yet this topic is n o t considered an "application," because students are not convinced they will ever have to calculate such an area or volume. It is hard to find real applications of calculus that students can identify as important. Finally, how do we change the study habits of the average student? Or should we try? The organization of most texts works against change; unless we develop another system we will be locked into the one we have.

Mathematics Department University of California Berkeley, CA 94720 USA

Calculus and Computers At the present time, using computers as an aid in a large calculus course is very expensive. We have an experimental program at Berkeley in which some students have access to computers. Expanding this to the thousands in our regular program is not possible with the resources available. Besides, it remains to be seen whether computer use clarifies the basic concepts of calculus or actually makes it more difficult for the student to grasp what is going on. We need a lot of experimentation with new texts and much software. I THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990 9

Karen V. H. P arshall*

The One-Hundredth Anniversary of the Death of Invariant Theory? Karen V. H. Parshall In his article surveying "Fifty Years of Algebra in tificate of invariant theory effectively reads 15 FebAmerica, 1888-1938," that ever-colorful mathemati- ruary 1890. On that date, the twenty-eight-year-old cian-turned-historian, Eric Temple Bel![recounted an David Hilbert signed off on his paper, "Uber die ironic twist in the history of the theory of algebras. In Theorie der algebraischen Formen" and presented his 1907, under the auspices of no less than the Carnegie proof of the so-called finite basis theorem to the readInstitution of Washington, James Byrnie Shaw pro- ership of the Mathematische Annalen, a theorem and duced a comprehensive Synopsis of Linear Associative proof that killed an entire area. Two and a half years Algebra [1] designed both to codify the theory as then later, he completed yet another invariant-theoretic understood and to direct its future course. Unfortu- work, entitled "Uber die vollen Invariantensysteme" nately, Shaw's timing could not have been worse, for [4] and put an end to any lingering hopes of the also in 1907, Joseph H. M. Wedderburn published his theory's resurrection. Thus, after fifty years of vigpaper "On Hypercomplex Numbers" [2] and moved orous life, one of the nineteenth century's major areas the theory of algebras in directions totally unforeseen of mathematical research abruptly ceased to exist. Whereas the origins of most myths remain forever by Shaw. In Bell's inimitable w o r d s , Shaw had shrouded in the long-forgotten past, this story's in" . . . all but immortalized the subject like a perfectly preserved green beetle in a beautiful tear of fossilized ception most probably dates to the 1939 publication of amber. Had this exhaustive synopsis been the last Hermann Weyl's book The Classical Groups: Their Inwords on linear associative algebra, it would have variants and Representations. There, Weyl offered the made a noble epitaph. Instead of submitting to prema- opinion that Hilbert's two articles " . . . mark a turning ture mummification and honorific burial, however, point in the history of invariant theory. He solves the the subject insisted on getting itself reborn, or its soul main problems and thus almost kills the whole subtransmigrated immediately, in W e d d e r b u r n ' s pa- ject" [5]. Note, h o w e v e r , the use of the w o r d s "turning point" and "almost." Weyl went on to add p e r . . . " [3]. Although writing about the theory of algebras as it that the theory " . . . lingers on, however flickering, stood in 1907, Bell might just as well have been de- during the next d e c a d e s . . . " and that by the 1930s it scribing the state of invariant theory in the early 1890s. " . . . has begun to blossom again . . ." [6] thanks to cross-fertilization both from mathematical physics in F u r t h e r m o r e , the p u r p l e - t i n g e d prose n o t w i t h standing, he would have presented a much more ac- the form of relativity and quantum theory and from curate characterization of that state than that which mathematics itself in the form of group representation has become entrenched in twentieth-century mathe- theory. Thus, in Weyl's view, invariant theory took a matical folklore. In this modern legend, the death cer- crucial turn in the early 1890s due to Hilbert's work and lost its m o m e n t u m for a time as a result of its reorientation, but the theory's basic underlying notions of invariance and the possibility of its mathematical description and interpretation endured. In his * Column Editor's address: Departments of M a t h e m a t i c s a n d History, University of V i r g i n i a , C h a r l o t t e s v i l l e , V A 22903 USA. book, in fact, he aimed to redirect invariant theory 10

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along a trajectory d e t e r m i n e d by r e p r e s e n t a t i o n theory. Clearly, he felt the theory was far from dead, yet from his words the legend seemed to grow. A word omitted here, a meaning obscured there, and Hilbert slew invariant theory in 1890. As with all myths, though, this one does contain certain elements of truth. The ever-strengthening emphasis on axiomatization and algebraic structures after the advent of the twentieth century guided areas like invariant theory away from the micro-level of calculation and toward the macro-level of more abstract and general theorization. Hilbert's work in the early 1890s exemplified such a shift to some extent, but it by no means turned things around overnight. Emmy Noether, one of the key figures credited with establishing the so-called modern algebra, wrote a doctoral dissertation [7] on the invariant theory of ternary biquadratic forms in the best nineteenth-century calculational style, this in 1907 fully fifteen years after Hilbert's papers had appeared. It would still take another ten to fifteen years before her ideas and work would evolve into the abstract style for which she is now remembered. In two articles written in the late 1960s, Charles S. Fisher explained the apparent contradiction between the "death hypothesis" and Emmy Noether's career in sociological terms. He argued that invariant theory could be interpreted sociologically on two different levels: "In the first, 'the Theory of Invariants' is taken to be a social category within the world of mathematics. As a category its existence is constituted in terms of the opinions mathematicians have about the theory and the actions to which those opinions lead. 9 . . [In] the second mode of analysis, Invariant Theory [is] treated as an intellectual tradition which, if it is to be sustained, must be transmitted to future generations of researchers" [8]. By tracing the careers of the students of several of the key nineteenth-century invariant t h e o r i s t s - - m o s t notably James Joseph Sylvester and Paul G o r d a n - - F i s h e r s h o w e d that invariant theory persisted well into the 1920s. Then, he claimed, due to sociological conditions--mathematical isolation, positions at institutions which did not support research and/or graduate students, etc.--these disciples largely failed to perpetuate their invarianttheoretic heritage. Their failure resulted in the termination of the "pure" history of the theory [9]. Furthermore, Fisher contended that various mathematical constituencies crafted historical m y t h s in which invariant theory no longer enjoyed the status of an a u t o n o m o u s subdiscipline. On the one hand, mathematicians like Weyl followed the origins of their inquiries back to nineteenth-century invariant theory but subsumed the subject into a new mathematical subdiscipline, group representation theory in Weyl's case. On the other hand, the " m o d e r n algebraists" like Hilbert and later Noether wrote invariant theory

out of their history altogether. For them, modern algebra grew out of a tradition going back to Richard Dedekind and Leopold Kronecker, not to Sylvester and Gordan. Although both Hilbert and Noether worked on questions stemming from traditional computationally oriented work, the school of modern algebraists ignored this part of its past and instead traced a straight-line progression through progenitors w h o m it viewed as more abstract and so more respectable. In both cases, invariant theory, as pursued by self-proclaimed invariant theorists, had ceased to exist by the 1930s [10]. So w h e n did invariant theory die? Perhaps this question will yield to a more properly historical approach, that is, maybe an answer will emerge through a brief survey of the subject's nineteenth-century history followed by a look at Hilbert's novel results of the early 1890s and a glance at subsequent lines of invarianttheoretic research 9 Although Gauss had observed a special case of algebraic invariance, the discriminant of a binary quadratic form, in 1801, the credit for actually isolating the phenomenon as a subject worthy of independent and sustained study must go to George Boole, on the basis of his 1841 "Exposition of a General Theory of Linear Transformations" [11]. In this two-part paper, Boole tackled the general problem of determining algebraic relationships among the coefficients of homogeneous polynomials of degree n in m u n k n o w n s which remained invariant under a (nonsingular) linear transformation. To clarify matters, consider the simplest case of a binary quadratic form Q = ax2 + 2bxy + cy2. Here, Boole's elimination method involved calculating the partial derivatives oxQ and 3yQ of Q and working with the system of equations 3xQ = 0, O~Q = 0 to get what he called 0(Q). In this setting, O(Q) = b2 - ac, the discriminant of Q. Boole then applied the (nonsingular) linear transformation: x--* mx' + ny' y --* m'x' + n'y'

(where m,n,m',n' ~ R) to Q to get a new binary quadratic form R = Ax '2 + 2Bx'y' + Cy '2. Clearly, the elimination method applied to R yielded 0(R) = B2 AC, or the discriminant of R. The questions then arose: Are 0(Q) and 0(R) related, and, if so, how? Although he proved his subsequent result only in the cases of binary quadratic and binary cubic forms, Boole gave a general (and correct) answer to these two questions: 0(Q) and 0(R) were indeed related, and, in fact, they were equal up to a power of the determinant of the linear transformation above [12]. In the language Sylvester would invent only in 1851, Boole's elimination process had generated an "invariant" of the homogeneous p o l y n o m i a l o f degree n in m unTHE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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knowns (or the m-ary n-ic form), and Boole himself had launched "invariant theory" as a potent n e w mathematical area [13]. Not long after Boole's work had appeared on the pages of the C a m b r i d g e M a t h e m a t i c a l Journal, it caught the attention of the 1842 Cambridge Senior Wrangler, Arthur Cayley. By 1845, he had published his first of voluminously m a n y publications in invariant theory. In his article " O n the Theory of Linear Transformations," Cayley presented a rather gruesome calculational technique for generating invariants distinct from Boole's elimination method. In particular, relative to the binary quartic form U = oLx4 + 413x3y + 6~x2y2 + 48xy3 + ~y4, Cayley's method yielded the invariant I = o~e - 4~8 + 3~2, whereas Boole's process had generated a very complicated invariant K = 0(U). On receiving Cayley's letter of 11 N o v e m b e r 1844 announcing his curious discovery, Boole, using yet ano t h e r calculational t e c h n i q u e , g e n e r a t e d a still different invariant J = o~e - c~ 2 - ~ 2 _ ~3 + 2~/~ and noticed something even more peculiar: His original invariant K equaled/3 _ 27]2 [14]. Again, using the terminology Sylvester would coin later, this phen o m e n o n s u g g e s t e d to C a y l e y the p r o b l e m of "syzygies" or polynomial relations between invariants [15]. Based on his subsequent work as well as on Boole's observation, by 1846 Cayley had posed what would become the two motivating problems of nineteenthcentury invariant theory: 1) "[t]o find all derivatives [i.e., invariants] of any number of functions, which have the property of preserving their form unaltered after any linear transformation of the variables," and 2) to determine " . . . the independent derivatives, and the relations between these and the remaining ones" [16]. Thus, to Cayley's way of thinking, the goal was to elaborate a theory that would allow for the production of some sort of minimal set of invariants for a given form. Via a related theory of syzygies, all invariants of the given form could then be generated from this minimal set. As a result of the lifelong friendship and mathematical exchange which ensued after their meeting in 1846, James Joseph Sylvester also pursued the theory of invariants within the framework Cayley had defined. A prolific mathematician and, later in life, an influential teacher, Sylvester joined forces with Cayley from 1850 on in establishing what would come to be recognized as a British school of invariant theory. As part of its membership, this school would claim, among others, fellow Englishmen, E. B. Elliott and H. W. Turnbull; the Irish m a t h e m a t i c i a n George Salmon; and the American Fabian Franklin. Roughly concurrent with and initially independent of the growth of this British school, another rather distinct approach to invariant theory developed on the other side of the English Channel in Germany. Unlike 12

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Boole's " p u r e " invariant theory as taken over by Cayley and Sylvester, German invariant theory grew primarily out of the geometrical researches of Otto Hesse [17]. In his efforts to build a theory for studying third-order plane curves, Hesse fashioned the determinant into a tool for calculating critical points in his 1844 paper, "Uber die Elimination der Variabeln aus drei algebraischen Gleichungen von zweiten Grade mit zwei Variabeln" [18]. Presenting his new construct in general terms and studying it algebraically, Hesse took a homogeneous polynomial f of degree m in n unknowns x 1. . . . . x n and defined the functional determinant ~b = 132f/Ox,Oxjt,

1 ~ i,j ~ n.

(1)

He then observed that if ~b denoted the functional determinant as in (1) of f, the function resulting from f under the action of the linear transformation x i ~ Yi = ailx I q- ai2x 2 + " ' " + ainXn, then ~ = r2~b, where r is the d e t e r m i n a n t of that transformation. In other words, independently of the work of Boole, Hesse had discovered that (1), the so-called Hessian, satisfied the invariantive property [19]. Following his teacher's lead, Siegfried Aronhold began pursuing the invariance question apart from its geometrical interpretation in 1849. By 1858, he had not only articulated an invariant theory of ternary cubic forms but also discovered the work of the British school and adopted much of its terminology. Still, terminology aside, invariant theory in the hands of Aronhold and his followers, Alfred Clebsch and Paul Gordan, looked very different from its British embodiment. In his work between 1849 and 1863, Aronhold developed a sophisticated notation and accompanying calculus for expressing and manipulating invariants (and covariants) of general forms [20]. This so-called symbolic method characterized the German school of invariant theory and allowed its members to skim just above the level of explicit calculation in their work. Their goals were the same as those of the British, however: to calculate a complete set of covariants for an arbitrary m-ary n-ic form and to construct all other covariants from the covariants in this set. Thus, both schools focused on calculation, yet the British insisted on expressing their findings in what Tony Crilly has termed their "Cartesian form," while the Germans contented themselves with the more abstract symbolic form [21]. The extra measure of flexibility inherent in the German approach made the crucial difference. In 1868, Paul Gordan established the theoretical superiority of the symbolic m e t h o d w h e n he used it to prove, in the case of binary forms, what is now known as t h e First F u n d a m e n t a l T h e o r e m of I n v a r i a n t Theory: all covariants are explicitly constructible as polynomials in a finite number of them (with coefficients in the underlying field) [22].

Top row (left to right): Carl Friedrich Gauss, Leopold Kronecker, Emmy Noether; bottom row (left to right): Arthur Cayley, Richard Dedekind, Hermann Weyl.

In his "Second memoir upon quantics" of 1856, Cayley [23] had presented an algorithm for calculating the number of independent covariants of a given degree and order in the fundamental set of a binary form. He also thought he had shown that while the fundamental sets of the binary quadratic, cubic, and quartic forms were finite, that of the binary quintic was infinite [24]. Undaunted by this result, he and Sylvester spent the next twelve years churning out covariants and developing their theory further. With Gordan's proof of the finiteness theorem in 1868, they readjusted their theory and tried, unsuccessfully, to reprove his result using their techniques. Although Gordan's work represented a triumph of the German over the British school of invariant theory, it only settled one of the theory's key questions and then only for binary forms. The syzygy problem still remained unresolved, and little headway had been made on a general theory of forms in more than two variables. Surveying the invariant-theoretic landscape in 1892 under the auspices of the newly founded Deutsche Mathematiker-Vereinigung, Wilhelm-Franz Meyer [25] detailed the progress made on these and many other open questions in the field. As James

Byrnie Shaw would do fifteen years later for the theory of algebras, Meyer also indicated, via his analysis of the open questions, possible future directions for invariant-theoretic research. Among the work he surveyed was David Hilbert's now infamous paper of 1890, and while he recognized it as a major breakthrough in the area, he hardly saw it as the end of the story. In fact, he sketched broad vistas of projective and differential invariants which, although they might ultimately be informed by Hilbert's ideas, represented new fields ripe for inquiry. Viewing Hilbert's work of 1890 from the perspective of his contemporaries rather than from a prejudiced, twentieth-century optic, what did he achieve in it? As early as 1888, Hilbert had announced some surprising new results on the pages of the G6ttinger Nachrichten [26] that, as he pointed out, not only gave a new proof of Gordan's theorem for binary forms but also generalized it to n-ary forms. By 1890, his full exposition of these ideas and their implications for invariant theory had appeared in the Mathematische Annalen, although not without some dissension on the part of one of the referees, Paul Gordan [27]. There, Hilbert proved what is now known as Hilbert's Basis THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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Theorem, namely, "[f]rom among the forms of an arbitrary module one can always choose a finite number such that every other form in the module can be obtained from the chosen ones by linear combination" [28]. To put this in more modern terminology, if R is a field, then any ideal of the polynomial ring R[x I . . . . . xn] is finitely generated [29]. After pausing to establish the results of section one in the more number-theoretic setting of the ring of integers, Hilbert tackled the problem of syzygies in his lengthy third section. As early as 1846, Cayley had warned that this phenomenon would not yield easily to theoretical explanation and description, and history proved him correct. Although he and Sylvester as well as the German school had made some progress in this direction by 1890, the matter was far from settled until Hilbert presented his Theorem III in that same year. As a simple consequence of the Basis Theorem, Hilbert first proved that the algebraic relations holding among a finite set of invariants were finitely generated. Then, based on an argument that he himself admitted was "not easy [nicht mfihelos],'" he obtained the much deeper result that the so-called "higher syzygies" eventually vanished [30]. He continued his analysis of syzygies with the introduction of the "Hilbert polynomials" before applying the more abstract method inherent in the proof of the Basis Theorem to establish Gordan's Theorem for n-ary forms, a generalization Gordan's highly constructive t e c h n i q u e (called " U b e r s c h i e b u n g " or "transvection") could not accomplish. The paper finally closed with allusions to the ideas Hilbert would explore in the sequel of 1893 and with a sketch of possible future lines of invariant-theoretic research [31]. In particular, Hilbert called for linking the subject more closely to the work of Felix Klein and Sophus Lie. As he remarked [32]: Till now, we have defined an invariant as a homogeneous polynomial function of the coefficients of the ground-form that is invariant under all linear transformations of the variables. But now, following the more general concept, we choose a definite subgroup of the general group of linear transformations and ask for the homogeneous polynomial functions of the coefficients of the ground-forms which are invariant only under the substitutions of the chosen subgroup. Although all the invariants in the previous sense are obviously found among these new invariants, still it [does not] follow from our previous propositions on the finitude of the complete systems of invariants that one can always choose finitely many of the invariants in the extended sense such that every other invariant of the same kind is a polynomial in the chosen ones. Hilbert would reformulate this remark in 1900 as the fourteenth problem in his famous lecture before the International Congress of Mathematicians in Paris [33]. Given the influence of Hilbert's lecture on twentieth-century mathematics and given that its four14

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teenth problem had essentially been stated as early as the 1890 paper, the myth that that paper killed invariant theory would seem completely at odds with reality. As the accompanying table shows, the pursuit of solutions to this and related questions has more than kept invariant theory alive throughout the twentieth century. Also in light of this table, what do we make of Fisher's conclusions that invariant theory had died sociologically by the 1930s? In closing his article on " T h e Last I n v a r i a n t Theorists," Fisher provided one last caveat. Although the area had died sociologically, "[t]his does not mean it must necessarily remain so. The documents still exist. Present and future generations of mathematicians can read them. There is no reason why a new generation could not arise carrying forward its mathematical activities u n d e r the b a n n e r of Invariant Theory" [34]. Thus, if sociological death is only temporary, maybe the " d e a t h metaphor" falls short in characterizing the situation. While the sociological model may shed light on certain aspects of the history of invariant theory, it hardly seems unproblematic. Given the evidence of continuing work in the area since the 1890s and its resurgence since the 1960s, p e r h a p s an historical s t u d y that is i n f o r m e d by broader external issues but that also takes into account the technical internal developments of mathematics-and particularly of abstract algebra--would yield an account of the development of invariant theory free of the ultimate contradiction with which Fisher's study ends.

References 1. James Byrnie Shaw, Synopsis of Linear Associative Algebra: A Report on Its Natural Development and Results Reached up to the Present Time, Washington, D.C.: Carnegie Institution of Washington (1907). 2. Joseph H. M. Wedderburn, On hypercomplex numbers, Proceedings of the London Mathematical Society, 2d ser., 6 (1907), 77-118. 3. Eric Temple Bell, Fifty Years of Algebra in America, 1888-1938, Semicentennial Addresses of the American Mathematical Society, 2 vols., New York: American Mathematical Society, 1938; reprint ed., New York: Arno Press, 1980, 2:1-34 on p.. 30. 4. David Hilbert, Uber die Theorie der algebraischen Formen, Mathematische Annalen 36 (1890), 473-534; and "Uber die vollen Invariantensysteme," op. cit., 42 (1893), 313-373. Both of these papers have been translated by Michael Ackermann and commented upon by Robert Hermann in Hilbert's Invariant Theory Papers, Lie Groups: History, Frontiers, and Application, vol. 8, Brookline: Math Sci Press (1978). All English quotations are taken from this source. 5. Hermann Weyl, The Classical Groups: Their Invariants and Representations, Princeton: University Press (1939), 27. Of course, Hilbert also closed his 1893 paper with the remark: "With this, I believe, we have attained the most important general goals of a theory of the function fields formed by the invariants. [Ackermann, trans., p. 301.]" 6. Weyl, 28.

7. Emmy Noether, Uber die Bildung des Formensystems der tern/iren biquadratischen Formen, Journal fiir die reine und angewandte Mathematik 134 (1908), 23-90 in Emmy

Noether: Gesammelte Abhandlungen Collected Papers, (Nathan Jacobson, ed.), New York: Springer-Verlag (1983), 31-99. Given that Noether wrote her dissertation under the direction of Paul Gordan, its traditional quality should perhaps come as no surprise. Here, the term ternary biquadratic form means a homogeneous polynomial of degree four in three unknowns. 8. Charles S. Fisher, The last invariant theorists: a sociological study of the collective biographies of mathematical specialists, Archives europdennes de Sociologie 8 (1967), 216-244 on pp. 217-218.

9 . Ibid. See, especially, his conclusion on pp. 242-243. 10. Charles S. Fisher, The death of a mathematical theory: a study in the sociology of knowledge, Archive for History of Exact Sciences 3 (1966), 137-159. 11. Carl Friedrich Gauss, Disquisitiones arithmeticae (Arthur A. Clarke trans.), New Haven: Yale University Press (1966), 111-112; and George Boole, Exposition of a general theory of linear transformations, Cambridge Mathematical Journal 3 (1841-1842), 1-20, 106-119. What follows is a necessarily abbreviated and sketchy history of nineteenth-century invariant theory. For a more detailed account, see Tony Crilly, The rise of Cayley's invariant theory (1841-1862), Historia Mathematica 13 (1986), 241-254; Tony Crilly, The decline of Cayley's in-

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variant theory (1863-1895), op. cit., 15 (1988), 332-347; and Karen Hunger Parshall, Toward a history of nineteenth-century invariant theory, (David E. Rowe and John McCleary, eds.), The History of Modern Mathematics, 2 vols., Boston: Academic Press (1989), 1: 157-206. 12. Boole, p. 19. Note here that Boole only tacitly assumed the nonsingularity of his linear transformation, and all of his calculations were explicit. In what follows, I adhere to the historical presentation as much as possible, while still rendering it comprehensible to the modern reader. Thus, I do not use modern formulations and notations when they would trivialize or otherwise distort the nineteenth-century work. 13. James Joseph Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants, Philosophical Magazine 2 (1851), 391-410, or The Collected Mathematical Papers of James Joseph Sylvester, (H. F. Baker, ed.), 4 vols., Cambridge: University Press, 1904-1912; reprint ed., New York: Chelsea Publishing Co. (1973), 1, 265-283 on p. 273. (Math. Papers JJS.) 14. Arthur Cayley, On the theory of linear transformations, Cambridge Mathematical Journal 4 (1845), 193-209, or The Collected Mathematical Papers of Arthur Cayley (Arthur Cayley and A. R. Forsyth, ed.), 14 vols., Cambridge: University Press (1889-1898), 1:80-94 on pp. 93-94. (Math. Papers AC.) There is an inaccuracy on this point in Parshall, p. 164. 15. James Joseph Sylvester, On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure, Philosophical Transactions of the Royal Society of London 143 (1853), 407-548, or Math. Papers JJS 1: 429-586. See the glossary of "new and unusual terms" on pp. 580-586. 16. Arthur Cayley, On linear transformations, Cambridge and Dublin Mathematical Journal 1 (1846), 104-122, or Math. Papers AC, 1:95-112 on p. 95. Cayley's emphasis. 17. Some impetus also came from the number-theoretic work of Gotthold Eisenstein. 18. Otto Hesse, Ober die Elimination der Variabeln aus drei algebraischen Gleichungen von zweiten Grade mit zwei Variabeln, Journal fiir die reine und angewandte Mathematik 28 (1844), 68-96. 19. Ibid., 89. Note that the Hessian is an example of a "covariant," that is, an expression in the coefficients and variables of a given form which remains invariant under a linear transformation of the variables of the original form. Clearly, invariants are just special cases of covariants. 20. See, in particular, Siegfried Aronhold, Zur Theorie der homogenen Functionen dritten Grades von drei Variabeln, Journal fiir die reine und angewandte Mathematik 39 (1849), 140-159; Theorie der homogenen Functionen dritten Grades von drei Ver/inderlichen, op. cit., 55 (1858), 97-191; and Ueber eine fundamentale Begri~ndung der I n v a r i a n t e n s y s t e m e , op. cit., 62 (1863), 281-345. 21. The Cartesian form explicitly exhibits a covariant, for example, as a sum of monomials with coefficients specified. Without defining the notation any further, the symbolic form of a covariant might be (ab)~(ac)~ 999 a~b8 9 9 9 d X.

22. Paul Gordan, Beweis, dass jede Covariante und Invariante einer bin~iren Forme eine ganze Function mit numerische Coef~ienten einer endlichen Anzahl solchen Formen ist, Journal fiir die reine und angewandte Mathematik 69 (1868), 323-354. See, particularly, 341-343. Gordan's proof, in keeping with his philosophy of math16

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ematics, was constructible and not merely existential. 23. Arthur Cayley, A second memoir upon quantics, Philosophical Transactions of the Royal Society of London 146 (1856), 101-126, or Math. Papers AC, 2: 250-281. Here, "degree" refers to the degree of homogeneity in the coefficients of the original form and "order" refers to the degree of homogeneity in the variables. 24. Ibid., 269-270. 25. Wilhelm-Franz Meyer, Bericht iiber den gegenw/irtigen Stand der Invariantentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 1 (1892), 79-292. 26. David Hilbert, Zur Theorie der algebraische Gebilde I, G6ttinger Nachrichten (1888), 450-457, in David Hilbert, Gesammelte Abhandlungen, 3 vols., Berlin: Verlag von Julius Springer (1933), 2: 176-183. (Ges. Ab. DH.) He followed this up with two further notes, also in the G6ttinger Nachrichten. See Ges. Ab. DH, 2: 184-191, 192-198. 27. For the exchange of letters between Hilbert and Klein relative to Hilbert's 1890 paper and Gordan's reaction to it, see (Giinther Frei, ed.), Der Briefwechsel David HilbertFelix Klein (1886-1918), G6ttingen: Vandenhoeck & Ruprecht (1985), 61-65. On 24 February, 1890, Gordan wrote to Klein that he was "very dissatisfied [sehr unzufrieden]" with Hilbert's work because of its merely "existential" as opposed to "constructive" nature. 28. Ackermann, trans., p. 150. 29. Hilbert's proof of this theorem, which is not unlike that in most standard algebra textbooks, also goes through under the more general hypothesis of a ring R in which every ideal is finitely generated. See, for example, Nathan Jacobson, Basic Algebra II, San Francisco: W. H. Freeman & Co. (1980), 417-418. 30. Ackermann, trans., 165-183. For a more modern statement, see Weyl, 36; and Peter Hilton and Urs Stammbach, A Course in Homological Algebra, New York: Springer-Verlag (1971), 251-254. It is very important to note that, while Hilbert established the finiteness here, he did not provide any sort of a constructive method of producing syzygies. 31. Among other things in this 1893 paper, Hilbert proved his famous Nullstellensatz (in section three). (See note [4] above.) This p a p e r is generally credited with ushering in modern algebra, although as this article hopefully makes clear relative to invariant theory, these sorts of statements generally beg for deeper historical inquiry. 32. Ackermann, trans., 221-222. Hilbert's emphasis. 33. David Hilbert, Mathematical problems: lecture delivered before the International Congress of Mathematicians at Paris in 1900, Bulletin of the American Mathematical Society 8 (1902), 437-479, in Mathematical Developments Arising From Hilbert Problems (Felix Browder, ed.), 2 vols., Providence: American Mathematical Society (1976), 1-34. 34. Fisher, The last invariant theorists, 243.

The War of the Frogs and the Mice, or the Crisis of the Mathematische Annalen D. van Dalen*

Will no one rid me of this turbulent priest? Henry II On 27 October 1928, a curious telegram was delivered to L. E. J. Brouwer, a telegram that was to plunge him into a conflict that for some months threatened to split the German mathematical community. This telegram set into motion a train of events that was to lead to the end of Brouwer's involvement in the affairs of German mathematicians and indirectly to the conclusion of the Grundlagenstreit. The story of the ensuing conflict that upset the mathematical world is not a pleasant one; it tells of the foolishness of great men, of loyalty, and of tragedy. There must have been an enormous correspondence relating to the subject. Only a part of that was available to me, but I believe that enough of the significant material could be consulted so as to warrant a fairly accurate picture. The telegram was dispatched in Berlin, and it read1: Professor Brouwer, Laren N.H. Please do not undertake anything before you have talked to Carath6odory who must inform you of an unknown fact of the greatest consequence. The matter is totally different from what you might believe on the grounds of the letters received. Carath6odory is coming to Amsterdam on Monday. Erhard Schmidt.

from G6ttingen and waited for the arrival of Constantin Carath6odory. The letters were still unopened when Carath6odory arrived in Laren 2 on the thirtieth of October.

Bearer of Bad News Carath6odory's visit figures prominently in the history that is to follow. In order to appreciate the tragic quality of the following history, one must be aware that Brouwer was on friendly terms with all the actors in this small drama, with the exception of David Hilbert; some of them were even intimate friends, for example Carath6odory and Otto Blumenthal. Carath6odory found himself in the embarrassing position of being the messenger of disturbing, even offensive, news, and at the same time disagreeing with its contents. It was regrettable, he said, that the two unopened letters had been written. The first letter

A message of this kind could hardly be called reassuring. Brouwer duly collected two registered letters

* The research for this p a p e r w a s partially s u p p o r t e d b y t h e Netherl a n d s O r g a n i z a t i o n for A d v a n c e m e n t of Pure Research (Z.W.O.) u n d e r g r a n t R 60-19. 1 All t h e c o r r e s p o n d e n c e in t h e Annalen affair w a s in G e r m a n ; in the translations I h a v e exercised s o m e f r e e d o m in t h o s e cases w h e r e a literal translation w o u l d h a v e resulted in overly a w k w a r d English. 2 Brouwer lived in a small t o w n , Laren, s o m e distance from A m s t e r d a m . He also o w n e d a h o u s e in Blaricum. THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4 9 1990Springer-VerlagNew York 17

contained a statement that should have carried more signatures, or at least Blumenthal's signature. Carath6odory's name was used in a manner not in accordance with the facts, although he would not disown the letter should Brouwer open it. Finally, the sender of the letter would probably seriously deplore his action within a couple of weeks. The second letter was written by C a r a t h 6 o d o r y himself, a l t h o u g h Blumenthal's name was on the envelope. He, Carath6odory, regretted the contents of the letter. Thereupon, Brouwer handed the second letter over to Carath6odory, who proceeded to relate the theme of the letters. The contents of the second can only be guessed, but the first letter can be quoted verbatim. It was written by Hilbert, and copies were sent to the other actors in the tragedy that was about to fill the stage for almost half a year. Hilbert's letter was a short note: Dear Colleague, Because it is not possible for me to cooperate with you, given the incompatibility of our views on fundamental matters, I have asked the members of the board of managing editors of the Mathematische Annalen for the authorization, which was given to me by Blumenthal and Carath6odory, to inform you that henceforth we will forgo your cooperation in the editing of the Annalen and thus delete your name from the title page. And at the same time I thank you in the name of the editors of the Annalen for your past activities in the interest of our journal.

Respectfully yours, D. Hilbert The meeting of the old friends was painful and stormy; it broke up in confusion. Carath6odory left in d e s p o n d e n c y a n d Brouwer was dealt one of the roughest blows of his career.

The Annalen The Mathematische Annalen was the most prestigious mathematics journal at that time. It was founded in 1868 by A. Clebsch and C. Neumann. In 1920 it was taken over from the first publisher, Teubner, by Springer. For a long period the name of Felix Klein and the Mathematische Annalen were inseparable. The authority of the journal was mostly, if not exclusively, based on his mathematical fame and management abilities. The success of Klein in building up the reputation of the Annalen was largely the result of his choice of editors. The journal was run, on Klein's instigation, on a rather unusual basis; the editors formed a small exclusive society with a remarkably democratic practice. The board of editors met regularly to discuss the affairs of the journal and to talk mathematics. Klein did not use his immense status to give orders, but the editors implicitly recognized his authority. Being an editor of the Mathematische Annalen was considered a token of recognition and an honour. 18 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990

Through the close connection of Klein--and after his resignation, of Hilbert--with the Annalen, the journal was considered, sometimes fondly, sometimes less than fondly, to be " o w n e d " by the Gbttingen mathematicians. Brouwer's association with the Annalen went back to 1915 and before, and was based on his expertise in geo m e t r y and topology. In 1915 his name appeared under the heading "With the cooperation of" (Unter Mitwirkung der Herren). Brouwer was an active editor indeed; he spent a great deal of time refereeing papers in a most meticulous way. The status of the editorial board, in the sense of bylaws, was vague. The front page of the Annalen listed two groups of editors, one under the head Unter Mitwirkung yon (with cooperation of) and one under the head Gegenw~rtig herausgegeben von (at present published by). I will refer to the members of those groups as associate editors and chief editors. The contract that was concluded between the publisher, Springer, and the Herausgeber Felix Klein, David Hilbert, Albert Einstein, and Otto Blumenthal (25 February 1920) speaks of Redakteure, but does not specify any details except that Blumenthal is designated as managing editor. The loose formulation of the contract would prove to be a stumbling block in settling the conflict that was triggered by Hilbert's letter. At the time of Hilbert's letter the journal was published by David Hilbert, Albert Einstein, Otto Blumenthal, and Constantin Carath6odory, with the cooperation of (unter Mitwirkung von) L. Bieberbach, H. Bohr, L. E. J. Brouwer, R. Courant, W. v. Dyck, O. H61der, T. von Karm~n, and A. Sommerfeld. The daily affairs of the Annalen were managed by Blumenthal, but the chief authority undeniably was Hilbert.

Brouwer and Hilbert Nowadays the names of Brouwer and Hilbert are automatically associated as the chief antagonists in the most prominent conflict in the mathematical world of this century, the notorious Grundlagenstreit. But things had not always been like that; some twenty years earlier Brouwer had met Hilbert, who was nineteen years his senior, in the fashionable seaside resort Scheveningen and had instantly admired "the first mathematician of the world. ''3 Hilbert obviously recognized the 3 From a letter of Brouwer to the D u t c h poet C. S. A d a m a van Scheltema (9 N o v e m b e r 1909): "This s u m m e r the first mathematician of the world was in Scheveningen, I was already acquainted with him t h r o u g h m y work; n o w I have repeatedly walked with him, a n d talked to him as a y o u n g apostle to a prophet. He was 46, but y o u n g in heart a n d body, he s w a m vigourously and enjoyed climbing over walls and fences with barbed wire" [2, p. 100].

Constantin Carath4odory

David Hilbert

L. E. J. Brouwer

genius of the young man and on the whole accepted and respected him. Brouwer's letters to Hilbert for a prolonged period were written in a warm and friendly tone. Already in his dissertation of 1907 Brouwer was markedly critical of Hilbert's formalism; this caused, however, no observable friction, probably because the dissertation was written in Dutch and thus escaped Hilbert's attention. The relationship remained friendly for a long time; G6ttingen was Brouwer's second scientific home, and Hilbert wrote a warm letter of recommendation in 1912 when Brouwer was considered for a chair at the University of Amsterdam. In 1919 Hilbert went so far as to offer Brouwer a chair in G6ttingen, an offer that Brouwer turned down. The initially warm relationship between Hilbert and Brouwer began to cool in the twenties, when Brouwer started to campaign for his foundational views. Hilbert accepted the challenge--he took the threat of an intuitionistic revolution seriously. Brouwer lectured successfully at meetings of the German Mathematical Society. His series of Berlin lectures in 1927 caused a considerable stir; there was even some popular reference to a Putsch in mathematics. In March 1928 Brouwer gave talks of a mainly philosophical nature in Vienna (tradition has it that these talks were instrumental in Wittgenstein's return to philosophy). On the whole the future of intuitionism looked rosy. Gradually the scientific differences between the two adversaries t u r n e d into a personal animosity. The Grundlagenstreit is in part the collision of two strong characters, both convinced that they were under a personal obligation to save mathematics from destruction. Brouwer's involvement in the national affairs of the German mathematicians also played a role. In so far as

Brouwer had any political views, they could not be called sophisticated. From the end of the first world war, Brouwer had taken up the cause of the German mathematicians, subjected as they were to harsh measures and an international boycott. 4 For example, he forcefully opposed the participation of certain French mathematicians in the Riemann memorial volume of the Mathematische Annalen, much to the chagrin of Hilbert. His latest exploit in this area was his campaign against the participation of German mathematicians in the International Congress of Mathematicians at Bologna in August 1928. Hilbert put the full weight of his authority to bear on this matter, with the result that a sizable delegation followed Hilbert to Bologna [4, p. 188] .5

Hilbert's D e c i s i o n The stage was set for the final act, and the letter of dismissal was the signal to raise the curtain. It is hard to imagine what Hilbert had expected; he could not have counted on a calm, resigned acquiescence from the highly strung emotional Brouwer. In Brouwer's eyes (and quite a few colleagues would have taken the same view) a dismissal from the Annalen board was a gross insult.

4 Brouwer's views and actions in this area can easily be (and have been) misrepresented; they deserve a more detailed treatment. The matter will be covered in a forthcoming biography. s It was felt by a n u m b e r of Germans, and by Brouwer, that the G e r m a n s were tolerated only as second-rat~7"participants at the Bologna conference. Rather than suffer such an insult, they advocated a boycott of the conference. This topic has also received some degree of notoriety, and is in need of a more balanced treatment. It will find a place in the forthcoming biography. THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990 19

Carath6odory m u s t have revealed some of the underlying motive to Brouwer, w h o wrote in his letter of 2 N o v e m b e r to Blumenthal: Furthermore Carath6odory informed me that the Hauptredaktion of the Mathematische Annalen intended (and felt legally competent) to remove me from the Annalenredaktion. And only for the reason that Hilbert wished to remove me, and that the state of his health required giving in to him. Carath6odory begged me, out of compassion for Hilbert, who was in such a state that one could not hold him responsible for his behaviour, to accept this shocking injury in resignation and without resistance. Hilbert himself was explicit; in a letter of 15 October he asked Einstein for his permission (as a Mitherausgeber) to send a letter of dismissal (the draft to the chief editors did not contain any explanation) a n d a d d e d Just to forestall misunderstandings and further ado, which are totally superfluous under the present circumstances, I would like to point out that my decision--to belong under no circumstances to the same board of editors as Brouwer--is firm and unalterable9 To explain my request I would like to put forward, briefly, the following: 1. Brouwer has, in particular by means of his final circular letter to German mathematicians before Bologna, insulted me and, as I believe, the majority of German mathematicians9 2. In particular because of his strikingly hostile position via-dl-vis sympathetic foreign mathematicians, he is, in particular in the present time, unsuitable to participate in the editing of the Mathematische Annalen. 3. I would like to keep, in the spirit of the founders of the Mathematische Annalen, G6ttingen as the chief base of the Mathematische Annalen--Klein, who earlier than any of us realized the overall detrimental activity of Brouwer, would also agree with me. In a postscript he added: "I myself have for three y e a r s b e e n afflicted b y a g r a v e illness ( p e r n i c i o u s anemia); even t h o u g h the deadly sting of this disease has b e e n t a k e n b y an American i n v e n t i o n , 6 I h a v e been suffering badly from its s y m p t o m s . " Clearly, Hilbert's position was that the Herausgeber (chief editors) could appoint or dismiss the Mitarbeiter (associate editors). As such he n e e d e d the approval of Blumenthal, Carath6odory, and Einstein. Blumenthal had complied with Hilbert's wishes, b u t for Carath6odory, consent was problematic; a p p a r e n t l y he did not wish to u p s e t Hilbert b y contradicting him, but neither did he w a n t to authorize him to dismiss Brouwer. Hilbert m a y easily have mistaken Carath6odory's evasive attitude for an implicit approval. Carath6odory had landed in an a w k w a r d conflict b e t w e e n loyalty and fairness. H e obviously tried hard to reach a compromise. In view of Hilbert's firmly fixed conviction, he a c c e p t e d the u n a v o i d a b l e conclusion that B r o u w e r h a d to go; b u t at least B r o u w e r s h o u l d go w i t h honour. 6 The w o r k of G. H. W h i p p l e , F. S. Robscheit-Robbins a n d of G. R. Minot, cf. [4, p. 179].

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THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990

Einstein's

Neutrality

Being caught in the middle, Carath6odory sought Einstein's advice. In a letter of 16 October he wrote "It is m y o p i n i o n that a letter, as c o n c e i v e d by Hilbert, c a n n o t possibly be sent off." H e p r o p o s e d instead, t o send a letter to Brouwer explaining the situation and suggesting that Brouwer should voluntarily h a n d in his resignation 9 Thus a conflict w o u l d be avoided a n d one could do Brouwer's w o r k justice: " B r o u w e r is one of the foremost mathematicians of our time and of all the editors he has done most for the M.A." The second letter we m e n t i o n e d above m u s t have b e e n the concrete result of Carath6odory's plan. Eins t e i n a n s w e r e d : " I t w o u l d be b e s t to i g n o r e this Brouwer-affair. I would not have t h o u g h t that Hilbert was p r o n e to such emotional outbursts" (19 October 1928). The m a n a g i n g editor, Blumenthal, must have b e e n in an e v e n greater conflict of loyalties, being a close, personal friend of Brouwer a n d the first Ph.D. s t u d e n t (1898) of Hilbert, w h o m he revered. Einstein did not give in to Hilbert's request. In his a n s w e r to Hilbert (19 October 1928) he wrote: I consider him [Brouwer], with all due respect for his mind, a psychopath and it is my opinion that it is neither objectively justified nor appropriate to undertake anything against him. I would say: "Sire, give him the liberty of a jester (Narrenfreiheit)!'" If you cannot bring yourself to this, because his behaviour gets too much on your nerves, for God's sake do what you have to do. I, myself, for the above reasons cannot sign such a letter9 Carath6odory, however, was seriously troubled and could not let the matter rest. H e again t u r n e d to Einstein (20 October 1928): 9 . . Your opinion would be the most sensible, if the situation would not be so hopelessly muddled9 The fight over Bologna . . . seems to me a pretext for Hilbert's action. The true grounds are deeper--in part they go back for almost ten years9 Hilbert is of the opinion that after his death Brouwer will constitute a danger for the continued existence of the M.A. The worst thing is that while Hilbert imagines that he does not have much longer to l i v e . . , he concentrates all his energy on this one matter . . . . This stubbornness, which is connected with his illness, is confronted by Brouwer's unpredictability . . . . If Hilbert were in good health, one could find ways and means, but what should one do if one knows that every excitement is harmful and dangerous? Until now I got along very well with Brouwer; the picture you sketch of him seems me a bit distorted, but it would lead too far to discuss this here.

This letter m a d e Einstein, w h o in all public matters practised a high standard of moral behaviour, realize that these were d e e p waters i n d e e d (23 October 1928): I thought it was a matter of mutual quirk, not a planned action9 Now I fear to become an accomplice to a proceeding that I cannot approve of, nor justify, because my n a m e - - b y the way, totally unjustifiedly--has found its way to the title page of the Annalen . . . . My opinion, that

Brouwer has a weakness, which is wholly reminiscent of the Prozessbauern,7 is based on many isolated incidents. For the rest I not only respect him as an extra'ordinarily clear visioned mind, but also as an honest man, and a man of character. From these letters, even before the real fight h a d started, it clearly appears that Einstein was firmly res o l v e d to r e s e r v e his n e u t r a l i t y . E i n s t e i n called B r o u w e r " a n i n v o l u n t a r y p r o p o n e n t of L o m b r o s o ' s t h e o r y of the close relation b e t w e e n g e n i u s a n d ins a n i t y , " but Einstein was well a w a r e of B r o u w e r ' s greatness, and did not wish him to be victimized. It is n o t clear w h e t h e r Einstein's o p i n i o n was b a s e d o n personal observation or on hearsay; there are no reports of personal contacts b e t w e e n Brouwer and Einstein, but one m a y conjecture that they had met at one of the m a n y meetings of the Naturforscherverein or in Holland during one of Einstein's visits to Lorentz.

Unsound Mind It did not take Brouwer long to react. Brouwer was a m a n of great sensitivity, and w h e n emotionally excited he was frequently subject to nervous fits. According to one report (a letter from Dr. Irmgard G a w e h n to v o n Mises), Brouwer was ill and feverish for some days following Carath6odory's visit. O n 2 N o v e m b e r Brouwer sent letters to Blumentha| and Carath6odory, from which only the copy of the first one is in the Brouwer a r c h i v e - - i t contained a report of Carath6odory's visit. The letter stated that "in calm deliberation a decision on C a r a t h 6 o d o r y ' s request was r e a c h e d . " The a n s w e r to Carath6odory, as r e p r o d u c e d in the letter to Blumenthal, was short: Dear Colleague, After close consideration and extensive consultation I have to take the position that the request from you to me, to behave with respect to Hi[bert as to one of unsound mind, qualifies for compliance only if it should reach me in writing from Mrs. Hilbert and Hi[bert's physician. Yours L. E. J. Brouwer This solution, a l t h o u g h perhaps a clever m o v e in a political game of chess, was of course totally unacceptable-even w o r s e , it w a s a m i s j u d g m e n t of the m a t t e r . In a m o r e or less f o r m a l i n d i c t m e n t , Blumenthal declares concerning "this frightful and repulsive letter" that a p p a r e n t l y Brouwer had picked from Carath6odory's statements and e n t r e a t m e n t s the ugliest interpretation. "'I m u s t confess, a n d Cara has written me likewise, that I have been t h o r o u g h l y de7 This probably refers to the troubles in Schleswig-Holstein during roughly the same period, when farmers resisted the tax policies of the government. Hans Fallada has sketched the episode in his Bauern, Bonzen und Bomben.

a.<

Albert Einstein ceived in B r o u w e r ' s character a n d that Hilbert has k n o w n a n d j u d g e d him better than we d i d . " So Brouwer's first action only served to rob him of his potential support. The conflict had p r e s e n t e d itself so s u d d e n l y and so totally u n e x p e c t e d l y to Brouwer that he failed to realize to w h a t e x t e n t H i l b e r t saw h i m as a d e a d l y d a n g e r for mathematics, a n d as the bane of the Mathematische Annalen. His belief that the a n n o u n c e d dismissal was the w h i m of a sick a n d t e m p o r a r i l y der a n g e d m a n emerges from a letter he dispatched to Mrs. Hilbert three days later: I beg you, use your influence on your husband, so that he does not pursue what he has undertaken against me. Not because it is going to hurt him and me, but in the first place because it is wrong, and because in his heart he is too good for this. For the time being I have, of course, to defend myself, but I hope that it will be restricted to an incident within the board of editors of the Annalen, and that the outer world will not notice anything. A c o p y of this letter w e n t to C o u r a n t with a friendly note, asking him (among other things) to keep an eye on the matter: "As a matter of course, I count especially on y o u to bring Hilbert to reason, and to make sure that a scandal will be a v o i d e d " (6 N o v e m b e r 1928). C o u r a n t , a f t e r v i s i t i n g Mrs. H i l b e r t , r e p l i e d to Brouwer (10 N o v e m b e r 1928) that Hilbert was in this matter u n d e r n o b o d y ' s influence, a n d that it was impossible to exert any influence on him. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

21

Ferdinand Springer. This photograph was taken on Hilbert's sixtieth birthday, 23 January 1922. A p a r t f r o m Einstein, w h o k e p t a strict neutrality, all the editors (mostly reluctantly) did take sides--the m a j o r i t y w i t h Hilbert, b u t Hilbert h i m s e l f n o l o n g e r t o o k p a r t in t h e conflict. His p o s i t i o n w a s fixed o n c e a n d for all, a n d in v i e w of his illness t h e d e v e l o p m e n t s w e r e as far as p o s s i b l e k e p t f r o m h i m (e.g., Blum e n t h a l to C o u r a n t o n 4 N o v e m b e r 1928: " H i l b e r t m u s t n o t find o u t a b o u t C a r a ' s trip to B r o u w e r " ) . O n e m i g h t w o n d e r w h e t h e r B r o u w e r , as a relative o u t s i d e r (one of t h e t h r e e n o n - G e r m a n s a m o n g the editors), s t o o d a c h a n c e f r o m t h e b e g i n n i n g ; his letter of 2 N o v e m b e r to C a r a t h 6 o d o r y , h o w e v e r , definitely lost h i m a g o o d deal of s y m p a t h y a n d p r o v e d a w e a p o n to his o p p o n e n t s .

The R i p p l e s S p r e a d In a circular letter of 5 N o v e m b e r 1928 B r o u w e r app e a l e d d i r e c t l y to t h e p u b l i s h e r s a n d e d i t o r s , t h u s w i d e n i n g the circle of p e r s o n s i n v o l v e d : To the publisher and the editors of the Mathematische Annalen. From information communicated to me by one of the chief editors of the Mathematische Annalen at the occasion of a visit on 30-10-1928 1 gather the following: 1. That during the last years, as a consequence of differences between m y opinion and that of Hilbert, which had nothing to do with the editing of the Mathematische Annalen (my turning d o w n the offer of a chair in G6ttingen, conflict between formalism and intuitionism, difference in opinion concerning the moral position of the Bologna congress), Hilbert had developed a continuously increasing anger against me. 22 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990

2. That lately Hilbert had repeatedly announced his intention to remove me from the board of editors of the Mathematische Annalen, and this with the argument that he could no longer "cooperate" (zusammenarbeiten) with me. 3. That this argument was only a pretext, because in the editorial board of the Mathematische Annalen there has never been a cooperation between Hilbert and me (just as there has been no cooperation between me and various other editors). That I have not even exchanged any letters with Hilbert for many years and that I have only superficially talked to him (the last time in July 1926). 4. That the real grounds lie in the wish, dictated by Hilbert's anger, to harm and damage me in some way. 5. That the equal rights among the editors (repeatedly stressed by the editorial board within and outside the board t) allow a fulfillment of Hilbert's will only in so far that from the total board a majority should vote for my expulsion. That such a majority is scarcely to be thought of, since I belong among the most active members of the editorial board of the Mathematische Annalen, since no editor ever had the slightest objection against the manner in which I fulfill my editorial activities, and since my departure from the board, both for the future contents and for the future status of the Annalen, would mean a definite loss. 6. That, however, the often proclaimed equal rights, from the point of view of the chief editors, was only a mask, now to be thrown down. That as a matter of fact the chief editors wanted (and considered themselves legally competent) to take it upon themselves to remove me from the editorial board. 7. That Carath6odory and Blumenthal explain their cooperation in this undertaking by the fact that they estimate the advantages of it for Hilbert's state of health higher than my rights and honor and freedom of practice (Wirkungsm6glichkeiten) and than the moral prestige and scientific contents of the Mathematische Annalen that are to be sacrificed. I n o w appeal to your sense of chivalry and most of all to your respect for Felix Klein's memory and I beg you to act in such a way, that either the chief editors abandon this undertaking, or that the remaining editors separate themselves [from the chief editors, v.D.] and carry on the tradition of Klein in the managing of the journal by themselves. Laren, 5. November 1928 L. E. J. Brouwer

t From the editorial obituary of Felix Klein, written by Carath~odory: " H e (Klein) has taken care that the various schools of mathematics were represented in the editorial board and that the editors

operated with equal rights alongside of himself--He h a s . . , never heeded his own person, always had kept in view the goal to be achieved." From a letter from Blumenthal to me, 13-9-1927: "I believe that you overestimate the meaning of the distinction between editors in large and small print. It seems to me that we all have equal rights. In particular we can speak for the Annalenredaktion if and only if we have made sure of the approval of the editors interested in the matter under consideration.--Although I too take the distinction between the two kinds of editors more to be typographical than factual (I make an exception for myself as managing editor), I very well understand your wish for a better typographical make-up. You know that I personally warmly support it. However, we can for the time being, as long as the state of Hilbert's health is as shaky as it is now, change nothing in the editorial board. I thus cordially beg you to put aside your wish. In good time I will gladly bring it out."

The above circular letter was dispatched on the same day as Brouwer's plea to Mrs. Hilbert; the two letters are in striking contrast. One letter is written on a conciliatory note, the other is a determined defence and closes with an unmistakable incitement to mutiny. Blumenthal immediately took the matter in hand; he wrote to the publisher and the editors (16 November) to ignore the letter until he had prepared a rejoinder. The draft of the rejoinder was sent off to Courant on 12 November, with instructions to wait for Carath6odory's approval and to send copies to Bieberbach, H61der, von Dyck, Einstein, and Springer. It appears from the accompanying letter that Carath6odory had already handed in his resignation, although he had given Blumenthal permission to p o s t p o n e its announcement, so that it would not give food to the rumour that Carath6odory had turned against Hilbert. In the meantime Brouwer had travelled to Berlin to talk the matter over with Erhard Schmidt and to exp l a i n his p o s i t i o n to the p u b l i s h e r , F e r d i n a n d Springer. Brouwer, a c c o m p a n i e d by Bieberbach, called at the Berlin office of Springer, who reported the discussion in an Aktennotiz " U n a n n o u n c e d and surprising visit of Professor Bieberbach and Professor Brouwer" (13 November 1928). As Springer wrote, his first idea was to refuse to receive the gentlemen, but he then realized that a refusal would provide propaganda material for the opposition. Springer opened the discussion with the remark that he was firmly resolved not to mix in the skirmishes and that he did not consider the Annalen the sole property of the Company (like other journals), but that the proper Herausgeber, Klein and Hilbert, had been in a sense in charge. Moreover, he would choose Hilbert's side out of friendship and admiration, if he would be forced to choose sides. The unwelcome visitors then proceeded to inquire into the legal position of Hilbert, a topic that Springer was not eager to discuss without the advice of his friends and which he could not enter into without consulting the contract. Thence the two gentlemen proceeded to "threaten to damage the Annalen and my business interests. Attacks on the publishing house, which could get the reputation of lack of national feeling among German mathematicians, could be expected.'" This threat was definitely in bad taste, not in the last place because the Springer family had Jewish ancestry. Bieberbach's later political views have gained a good measure of notoriety (cf. [3]); it certainly is true that already before the arrival of the Third Reich he held extreme nationalistic views. Brouwer's position in this matter of Nationalgefiihl was rather complex, it was not based on a political ideology but rather on his moral indignation at the boycott of German science. Be that as it may, this particular approach was not likely to mollify Springer, who calmly answered that

he would deplore damage resulting from this quarrel, but that he would bear it without complaints under the present circumstances. Thus rejected, Bieberbach and Brouwer asked if Springer could suggest a mediator, u p o n w h i c h Springer answered that he was not sufficiently familiar with the personal features involved, b u t that two deutschfreundliche foreigners like Harald Bohr and G. H. Hardy might do.8 Before leaving, Brouwer threatened to found a new journal with De Gruyter, and Bieberbach declared that he would resign from the board of editors if it definitively came to the exclusion of Brouwer. In a letter to Courant (13 November 1928) Springer dryly commented "On the whole the founding of a n e w journal, wholly under Brouwer's supervision, would be the best solution out of all difficulties. 9 He also conveyed his impression of the visit: "I would like to add that Brouwer, as a matter of fact, does make a scarcely pleasant (unerfreulich) impression. It seems, moreover, that he will carry the fight to the bitter end

(der Kampf bis aufs Messer fiihren wird). The Case for the Prosecution In Aachen Blumenthal was preparing his defense of the intended dismissal of Brouwer and, following an old strategic tradition, he took to the attack. After consulting Courant, Carath6odory, and Bohr he drew up a kind of indictment. I have not seen the draft of 12 November, but from a letter from Bohr and Courant to Blumenthal (14 November 1928), one may infer that it was harder in tone and more comprehensive than the final version. There is mention of a detailed criticism of Brouwer's editorial activities and of matters of formulation ( " . . . leave out Schrullenhaftigkeit [capric i o u s n e s s ] . . . . "). M o r e o v e r Bohr and C o u r a n t warned Blumenthal: To what extent Brouwer exploits without consideration every tactical advantage that is offered to him, and how dangerous his personal influence is (Bieberbach), can be seen from the enclosed notice which Springer has just sent us [the above-mentioned Aktennotiz]. The c o r r e s p o n d e n c e of Blumenthal, Bohr, and Courant shows an unlimited loyalty to Hilbert, which it would be unjust to ascribe to Hilbert's state of health alone. There is no doubt that Hilbert as a man and a

S This suggestion of the publisher encouraged the impression that the conflict h a d a political origin. B l u m e n t h a l complained to Courant (letter of 18 November 1928) " . . . the bad thing is, that Brouwer managed to move everything on to the political plane, just what Carath6odory thought he had prevented." The idea of mediation was not pursued. 9 Brouwer indeed founded a new journal, the Compositio Mathematica, with the Dutch publisher Noordhoff. THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990 23

scientist inspired a great deal of loyalty in others, let alone in his students. Sentences like "We don't particularly have to stress that we are, like you, wholly on Hilbert's side, and also, when necessary, prepared for action" (same letter), illustrate the feeling among Hilbert's students. A revised version of Blumenthal's letter is dated 16 November, and it is this version that was in Brouwer's possession. It i n c o r p o r a t e d remarks of Bohr a n d Courant, but not yet those (at least not all of them) of Carathrodory. It contained a concise resumd of the affair so far, and proceeds to answer Brouwer's points (from the letter of 5 November 1928). As Blumenthal put it, he partly based his handling of the matter on letters from Hilbert, Carath~odory, and Brouwer, partly on an extensive conversation with Hilbert in Bologna. The contents of the latter conversation remain a matter of conjecture, but it may be guessed that in August at the conference Hilbert had made clear his objections to Brouwer--in particular after Brouwer's opposition to the German participation in the conference. From Blumenthal's circular letter, the editors--and also Brouwer--learned the contents of Hilbert's letter of 25 October. In answering Brouwer's points Blumenthal quite correctly stressed that Brouwer interpreted "cooperation" in a too narrow fashion. Hilbert, he said, found it impossible to justify his sharing responsibilities in an editorial board t o g e t h e r with Brouwer. As to point 4 of Brouwer's letter, "the motivation is ugly and thus needs no answer." The scientific opposition in foundational matters had not played a role, according to Blumenthal. Even Brouwer's circular letter concerning the Bologna Congress " b y which statements Hilbert felt insulted" had, according to Blumenthal, only acted in a catalytic way on his decision: "The motives lie much deeper." Concerning Klein's position, Blumenthal remarked that Klein always acted as a kind of higher authority, to which one could appeal. After Klein's death Hilbert felt obliged to take on Klein's role. "Hilbert has recognized in Brouwer a stubborn, unpredictable, and ambitious (herrschsfichtig) character. He has feared that when he should eventually resign from the editorial board, Brouwer would bend the editorial board to his will and he had considered this such a danger for the Annalen that he wanted to oppose him as long as he still could do so." How strongly Blumenthal and Carathrodory wished to spare Brouwer, while complying with Hilbert's wishes, can be seen from the following paragraph: Cara and I, who were associated with Brouwer in a long-standing friendship, had objectively to recognize Hilbert's objections to Brouwer's editorial activity. True, Brouwer was a very conscientious and active editor, but he was quite difficult in his dealings with the

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THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

managing editor and he subjected the authors to hardships that were hard to bear. E.g., manuscripts that were submitted for refereeing to him lay around for months, while in principle he had prepared a copy of each submitted paper (I recently had an example of this practice). Above all there is no doubt that Klein's premature resignation from the editorial board is to be traced back to Brouwer's rude behaviour (in a matter in which Brouwer was formally right). The further course of events has shown that Hilbert was even far more right than we thought at the time. Since we could not reject the objective justification of Hilbert's point of view, and were confronted by his immutable will, we have given our permission for the removal of Brouwer from the editorial board [at this point it should be made clear that Carathrodory had not given his permission, as he wrote to Courant (14 November) in his letter with corrections to the draft]. We only wished--unjustifie G" as I now realize--a milder form, in the sense that Brouwer should be prevailed upon to resign. Hilbert could not be induced to this procedure, so we finally, though reluctantly, have decided to give in to him (den Weg freigeben). Mr. Einstein did not comply, with the argument that one should not take Brouwer's peculiarities seriously. To what the reader already knows about Carath4odory's trip to Brouwer, Blumenthal's letter adds the following: in Grttingen on 26 and 27 October Blumenthal and Carath6odory discussed the situation. In a last a t t e m p t to bring the matter to a good e n d through a mitigation of the categorical form of the statement of notice, Carathrodory travelled to Berlin and discussed the matter, as it appears, with Erhardt Schmidt. The result was, as we know, the request to Brouwer not to take action before Carath6odory's arrival. Finally, Blumenthal proceeded to reproduce the text of Brouwer's letter to Carath6odory of 2 November, with the " u n s o u n d mind" phrase, concluding that "I have thoroughly misjudged Brouwer's character and that Hilbert has known and judged him better than we have." The letter ended with the request to the editors for permission to delete Brouwer's name from the title page of the Annalen.

Defence of the Underdog A few editors responded to Blumenthal's letter in writing, but the majority remained silent. Only von Dyck, Hrlder, and Bieberbach sent their comments. Von Dyck could "neither justify Brouwer's views nor Hilbert's action" and he hoped that a peaceful solution could be found. Hrlder was of the opinion that he could not approve of a removal of Brouwer by force (27 November). Bieberbach's letter showed a thorough appreciationof the situation. And he at least was willing to take up the case of the underdog. In view of his later political extremism one might be inclined to question the purity of his motives; however, in the present letter there

Otto Blumenthal (left) and L. E. J. Brouwer (right) in happier days (1920?). is no reason not to take his arguments at their face value. Like Brouwer, and probably the majority if not the whole of the editorial board, Bieberbach contested the right of the Herausgeber to decide matters without the support of the majority of all the editors, let alone without consultation. Indeed, this seems to be a shaky point in the whole procedure. As a matter of fact, the contract between Springer and the Herausgeber (25 February 1920) is not very concrete in this particular point. It states: "Changes in the membership of the editorial board require the approval of the publisher." The correspondence does not lead me to believe that Hilbert observed this rule. Bieberbach observed that a delay in handling papers cannot be taken seriously as grounds for dismissal: such things o u g h t to be d i s c u s s e d in the annual meeting of the board. Bieberbach's comments on Hilbert's annoyance (to say the least) with Brouwer's actions in the matter of the Bologna conference are of a rather scholastic nature and border on nit-picking: the o b j e c t i o n a b l e s t a t e m e n t s of B r o u w e r c o n c e r n e d Germans who were to attend the Unionskongress in Bologna, and since Hilbert denied that the meeting in Bologna was a Unionskongress, the statements did not apply to him. Bieberbach correctly spotted a serious flaw in Blumenthal's charge involving Brouwer's "terrifying and repulsive" letter: Finally, I hold it totally unjustified to forge material against Brouwer from letters that he wrote after learning about the action that was mounted against himself. For it is morally impossible to use actions, to which a person is driven in a fully understandable emotion over an injustice that is inflicted on him, afterwards as a justification of this injustice itself. The point is well taken. It does not exonerate Brouwer from hitting below the belt, but it at least

Richard Courant

makes clear that to use it against Brouwer is distasteful. Bieberbach explicitly stated that he would not support Brouwer's dismissal; on the contrary, he strongly sided with Brouwer, without, however, attacking Hilbert. The publisher reacted in a cautious way. Springer thought that Brouwer was "an embittered and malicious adversary" and that he should not receive a copy of the circular letter without the permission of the lawyer of the firm. Springer also concluded that the publisher should not state in writing that he officially agreed to Brouwer's dismissal, because it would imply a recognition of Brouwer's membership on the board of editors in the sense of the contract. In short, Springer abstained from voting on Blumenthal's proposal. Froschen-Miiusekrieg At this point the w h o l e action against Brouwer seemed to reach a climax. One may surmise a good deal of activity in the camp of the G6ttinger, as the sympathizers of Hilbert were called. A certain amount of animosity between G6ttingen and Berlin mathematicians was a generally acknowledged fact. The Berlin faction had suffered a setback in the matter of the Bologna boycott, where Hilbert had undeniably carried the day. Born, in his letter to Einstein (20 November)j, quotes von Mises (a Berliner), "the G6ttinger simply run after Hilbert, who is not completely responsible for his actions (sei wohl nicht mehr ganz zurechnungsffihig).'" The friction between Berlin and G6ttingen was a weighty reason to settle the Annalen conflict as speedily and quietly as possible. If there was any risk of a rift in the German mathematical community, it was here. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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A key figure, in view of his immense scientific and moral prestige, was Albert Einstein. If he could be persuaded to side with Hilbert the battle would be half won. In spite of personal pressure from Born (20 November 1928) on behalf of Hilbert, Einstein remained stubbornly neutral. In his letters to Born and to Brouwer and Blumenthal one may recognize a measure of disgust behind a facade of raillery. In the letter to Born (27 November) the apt characterization of "Frosch-Mdusekrieg'" (war of the frogs and the mice) 1~ was introduced. After declaring his strict neutrality Einstein continued: If Hilbert's illness did not lend a tragic feature, this ink war would for me be one of the most funny and successful farces performed by that people who take themselves deadly seriously. Objectively I might briefly point out that in my opinion there would have been more painless remedies against an overly large influence on the managing of the Annalen by the somewhat mad (verr~ickt)Brouwer, than eviction from the editorial board. This, however, I only say to you in private and I do not intend to plunge as a champion into this frog-mice battle with another paper lance. Einstein's letter to Brouwer and Blumenthal (25 November) is even more cutting and reproving. I am sorry that I got into this mathematical wolf-pack

(Wolfsherde) like an innocent lamb. The sight of the scientific deeds of the men under consideration here impresses me with such cunning of the mind, and I cannot hope, in this extra-scientific matter, to reach a somewhat correct judgment of them. Please, allow me thereforei to persist in my "booh-nor-bah" (Muh-noch-Mdh)position and allow me to stick to my role of astounded contemporary. With best wishes for an ample continuation of this equally noble and important battle, I remain Yours truly, A. Einstein

Deadlock The whole affair now rapidly reached a deadlock. A week before, Springer, who had at Blumenthal's urging sought legal advice, had optimistically written to Courant (17 November 1928) that the legal adviser of the firm, E. Kalisher, was of the opinion that it would suffice that those of the four chief editors who did not want to advocate Brouwer's dismissal actively would abstain from voting, thus giving the remaining chief editors a free hand. Apparently Springer did not realize that since two editors with a high reputation had already decided not to support Hilbert, the solution, even if it was legally valid, would lack moral credibility. If this solution should turn out to raise difficulties within the editorial board, the publisher could still fire 10 W a r of the frogs a n d t h e m i c e - - a Greek play of u n k n o w n authorship; a late m e d i e v a l G e r m a n version, Froschmeuseler, is from the h a n d of Rollenhagen. 26

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the whole editorial board and reappoint Hilbert and his supporters, so the advice ran. In the opinion of the legal adviser the publishing house was contractually b o u n d to the chief editors (Herausgeber) only; there was no contract with the remaining editors. Bieberbach's letter, mentioned above, apparently worried Carath6odory to the extent that he decided to ask a colleague from the law faculty for advice. This advice from M~iller-Erzbach (Munich) plainly contradicted the advice from the Springer lawyer. It made clear that 1. Brouwer and Springer-Verlag were contractually bound since Brouwer had obtained a fee. 2. Hilbert's letter was not legally binding. M~iller-Erzbach sketched three solutions to the problem: 1. Springer dismisses Brouwer. A letter of dismissal should, however, contain appropriate grounds. 2. The four chief editors and the publisher form a company (Gesellschaft)and dismiss Brouwer. 3. A court of law could count the "Mitarbeiter'" as editors. In that case the only way out would be to dissolve the total editorial board and to form a new one. Carath6odory considered the first two suggestions inappropriate because it would not be fair to saddle Springer with the internal problems of the editors. Hence he recommended the third solution (letter to Blumenthal, 27 November). Here, for the first time, appeared the suggestion that was to be the basis of the eventual outcome of the dispute. Hilbert, the main contestant in the Annalen affair, had quite sensibly withdrawn from the stage. The developments, had he k n o w n them, would certainly have harmed his still precarious health. In a short notice he had e m p o w e r e d Harald Bohr and Richard Courant to represent him legally in matters concerning the Mathematische Annalen. Thus the whole matter became more and more a shadow fight between Brouwer and an absentee. At this point the dispute had reached an impasse. Although Springer upheld in a letter to Bieberbach the principle that the chief editors could dismiss any of the other editors, the impetus of the attack on Brouwer seemed to ebb. A meeting between Carath6odory, Courant, Blumenthal, and Springer had repeatedly to be postponed and finally had been cancelled. Courant agreed with Carath6odory that the dissolution of the complete board would be a good solution (30 November 1928); however, it would require a voluntary action from the editors and the ultimate organization of the editorial board should not have the character of a legal trick with the sole purpose of rendering Brouwer's opposition illusory. Carath6odory, who, on the basis of M~iller-Erzbach's information, had come to the conclusion that the original plan of Hilbert, even in a modified form,

would not stand up in a court of law, expressed his willingness to assist "out of devotion to Hilbert" in the liquidation of the affair, but quite firmly refused to be involved in the future organization of the Annalen.

Dissolution The reluctance of Carath6odory to be involved in the matter beyond the bare minimal efforts to satisfy Hilbert and spare Brouwer (his friend) is throughout understandable. As far as we can judge from the correspondence, only Blumenthal exhibited an unbroken fighting spirit. He realized, however, that his circular had not furthered an acceptable solution (letter to Courant and Bohr, 4 December), and he leaned towards alternative solutions. In particular, Blumenthal wrote, the time was favourable to Carath4odory's plan. The Annalen were completing their hundredth volume, and it would present a nice occasion to open with volume 101 a "new series" or "second series" with a different organization of the editorial board. But at the present time he was facing a dilemma. Because Hilbert's letter clearly had no legal status, Brouwer was still a Mitarbeiter and his name should appear on the cover of the issue that was to a p p e a r - this, however, conflicted with Hilbert's wishes. Could Bohr and Courant, as proxies of Hilbert, authorize him to print Brouwer's name on the cover? Otherwise the publication would have to be postponed. The authorization probably was given. It seems that Bohr had also put forward a solution to the affair. From the correspondence of Carath~odory and Bohr with Blumenthal, one gets the impression that Bohr's proposal was a slight variant of Carath6odory's suggestion. The main difference was that Bohr advocated a total reorganization of the editorial board. In his proposal there would only remain Herausgeber, and no Mitarbeiter. So the solution would look like a fundamental change of policy, and hence it would no longer be recognizable as an act levelled against Brouwer. Apparently Bohr envisaged Hilbert, Blumenthal, Hecke, and Weyl as the members of a new board. And should Weyl decline, one might invite Toeplitz. Blumenthal questioned the wisdom of reinstating himself as an editor; it could easily be viewed as the old board of Herausgeber in disguise (letter to Bohr, 5 December). In his letter to Courant, the next day, he considered the dissolution of the editorial board at large as necessary, and he fully agreed that Hilbert should choose the new editors. From then on things moved smoothly; Springer accepted the dissolution of the editorial board and agreed to enter into a contract with Hilbert on the subject of the reorganized Annalen. By and large only matters of formulation and legal points remained to be solved.

One might wonder where Brouwer was in all t h i s - he was completely ignored. In a letter of 30 November to the editors and the publisher he confirms the receipt of Blumenthal's indictment, which had only just reached him. In a surprisingly mild reaction he merely asked the editors to reserve their judgment--blissfully unaware that nobody was going to take a v o t e - - f o r the composition of a defence would take some days. Because the dissolution of the editorial board had to be a voluntary act, it was a matter of importance to get Einstein's concurrence. The contract of 1920 presented an elegant loophole that would allow both parties to settle the matter without breaking the rules. In w the clauses for termination of the contract were listed, and one of them stipulated that if the editors (Redaktion) renounced the contract, without a violation from the side of the publisher, the latter could continue the Mathematische Annalen at will. Possibly Einstein's agreement could be dispensed with, but it is likely that a decision to ignore Einstein's vote would influence general opinion adversely; moreover, it would be wise to opt for a watertight procedure, as Brouwer would not hesitate to test the outcome in court. So pressure was brought to bear on Einstein. James Franck, a physicist and a friend of Born, begged him to listen to the new plan. He stressed the political side of the issue, "At this time . . . . . whether the mathematicians split into factions, or whether the affair is arranged smoothly, d e p e n d s on your decision. It would almost be an ill-chosen joke (ein nicht all zu guter Witz) if in this case you would be claimed for the nationalistic side" (undated). Franck was not the only person to discover a (real or imaginary) political aspect in the controversy at hand. Blumenthal had already complained to Courant (18 November) that Brouwer had managed to introduce the political element into the matter. Born also, in his letter to Einstein of 11 November, tied the conflict to the political issue of the German nationalists and the animosity of Berlin vs. G6ttingen. The successful conclusion of the undertaking was conveyed to Springer by Courant. In his letter of 15 December he announced the cooperation of Einstein, Carath6odory, Blumenthal, and Hilbert. At the same time he proposed that a n e w contract be made between Hilbert and the publisher, and that Hilbert get carte blanche for organizing the editorial board. Blumenthal should be invited to continue his activity as managing editor and, according to Courant, he would probably accept. A l s o - - a n d this is a surprising misjudgement of Einstein's m o o d - - C o u r a n t thought that there was a fifty per cent chance that Einstein would join the n e w board. As far as he himself was concerned, Courant thought it wiser to postpone his own introduction as an editor until the dust had settled (the matter apparently had been discussed earlier). THE MATHEMATICAL 1NTELLIGENCER VOL. 12, NO. 4, 1990 2 7

Mathematische Annalen, volume 100 (1928) and volume 101 (1929). Notice the change in editors from volume 100 to volume 101.

Finally Courant suggested that the publisher alone should inform all present editors of the collective resignation. With r e s p e c t to B r o u w e r , he a d v i s e d Springer to write a personal letter explaining the solution to the conflict, and to stress that he [Springer] would regret it if Brouwer were left with the impression that the whole affair would restrict his freedom of practice, and that the publishing house would be at his disposal should he wish to report on his foundational views. It is not known whether this letter was ever written, b u t Courant's attitude certainly w a s statesmanlike and conciliatory. "No Personal Motives"

Once the decision was taken, no time was wasted; after the routine legal consultations the publisher carried out the reorganization and the editors were informed of the o u t c o m e (27 December). In spite of Courant's considerations mentioned above, the letter was signed by Hilbert and Springer. Brouwer, like everybody else, was thanked for his work and was given the right to a free copy of the future Annalen 28

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issues. The matter would have been over, were it not for some rumblings among the former editors and for a desparate but hopeless rearguard action by Brouwer. Carath6odory had been considerably distressed during the whole affair; from the beginning he had been torn between his loyalty to Hilbert and his abhorrence of the injustice of Brouwer's dismissal. His efforts to mediate had only worsened the matter and the final solution was an immense relief to him. In a fit of despondency he wrote to Courant (12 December) " Y o u cannot imagine h o w d e e p l y worried I w a s during the last weeks. I envisioned the possibility that, after I had parted with Brouwer, the same thing w o u l d happen with all my other friends." He had even considered accepting a chair at Stanford that was offered to him. In his answer (15 December) Courant tried to set Carath6odory's mind at ease: he believed that he had succeeded in convincing Hflbert that Carath~odory, in his position, could not have acted differently; the matter was settled now "without fears of a residue of resentment on Hilbert's part." Two days later Courant wrote that the night before

he had discussed the whole matter with Hilbert, who asked Courant to tell Carath6odory that "he thinks that you would have done everything for him, as far as possible." Hilbert was completely satisfied with the result of the undertaking, and in his opinion the Annalen were even better protected now than through his original dismissal of Brouwer " . . . and by and by. it has become completely clear to me that in fact no personal motives have ~ s p i r e d Hflbert's first step . . . . " Carath6odory expressed his pleasure with Hilbert's attitude (9 December) but he was not wholly satisfied with Courant's evaluation of the motives behind Hilbert's move. "Now, he himself has given as the exclusive motive for his decision that he felt insulted by Brouwer; ! would find it unworthy of him, to construe after the fact, that only impersonal motives had guided him." The last remark could hardly be left unanswered by Courant. He had worked hard to pacify the participants in the affair, and here one of the former Herausgeber was lending support to the rumour that Hilbert was not completely devoid of some personal feelings of revenge. In an attempt to quench this source of dissent he and Bohr admonished Carath6odory. Courant calmly repeated his view (23 December) and referred to Hilbert's personal statements that he "fostered no personal feelings of hate, anger or insult against Brouwer." Even a bit of subtle pressure was brought to bear on Carath6odory: "Our responsibility to Hilbert at this point is even greater, as he is not yet filled in on the development of the conflict; in particular he does not surmise your visit to Laren and the disconcerting report of it by Brouwer." Bohr was less subtle in his approach (same letter); if Carath6odory were not convinced of Hilbert's impersonal motives, he should ask Hilbert himself. "For, that Hilbert--without being aware of it and without being able to defend himself--should first be considered 'of unsound mind' and then 'not to the point' (unzurechnungsfiihig... unsachlich), that is a situation, that I as a representative of Hilbert, cannot in the long run witness without action." In spite of Bohr's saber rattling, Carath6odory stuck to his guns: "To judge Hilbert's motives is a very complicated matter; I believe that I see through his motives because I have known his way of thinking for more than 25 years. It is true that the motivations that you indicate, and which H. also expounded in Bologna in discussion with Blumenthal, were there. The total complex of thoughts that caused the explosion of feeling of 15 October [cf. letter to Einstein, 15 October] was much more complicated." Who was right, Courant and Bohr, or Carath6odory? The matter will probably never be completely settled. There is no doubt that the question of "how to safeguard the Annalen from Brouwer's negative influence (real or imagined)" was uppermost in Hilbert's

mind. But who is to say that no personal motives were involved? There are Hilbert's o w n statements (e.g., to Blumenthal and Courant) to the effect that no personal grudge led to his action, but h o w much weight can be attached to them? In any case they contradict the letter of 15 October.

Last Ditch The whole problem seemed to have been settled sarisfactorily. Hilbert, who was only partially informed of the goings on, wrote to Blumenthal (Blumenthal to Courant, 31 December) "a triumphant letter, that everything was glorious." Courant had written a conciliatory letter to Brouwer (23 December) in which he expressed the hope that the solution to the matter satisfied Brouwer. He also wished to convince Brouwer that no personal motives had played a role in Hilbert's action, and definitely no motives " w h o s e existence were in conflict with the respect for your scientific or moral personality." Little did he know Brouwer! To begin with, Brouwer had not yet received the letter from Springer and Hilbert, so he was unaware that the matter had been settled (unless he was informed by one of the other editors). As a matter of fact Brouwer launched another appeal to the publisher and the editors the same day C o u r a n t was offering B r o u w e r the "forgive-andforget" advice. Brouwer insisted that in the interest of mathematics the total editorial board of the Mathematische Annalen should remain in function; as he realized that a written defense from his hand would inevitably wreck the unity of the editors, he was willing to postpone such a letter; moreover, Carath6odory, in a letter of 3 December, had promised him to do his utmost to find an acceptable solution, and had begged him to be patient for a couple of more weeks. Sommerfeld had also pressed Brouwer to wait for Carath6odory's intervention. The final solution, as formulated in the HilbertSpringer letter, did not satisfy Brouwer. He recognized that the reorganization of the Annalen was mostly, if not wholly, designed to get rid of him. Also, Brouwer had explicit views on the ideal organisafion of the Annalen. In a circular letter (23 January 1929) to the editors, Blumenthal and Hilbert excluded, Brouwer rejected the final solution. According to him, the Mathematische Annalen was a spiritual heritage, a collective property of the total editorial board. The chief editors were, so to speak, appointed by free election and they were merely representatives vis-a-vis the mathematical world. Thus, Brouwer argued, the contractual rights of the chief editors were not a personal but an endowed good. Hflbert and Blumenthal, in his view, had abstracted this good from their principals, and hence were guilty of embezzlement, even if this could by THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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to, and including, Hilbert's action. The views, he wrote, were not mine, but "views that during the aforementioned visit, came up between Carath6odory and me in mutual agreement, i.e., that were successively uttered by one of us and accepted by the other." He also elaborated the grounds for not acquiescing in the dismissal. He had told Carath6odory that he would consider a possible dismissal from the editorial board not only a revolting injustice, but also a serious damage to my freedom to act (Wirkungsm6glichkeit) and, in the face of public opinion, as an offending insult; that, if it really came to this unbelievable event, my honour and freedom of practice could only be restored by the most extensive flight into public opinion. At the end of the otherwise friendly visit of 30 October, Carath6odory had once more returned to the matter. At Brouwer's exclamations Carath6odory could only answer "What can one do?" and "I don't want to kill a person." The final farewell was accompanied by Brouwer's bitter "I don't understand you For, Klein had organised the board of editors of the any more," "I consider this visit as a farewell," and "I Mathematische Annalen in such a way that it formed really a am sorry for you." kind of Academy, in which each member had the same After attacking Blumenthal for his desire to remove rights as the others. That was in my opinion the main Brouwer from the board of editors, Brouwer went on reason why Annalen could claim to be the first mathematics journal in the world. Now it will become a journal to answer Blumenthal's points. Without repeating the argument verbatim, some points may be taken from it like all other ones. to represent Brouwer's side in the discussion. BluThe wisdom of severely restricting the size of the menthal accused Brouwer of rudeness; the latter aneditorial board was questioned. Already on 2 February swered that if Blumenthal meant by "rude" the "de1929 Blumenthal sent out a note on the future organisire for integrity (duty of every human) increased by zation of the Annalen, in which he drew the attention the will for clarity (the destiny of the mathematician)," to the decline of the journal c o m p a r e d to other there could have been cases of rudeness, in w h i c h - - i n journals. Since the Nebenredaktion had been eliminated Brouwer's words--neither the vanity of the author, (ausgeschaltet) one simply needed a larger staff: "the nor the wish of Blumenthal to appear pleasant, could increasing necessity of scientific advisers follows inevibe spared. These cases, moreover, were entrusted by tably from the increasing specialisation." In short BluBlumenthal to Brouwer as a trouble shooter, and thus menthal proposed to reinstate something like the old Blumenthal could not possibly find support among his Mitarbeiter under a different name. In the same letter fellow editors if and when he complained. he broached the question of the successor of Hilbert, The matter of the resignation of Klein was, acshould he step down. One finds it difficult to reconcile cording to Brouwer, misrepresented; an author had, this letter with the arguments that were put forward in after his paper had been turned down by Brouwer, favour of the solution to the conflict. appealed to Klein and made the contents plausible. When Brouwer afterwards showed Klein that the auParting Shot thor was ("not formally, but materially") wrong, Klein The Annalen settled down under the new regime. Due saw that he could not fulfill his promise to the author. to tactful handling of all publicity, the excitement in In the discussion with Brouwer, Klein then uttered the Germany died out, even, as Courant wrote to Hecke, opinion that the public was misled by the lists of edamong the colleagues in Berlin--and Brouwer was itors on the cover of the journal, and that, as far as he completely ignored. After waiting for m o n t h s - - a n d was concerned, he could no longer carry the responsiprobably realizing that the battle was over and that bility for this impression. He retired soon afterwards. The reproach concerning the long delay of papers at everybody had gone h o m e - - B r o u w e r fired his parting shot, the letter of defense against Blumenthal's indict- Brouwer's desk was dismissed by Brouwer as nonment of 16 November 1928. The letter is three-and-a- sense. Papers with lots of mistakes take t i m e - - a n d half folio sheets long and contains a report of the never a paper got lost, as happened with Hilbert, he events mentioned above as experienced by Brouwer. said. In any case, Blumenthal's reproaches had never In the first place he denied Blumenthal's claim that been uttered before. Brouwer had substituted his own interpretation for The battle being lost, Brouwer no longer attempted Carath6odory's version of the developments leading to r e v e r s e the reorganization of the Annalen. He sheer accident not be dealt with by law (the reader may hear a faint echo of Brouwer's objections to the principle of the excluded third [1]). Brouwer then proceeded to attack Blumenthal's role in the Annalen. He repeated Blumenthal's earlier views on the equal rights of all editors and referred to certain irregularities in the management of the Annalen in 1925 resulting in Blumenthal's promise to resign after the appearance of volume 100. Carath6odory also deplored the end of the old r4gime. When confronted with Hecke's comments on the practice of the past (letter from Courant to Carath6odory, 17 December): " . . . that Hecke, w h e n he learned about the organisation of the editorial board and the competency of the Beirat [the advisory editors] grasped his head and judged a revision and a more strict organization absolutely necessary," Carath6odory heartily disagreed (to Courant, 19 December 1928):

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merely challenged Blumenthal to open the archive of the Annalen, claiming the correspondence would fully vindicate Brouwer.

Not Just Another Battle The whole history of the Mathematische Annalen conflict was quietly incorporated into the oral tradition of European mathematics. Little is known of the aftermath; the G6ttinger had won the battle, and they may have been tempted to pick a bone or two with some of the minor actors. For instance, Harald Bohr drafted a letter to the effect that "Schmidt for once realizes that he is vulnerable and that it is dangerous just to make a telephone call to Brouwer" (letter to Courant, 31 December 1928). After some reflection the letter to Schmidt was never sent. From the gossip generated by the Annalen affair, a few rumours have surfaced in print. Only in one case could some evidence be unearthed, to wit the claim that Brouwer's dismissal was partly motivated by the fact that he had reserved the right to handle all papers from Dutch mathematicians [4, p. 187]. Professor Freudenthal told me that this was indeed commonly believed at the time of the conflict. By chance this particular rumour was confirmed in the draft of a letter from Felix Klein to the Dutch mathematician Schouten (13 March 1920). Klein wrote that "Prof. B r o u w e r . . . who at his entry in the editorial board of the Annalen has reserved the right to decide, in particular about Dutch papers . . . . " In general not much is known about the actual use Brouwer made of this prerogative; the letter of Klein dealt with a paper of Schouten that had received a negative evaluation from Brouwer. Looking back, without the emotions of the contemporaries, we can only say that the whole affair was a tragedy of errors. Hilbert's annoyance with Brouwer was understandable. There had been a long series of conflicts, the Grundlagenstreit, the GOttingen chair that was turned down, the Riemann volume of the Annalen, and finally the Bologna affair. In a sense there had been an ongoing battle and each antagonist was firmly convinced that the survival of mathematics depended on him. Hilbert's illness, with the real danger of a fatal outcome, must have influenced his power of judgment. I do not see h o w Brouwer could have marched the Annalen to its doom. One has to agree with Einstein: if Brouwer was a menace of some sort, there were other ways to safeguard the Annalen. The question of the real motives behind Hilbert's action remains a matter of conjecture. Most likely the letter to Einstein shows an unguarded Hilbert with personal motives after all. For Brouwer the matter had, in my opinion, far more serious consequences. His mental state could, under severe stress, easily come dangerously close to instability. Hilbert's attack, the lack of support from

old friends, the (real or imagined) shame of his dismissal, the cynical ignoring of his undeniable efforts for the Annalen; each and all of these factors drove Brouwer to a self-chosen isolation. Although it is most unlikely that intuitionism would have become the dominant doctrine of mathematics, there was a real possibility that it would develop into a recognized, although limited, activity. As it was, history took another turn, the development of intuitionistic (or constructive) mathematics suffered a setback, from which it recovered only some forty years later. After the Annalen affair, little zest for the propagation of intuitionism was left in Brouwer; he continued to work in the field, but on a very limited scale with only a couple of followers. Actually, his whole mathematical activity became rather marginal for a prolonged period. During the thirties Brouwer hardly published at all (only two small papers on topology); he undertook all kinds of projects that had nothing to do with mathematics or its foundations. For all practical purposes, 1928 marks the end of the Grundlagenstreit. Acknowledgments: The material used for this paper comes from various sources; the letters of Einstein are published with the permission of the Department of Manuscripts and Archives of the Jewish National and University Library at Jerusalem; the Nieders~ichsische Staats- und Universit/itsbibliothek G6ttingen gave permission for publication of the quotation from Klein's letter to Schouten; Professor Freudenthal kindly made some of the correspondence available, and material from the Brouwer Archive has been used. I have received advice and help from a great number of people and institutions to w h o m ! express my gratitude. I would like to thank in particular P. Forman, H. Freudenthal, H. Mehrtens, D. Rowe, and C. Smorynski.

References 1. L. E. J. Brouwer, 0her die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie. Journal fiir reine und angewandte Mathematik 154 (1924), 1-7. Also in Collected Works, Vol. 1 (A. Heyting, ed.), Amsterdam: North-Holland Pub|. Co. (1975), 268-274. 2. D. van Dalen (ed.), L. E. ]. Brouwer, C. S. Adama van Scheltema. Droeve snaar, vriend van mij. Amsterdam: Arbeiderspers (1984). 3. H. Mehrtens, Ludwig Bieberbach and "Deutsche Mathematik," Studies in History of Mathematics (E. R. Philips, ed.). The Mathematical Association of America (1987), 195-241. 4. C. Reid, Hilbert-Courant. Berlin: Springer-Verlag (1986). (Originally two separate vols. 1970, 1976) Mathematisch Instituut Rijksuniversiteit Utrecht Budapestlaan 6 3508 TA Utrecht The Netherlands THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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Mathematical Anecdotes Steven G. Krantz

In any field of human endeavor, the "great" participants are distinguished from everyone else by the arcana and apocrypha that surround them. Stories about Wolfgang Amadeus Mozart abound, yet there are few stories about his musical contemporaries. Mozart had the je ne sais quoi that made people want to tell stories about him. And so it goes with mathematicians. Over the years I have collected dozens of anecdotes about famous mathematicians (a necessary condition for being the subject of legend is fame; it is by no means sufficient). These stories are of several types: (i) incidents to which I have been witness (there are few of these); (ii) incidents related to me by someone w h o witnessed them (on statistical grounds, one expects a greater number of these); and (iii) incidents that have been passed down through iterated tellings and are therefore unverifiable. I shall not consistently classify the stories that I will relate here. In many cases I cannot remember which of the three types they are, and actually knowing would generally spoil the fun. In any event, I must bear the ultimate responsibility for the stories. In writing this article, I am running the risk that readers will think me flip, disrespectful, or (worse) that I am attacking people who cannot fight back. Let me set the record straight once and for all: to me, the mathematicians described here are among the gods of twentieth-century mathematics. Much of what we know, and certainly much of my own work, follows from their insights. The enormous scholarly reputations of these men sometimes cause their humanity to be forgotten. Bergman, Besicovitch, G0del, Lefschetz, and Wiener were not merely collections of theorems masquerading as people; they had feet of clay like the rest of us. In telling stories about them we bring them back to life and celebrate their careers. 32

Bergman Stefan Bergman (1898-1977) was a native of Poland. He began his career in the United States at Brown University. It is said that shortly after Bergman and his mistress arrived in the U.S. he took her aside and told her " N o w we are in the United States where customs are different. When we are with other people, you should call me 'Stefan.' But at home you should continue to call me 'Professor Doktor Bergman.' " Others who knew Bergman will say that he was not the sort of man w h o would have had a mistress. It is more likely that the man in q u e s t i o n w a s v o n Mises

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(Bergman's sponsor); there is general agreement that the woman was Hilda Geiringer. In fact another story holds that Norbert Wiener (more on him later) went to D. C. Spencer around this time and said "I think that we should call the FBI (Federal Bureau of Investigation)." Puzzled, Spencer asked why. "Because von Mises has a mistress," was the serious reply. After a few years Bergman moved to Harvard and then to Stanford, where he spent most of his career. Supported almost always on grants and other soft money, Bergman rarely taught. This fact may have contributed to the general murkiness of his verbal and written communications. Murkiness aside, Bergman was proud of his ability to express himself in many tongues. Said he, "I speak twelve languages--English the bestest." In fact Bergman had a stammer and was sometimes difficult to understand in any language. Once he was talking to Antoni Zygmund, another celebrated Polish analyst, in their native tongue. After a bit Zygmund said "'Please let's speak English: it's more comfortable for me.'" Although Bergman had many fine theorems to his credit (including the invention of a version of the ~ilov boundary), the crowning achievement of his mathematical work was the invention of the kernel function, now known as the Bergman kernel. He spent most of his life developing properties and applications of the Bergman kernel and the associated Bergman metric. It must have been a special source of pride and pleasure for him when, near the end of Bergman's life, Charles Fefferman (1974) found a profound application of the Bergman theory to the study of biholomorphic mappings. Fefferman's discoveries, coupled with related ideas of J. J. Kohn and Norberto Kerzman, created a renaissance in the study of the Bergman kernel. Indeed, a major conference in several complex variables was held in Williamstown in 1975 in which many of the principal lectures m e n t i o n e d or discussed the Bergman kernel. Bergman had always felt that the value of his ideas was not sufficiently appreciated. He attended the conference and commented to several people how pleased he was that his wife (also present) could see his work finally being recognized. I sat next to him at most of the principal lectures. In each of these, he listened carefully for the phrase "and in 1922 Stefan Bergman invented the kernel function." Bergman would then dutifully record this fact in his n o t e s - - a n d nothing more. I must have seen him do this twenty times during the three-week conference. There was a rather poignant moment at the conference. In the middle of one of the many lectures on biholomorphic mappings, Bergman stood up and said, "I think you people should be looking at representative coordinates (also one of Bergman's inventions)." Most of us did not know what he was talking about,

and we ignored him. He repeated the comment a few more times, with the same reaction. Five years later S. Webster, S. Bell, and E. Ligocka found astonishing simplifications and extensions of the known results about holomorphic mappings using--guess w h a t ? representative coordinates. Bergman was an extraordinarily kind and gentle man. He went out of his way to help many young people begin their careers, and he made great efforts on behalf of Polish Jews during the Nazi terror. He is remembered fondly by all who knew him. But he was a shark when it came to his mathematics. When he attended a lecture about a theorem he liked, he often went to the lecturer afterwards and said "I really like your theorem. It reminds me of my studies of the kernel function. Consider complex two-space . . ." And Bergman was off and running on his favorite topic. On another occasion a young mathematician gave Bergman a manuscript he had just written. Bergman read it and said "I like your result. Let's make it a joint paper, and I'll write the next one." Whenever someone proved a new theorem about the Bergman kernel or Bergman metric, Bergman made a point of inviting the mathematician to his house for supper. Bergman and his wife were a gracious host and hostess and made their guest feel welcome. However, after supper the guest had to pay the piper by giving an impromptu lecture about the importance of the Bergman kernel. Bergman's wife Edy was very devoted to him, but life with Stefan was sometimes trying. When they first got married, Bergman had just completed a difficult job search. In the days immediately following World War II, jobs were scarce, and Bergman wanted a position with no teaching. After a long period of disappointment, Shiffer got Bergman a position at Stanford; so the mood was high at the Bergmans' wedding reception. The reception took place in New York City, and Bergman was delighted that one of the guests was a mathematician from N e w York University with w h o m he had many mutual interests. They got involved in a passionate mathematical discussion and after a while Bergman announced to the guests that he would be back in a few hours: he had to go to NYU to discuss mathematics. On hearing this, Shifter turned to Bergman and said "I got you your job at Stanford; if you leave this reception I will take it away." Bergman stayed. Bergman thought intensely about mathematics and cared passionately about his work. One day, during the 1950 International Congress of Mathematicians in Cambridge, Bergman had a luncheon date with two Italian friends. Right on schedule they appeared at Bergman's office: the distinguished elder Italian mathematician Piccone (bearing a bouquet of flowers for Bergman!) and his younger colleague Sichera. This was Piccone's first visit to the United States, and he THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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spoke no English; Sichera acted as interpreter. After greetings were exchanged, Bergman asked Sichera whether he had read Bergman's latest paper. Sichera allowed that he had, and that he thought it was very interesting. However, he said that he felt that certain additional differentiability assumptions were required. Bergman said "No, no, you don't understand," and proceeded to explain on the blackboard. Piccone, understanding none of this, waited patiently. After the explanation, Bergman asked Sichera whether he now understood. Sichera said that he did, b u t he still thought that some differentiability hypotheses were required in a certain step. Bergman became adamant and a heated argument e n s u e d - - P i c c o n e comprehending none of it. After some time, Sichera said "Well, let's forget it and go to lunch." Bergman cried "No differentiability--no lunch!" and he remained in his office while the two Italians went to lunch. Piccone gave the flowers to the waitress. There is c o n s i d e r a b l e e v i d e n c e that B e r g m a n t h o u g h t a b o u t m a t h e m a t i c s constantly. Once he phoned a student, at the student's home number, at 2:00 a.m. and said "Are you in the library? I want you to look something up for me!" On another occasion, when Bergman was at Brown, one of Bergman's graduate students got married. The student planned to attend a conference on the West Coast, so he and his new bride decided to take a bus to California as a sort of makeshift honeymoon. There was a method in their madness: the student knew that Bergman would attend the conference, but that he liked to get where he was going in a hurry. The bus seemed the least likely mode of transportation for Bergman. But w h e n Bergman heard about the impending bus trip, he thought it a charming idea and purchased a bus ticket for himself. The student protested that this trip was to be part of his honeymoon, and that he could not talk mathematics on the bus. Bergman promised to behave. When the bus took off, Bergman was at the back of the bus and, just to be safe, Bergman's student took a window seat near the front with his wife in the adjacent aisle seat. But after about ten minutes Bergman got a great idea, wandered up the aisle, leaned across the scowling bride, and began to discuss mathematics. It wasn't long before the wife was in the back of the bus and Bergman next to his s t u d e n t - - a n d so it remained for the rest of the bus trip! The story has a happy ending: the couple is still married, has a son who became a famous mathematician, and several grandchildren. Presumably it was his preoccupation with mathematics that caused Bergman to appear to be out of touch with reality at times. For example, one day he went to the beach in northern California with a group of people, including a friend of mine w h o told me this yarn. Northern California beaches are cold, so when Bergman came out of the water he decided that he'd 34

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better put on his street clothes. As he wandered off to the parking lot, his friends noticed that he w a s heading in the wrong direction; but they were used to this sort of behavior and paid him no mind. In a while, Bergman returned--clothed--exclaiming "You know, there's the most unfriendly woman in our car!" Bergman was a prolific writer. Of course he worked in the days before the advent of word processors. His method of writing was this: First, he would write a manuscript in longhand and give it to the secretary. When she had it typed up he would begin revising, stapling strips of paper over the portions that he wished to change. Strips would be stapled over strips, and then again and again, until parts of the manuscript would become so thick that the stapler could no longer penetrate. Then the manuscript would be returned to the secretary for a retype and the whole cycle would begin again. Sometimes it would repeat ad infinitum. Bergman once told a student that "a mathematician's most important tool is the stapler." Bergman had a self-conscious sense of humor and a loud laugh. He once walked into a secretary's office and, while he spoke to her, inadvertently stood on her white glove that had fallen on the floor. After a bit she said "Professor Bergman, y o u ' r e standing on m y glove." He acted embarrassed and exclaimed "Oh, I thought it was a m o u s y . " (It should be mentioned here that a number of wildly exaggerated versions of this story are in circulation, but I got this version from a primary source.)

Besicovitch Abram S. Besicovitch (1891-1970) was a geometric analyst of extraordinary power. He b e c a m e worldfamous for his solution of the Kakeya needle problem. The problem was to find the planar region of least area with the property that a segment of unit length lying in the region can be m o v e d through all direction angles 0, 0 ~ 0 ~ 2~r, within the region. Besicovitch's surprising answer was that for any e > 0 there is such a region with area less than ~. Besicovitch, a Russian by birth, was a creature of the old world. After leaving Russia (a prudent move on account of his rumored black market dealings during World War I), Besicovitch ended up at Cambridge University in England. A dinner was given in his honor, at which the main course was some sort of game bird. In his thick Russian accent, Besicovitch asked the name of the tasty food that they were eating. When he heard the reply, he exclaimed, "In Russia, we are not allowed to eat the peasants!" Besicovitch was a smart man, so he quickly became proficient at English. But it was never perfect. He adhered to the Russian paradigm of never using articles before nouns. One day, during his lecture, the class chuckled at his fractured English. Besicovitch turned to the audience and said "Gentlemen, there are fifty million Englishmen speak English you speak; there are two hundred million Russians speak English I speak". The chuckling ceased. In another lecture series, on approximation theory, he announced "zere is no t in ze name Chebysh6v." Two weeks later he said "Ve now introduce ze class of T-polynomials. Zey are called T-polynomials because T is ze first letter of ze name Chebysh6v." Besicovitch, in spite of his apparent powers, was modest. On his thirty-sixth birthday, he convinced himself that his best and most intense years of research were over. He said "I have had four-fifths of my life." Twenty-three years later, when in 1950 he was awarded the Rouse Ball Chair of Mathematics at Cambridge, someone reminded him of this frivolous remark. He replied "Numerator was correct." In the 1960s, the Mathematical Association of America made a series of delightful one-hour films in each of which a great mathematician gave a lecture, for a general mathematical audience, about one of his achievements. One of these films starred Besicovitch, and he explained his solution of the Kakeya needle problem. Besicovitch was a natty dresser under any circumstances, and he wore to this lecture an attractive light beige suit. However the lights were ho~and, after a while he removed his jacket, revealing bright red suspenders! The producers were most surprised (this was thirty-five years ago, and nobody but firemen wore red suspenders), but the filming continued and the suspenders can be seen today.

At one point during the filming of Besicovitch, the aged professor had to blow his nose. He drew a large white handkerchief from his pocket and did s o - loudly. Later, when Besicovitch viewed the finished product, he objected to the noseblowing scene as und i g n i f i e d - h e w a n t e d it removed. The producers were able to replace the offending video segment, but it was decided that the sound should remain. As a result, if you view the film today, there comes a point in the action where the camera abruptly leaves Besicovitch and focuses on the side of the r o o m - - a n d you can hear Besicovitch blow his nose.

GOdel Kurt G6del (1906-1978) was one of the most original mathematicians of the twentieth century. Any thesaurus links "originality" with "eccentricity," and G6del had his fair share of both. Toward the end of his life, G6del became convinced that he was being poisoned, and he ended up starving himself to death. However, years before that, his peculiar point of view exhibited itself in other ways. Einstein was G6del's closest personal friend in Princeton. For several years Einstein, G6del, and Einstein's assistant Ernst Straus (who later became a wellknown combinatorial theorist) would lunch together. During lunch they discussed non-mathematical topics--frequently politics. One notable discussion took place the day after Douglas MacArthur was given a ticker-tape parade down Madison Avenue upon his return from Korea. G6del came to lunch in an agitated state, insisting that the man in the picture on the front page of the New York Times was not MacArthur but a n impostefr. The proof? G6del had an earlier photo of MacArthur and a ruler. He compared the ratio of the length of the nose to the distance from the tip of the nose to the point of the chin in each picture. These were different: Q.E.D. G6del spent a significant part of his career trying to decide whether the Continuum Hypothesis (CH) is independent of the Axiom of Choice (AC). In the early 1960s, a brash, young, and extremely brilliant Fourier analyst (student of the aforementioned Zygmund) named Paul J. Cohen (people w h o knew him in high school and college assure me that he was always brash and brilliant) chatted with a group of colleagues at Stanford a b o u t w h e t h e r he w o u l d become more famous by solving a certain Hilbert problem or by proving that CH is independent of AC. This (informal) committee decide that the latter problem was the ticket. [To be fair, Cohen had been interested in logic and recursive functions for several years; he may have conducted this seance just for fun.] Cohen went off and learned the necessary logic and, in less than a year, had proved the independence. This is certainly THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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one of the most amazing intellectual achievements of the twentieth century. Cohen's technique of "forcing" has become a major tool of modern logic, and Cohen was awarded the Fields Medal for the work. But there is more. Proof in hand, Cohen flew off to the Institute for Advanced Study to have his result checked by Kurt G6del. G6del was naturally skeptical, as Cohen was not the first p e r s o n to claim to have solved the problem; and Cohen was not even a logician! G6del was also, at this time, beginning his phobic period. When Cohen went to G6del's house and knocked on the door, it was opened six inches and a hoary hand snatched the manuscript and slammed the door. Perplexed, Cohen departed. However, two days later Cohen received an invitation for tea at G6del's home. His proof was correct: the master had certified it.

Lefschetz The story goes that Solomon Lefschetz (1884-1972) was trained to be an engineer. This was in the days, near the turn of the century, when engineering was part carpentry, part alchemy, and part luck (the prevon K~rm~n era). In any event, Lefschetz had the misfortune to lose both his hands in a laboratory accident. This mishap was lucky for us, for he subsequently, at the age of 36, became a mathematician. Lefschetz h a d t w o p r o s t h e s e s in place of his h a n d s - - t h e y looked like hands, loosely clenched, but they did not move or function in any way. Over each he wore a shiny black glove. A friend of mine was a graduate student of Lefschetz; he tells me that one of his daily duties was to push a piece of chalk into Lefschetz's hand each morning and to remove it at the end of the day. Lefschetz starred in one of the MAA films. He gave a lovely lecture, p u n c t u a t e d b y a c a c o p h o n y of s q u e a k y chalk, a b o u t his celebrated fixed point theorem. His feelings about the film were mixed: at one point he says on film "I hope this is clear; it's probably about as clear as m u d . " After his lecture comes a filmed round table discussion including John Moore, Lefschetz, and a few others. For ten or fifteen m i n u t e s t h e y r e m i n i s c e a b o u t t h e old d a y s at Princeton. One person reminds Lefschetz that in the late 1940s, during the heyday of the development of algebraic topology, they were on a train together. Lefschetz was asked the difference between algebra and topology. He is reported to have said "If it's just turning the crank, it's algebra; but if there is an idea present then it's topology." When Lefschetz was reminded of this story in the film, he became most embarrassed and said "I couldn't have said anything like that." With his artificial h a n d s , L e f s c h e t z could not operate a doorknob, so his office door was equipped 36 THEMATHEMATICALINTELLIGENCERVOL.12,NO.4, 1990

with a lever. Presumably he had difficulty with other routine daily matters, too--dialing a phone, turning on a light, etc. By the time I was a graduate student at Princeton, Lefschetz was 87. He was still mathematically sharp but he had trouble getting around. In those days Fine Hall, the mathematics building in Princeton, was having constant trouble with the elevators: push the button for the fifth floor and you're shot to the penthouse, d o w n to the basement, and ejected on seven; or variations on that theme. The receptionist kept a log of complaints so that she could report them to the person who came to repair the elevator. One day Lefschetz got into the elevator and it delivered him to the fourth floor "machine room"; this room houses the air conditioning equipment and is ordinarily only accessible with a janitor's key. Poor Lefschetz unwittingly wandered out into the room, only to have the elevator door shut behind him before he realized what was going on. He was trapped in total darkness, could not summon the elevator (no key), could not turn the doorknob to use the stairwell, and could not find a telephone (which, even had he found, he probably could not have dialed). The members of the mathematics department rode that elevator for several hours, not realizing that Lefschetz was missing, before someone finally heard Lefschetz's shouts and under-

stood what was going on. Fortunately, Lefschetz survived the incident unharmed. Speaking of the elevators at Princeton, one of my earliest memories as a graduate student was of the elevator emergency stop alarm going off three or four times a day. Especially puzzling was that everyone ignored it. Bear in mind that this alarm only sounds if someone inside the elevator sets it off. It is sometimes used by janitors to hold the elevator at a certain floor; but the janitors never used it during the day. After I had asked around for some time, someone finally told me the secret. When the mathematics department moved from old Fine Hall to new Fine Hall (sometimes called "Finer Hall," overlooking "Steenrod Square"), Ralph Fox, the famous topologist, was annoyed that there was no men's room on his floor. So, whenever he had to use the facilities, he would take the elevator to the next floor, set the emergency stop alarm, do what needed to be done, and then return to his floor. N o w I knew w h y everyone smiled w h e n the alarm went off. So much for boyhood memories: back to Lefschetz. Lefschetz was famous for his aggressive self-confidence. He could terrorize most other mathematicians easily. At committee meetings he would pound his fist on the table with terrifying effect. So it is with pleasurable surprise that one hears of exceptions. The one I have in mind is a certain unflappable graduate student at the time of the student's qualifying examination. The qualifying exams at Princeton are administered as one long oral exam: three professors and one graduate student locked in an office for about three and onehalf hours. The student is examined on real analysis, complex analysis, algebra, and two advanced topics of the student's choosing (subject to the approval of the Director of Graduate Studies). Our confident student h a d Lefschetz on his c o m m i t t e e . L e f s c h e t z w a s famous for, among other things, profound generalizations of Picard's theorems in function theory to several complex variables. So it came as no surprise w h e n Lefschetz asked the student "Can you prove Picard's Great Theorem?" Came the reply "No, can you?" Lefschetz had to admit that he could not remember, and the exam moved on to another topic. Lefschetz was one of those mathematicians, of w h o m we all know at least one, w h o w o u l d sleep during lectures and then wake up at the end with a brilliant question. At one colloquium, the speaker got stuck on a point about twenty minutes into his talk. A silence of several minutes ensued. This threw off Lefschetz's rhythm: he woke up, said "'Are there any questions? Thank you very much," and the seminar was ended with a round of applause. The "roasting" of an individual is a peculiarly American custom: A group of close friends holds a fancy dinner in honor of the victim, after which they stand up one by one and make a collection of (humor-

ously delivered) insulting remarks about him. Some anecdotes are in the nature of a roast. Here is an example. In the fifties, it was said in Princeton that there were four definitions of the word "obvious." If something is obvious in the sense of Beckenbach, then it is true and you can see it immediately. If something is obvious in the sense of Chevalley, then it is true and it will take you several weeks to see it. If something is obvious in the sense of Bochner, then it is false and it will take you several weeks to see it. If something is obvious in the sense of Lefschetz, then it is false and you can see it immediately. This last item reminds me of the old concept of "True in the sense of Henri Cartan." In the 1930s and 1940s, a theorem was "true in the sense of Cartan" if Grauert could not find a counterexample in the space of an hour. The discussion of "truth" and obviousness" raises the issue of standards. Perhaps the least delightful arena in which we all wrestle with standards is that of r e f e r e e s ' r e p o r t s . The Annals of Mathematics, Princeton's journal, has very high standards and exhorts its referees to be tough-minded. Lefschetz was instrumental in establishing the pre-eminence of the Annals. But I d o u b t that even he could have anticipated the following event. Many years ago, Gerhard Hochschild (who sets high standards for himself and everyone else) submitted a paper to the Annals. The referee's report said " G o o d enough for the Annals. Not good enough for Hochschild. Rejected."

Wiener The brilliant analyst Norbert Wiener (1894-1964) is a favorite subject of anecdotes. He is just m o d e r n enough that many living mathematicians knew him, and was just eccentric enough to be a never-ending object of stories and pranks. Born the son of a distinguished professor of languages, Wiener became one of America's first internationally recognized mathematicians. Because of antiSemitism in the American mathematical establishment, Wiener spent the early years of his career working in England. The story goes that when he met Littlewood he said "'Oh, so you really exist. I thought that 'Littlewood' was just a pseudonym that Hardy put on his weaker papers." Poor Wiener was so chagrined by this story that he denied it vehemently in his autobiography, thus inadvertently fueling belief in its validity. [In fairness to Wiener I should point out that another popular version of the story involves Edm u n d Landau: Landau so doubted the existence of Littlewood that he made a special trip to Great Britain to see the man with his own eyes.] After Wiener left Britain, he moved to M.I.T. where he stayed for more than twenty-five years. He developed a reputation all over campus as a brilliant scienTHE MATHEMATICAL INTELL1GENCER VOL, 12, NO. 4, 1990 3 7

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tist and a bit of a character. He was always w o r k i n g - either thinking or writing or reading. When he walked the halls of M.I.T. he invariably read a book, running his finger along the wall to keep track of where he was going. One day, engaged in this activity, Wiener passed a classroom where a class was in session. It was a hot day and the door had been left open. But of course Wiener was unaware of these details--he followed his finger through the door, into the classroom, around the walls (right past the lecturer) and out the door again. People who knew Wiener tell m e - - a n d this comes through clearly in his autobiography as well--that he struggled all his life with feelings of inferiority. These feelings applied to non-mathematical as well as to mathematical activities. Thus when he played bridge at lunch with a group of friends he would invariably say, every time he bid or played, "Did I do the right thing? Was that a good play?" His partner, Norman Levinson, would patiently reassure him each time that he couldn't have done any better. It is not a well-known fact that Wiener wrote a novel. The villain in the novel was a thinly disguised version of R. Courant. The hero was a thinly disguised version of Wiener himself. Friends were successful in discouraging him from publication. [Another version of the story is that the villain was Osgood. In the book, he proves a theorem that is a thinly disguised version of a celebrated theorem of Osgood, but in a different branch of mathematics; he ends up dying in China.] Students liked to play pranks on Wiener. He read the newspaper every day at the same time in a certain lounge at M.I.T. As Wiener sat with the newspaper spread open before him, a student would sneak up and set the bottom of the paper afire. The results were spectacular, and the joke was repeated again and again. And sometimes Wiener played jokes on his stu38

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dents, though he did not realize that he was doing so. On one occasion, a student asked him how to solve a certain problem. Wiener thought for a moment and wrote d o w n the answer. The student hadn't really wanted the answer, but wanted the method to be explained (this really was a long time ago!). So he said "But isn't there some other way?" Wiener thought for another moment, smiled, and said "Yes there i s " - and he wrote down the answer a second time. Probably the most famous Wiener story concerns a day w h e n the Wiener family was moving to a n e w home. Wiener's wife knew Norbert only too well. So on the night before, as well as the morning of, the moving day she reminded him over and over that they were moving. She wrote the new address for him on a slip of paper (the new house was just a few blocks away), gave him the new keys, and took away his old keys. Wiener dutifully put the new address and keys into his pocket and left for work. During the course of the day, Wiener's thoughts were elsewhere. At one point somebody asked him a mathematical question, and Wiener gave him the answer on the back of the slip of paper his wife had given him. So much for the new address! At the end of the day Wiener, as was his habit, walked h o m e - - t o his old house. He was puzzled to find n o b o d y home. Looking t h r o u g h the window, he could see no furnishings. Panic took over w h e n he discovered that his key would not fit the lock. Wild-eyed, he began alternately to bang on the door and to run around in the yard. Then he spotted a child coming down the street. He ran up to her and cried "Little girl, I'm very upset. My family has disappeared and my key won't fit in the lock." She replied, "Yes, daddy. Mommy sent me for you." My final Wiener story, indeed my final story, does not seem to be well known. Even inveterate Wienerologists proclaim it too good to be true. But it's not too good for this article. I believe that I heard it when I was a graduate student at Princeton. As I've mentioned, Wiener was quite a celebrated figure on the M.I.T. campus. Therefore, when one of his students spied Wiener in the post office, the student wanted to introduce himself to the famous professor. After all, how many M.I.T. students could say that they had actually shaken the hand of Norbert Wiener? However, the student wasn't sure h o w to approach the man. The problem was aggravated by the fact that Wiener was pacing back and forth, deeply lost in thought. Were the student to interrupt Wiener, who knows what profound idea might be lost? Still, the student screwed up his courage and approached the great man. " G o o d morning, Professor Wiener," he said. The professor looked up, struck his forehead, and said "That's it: Wiener!" Department of Mathematics Washington University St. Louis, MO 63130 USA

Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the card 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.

Pavel Samuilovi?: Urysohn F. Palmeira and M. A. Sh ubin Batz-sur-Mer is a small town by the Atlantic Ocean in northwestern France in the region called Bretagne. To find Batz one proceeds by car or train from Nantes 70 kilometers southwest. In 1924, the young Russian topologist P. Urysohn drowned in Batz. One can find his grave in the old cemetery (there are two in Batz). The inscription on the tombstone in French reads: Paul Urysohn, mathematician, creator of theories of general topology, Professor at the University in Moscow, born in Odessa on 3 February 1898, drowned while swimming on 17 August 1924, at 26 years of age, in Batz. The Hebrew inscription is formed by the first five letters of a religious phrase that can be translated as: "May his soul be bound up in the bond of everlasting life." The Russian Editor's a d d r e s s : Mathematics Institute, University of Warwick, Coventry CV4 7AL England.

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inscription at the bottom is just his full name: "Pavel Samuilovi~ Urysohn." Urysohn's name is familiar to anyone who has taken a course in general topology. The following separation theorem is known as Urysohn's Lemma: Given two disjoint closed sets in a normal (T4) topological space, there exists a continuous real-valued function that is 0 on one of the sets and 1 on the other. Urysohn's grave is easy to find. Turn right at the entrance of the cemetery and follow the wail. Turn left at the corner, keep following the wall, and you will find the grave on your right. Notice how well kept it is from the picture taken by M. A. Shubin. F. Palmeira Department of Mathematics Pontificia Universidade Cat6lica do Rio de Janeiro Rio de Janeiro, Brasil

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M. A. Shubin Department of Mathematics and Mechanics Moscow State University Moscow, USSR

Janusz Onyszkiewicz Krzysztof Ciesielski and Zdzislaw Pogoda

Janusz Onyszkiewicz: the mathematician who was said to build differential equations of the highest degrees. We conducted an experiment. On a sheet of paper we p u t t h e n a m e s of five Polish m a t h e m a t i c i a n s : Stanis•aw Ulam, Janusz Onyszkiewicz, W a d a w Sierpifiski, Kazimierz Kuratowski, and Stefan Banach and at a central point of the city of Krak6w we asked the passersby if they had heard these names. We asked 100 persons: 75 of them had heard about Onyszkiewicz, 8 about Banach, 6 about Sierpifiski, nobody about Ulam and Kuratowskl. If we asked the same question of any non-Polish mathematician, the correspondence of names and numbers would probably be reversed. In Poland, however, Janusz Onyszkiewicz has been the best known mathematician for about 10 years. This sounds very surprising. Who is this mysterious mathematician? Why is he known so well? The answer is simple. Onyszkiewicz is famous all over Poland, but not because he is a mathematician. He is famous because of his political and social activities. Since 1980 he has been one of the leaders of Solidarity. After the strikes of Polish workers in August 1980 there was created in Poland a free, independent trade union (the first since the Second World War). The leader of this u n i o n (called Solidarity) was Lech W ~ s a . Dr. Janusz Onyszkiewicz, a member of the staff of Warsaw University, began active work in this organization. He was the official spokesman of the Warsaw Branch of Solidarity and since spring 1981 the official spokesman of the whole union. No w o n d e r that he appeared frequently on television and radio and that his statements and declarations were often quoted in newspapers. At the first congress of Solidarity, in fall 1981, he was elected to the highest body of the organization. 40

The introduction of martial law in December 1981 broke the development of democracy in Poland. Many persons actively engaged in the work of Solidarity, including Lech WaX~sa, were arrested and sent to internment camps. This happened to Onyszkiewicz as well. During martial law, many Polish newspapers ceased to appear. Only some were being published, prepared by carefully selected journalists. One of the main subjects of articles was "exposing the true intentions of Solidarity" and the "obnoxious characters of the union's leaders." Of course, Onyszkiewicz was not overlooked. In particular, there was an article about Onyszkiewicz published on 8 January 1982 in the daily newspaper Zolnierz Wolnodci ("A Soldier of Freedom"), which quickly became famous among Polish mathematicians. It was written in a very unpleasant tone and contained many false accusations, so it shouldn't

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attract a n y o n e ' s attention. H o w e v e r , there was a fragm e n t definitely w o r t h mentioning. Let us quote: He worked for the anti-Polish and anti-workers political mafia . . . which used the deceitful name of 'Social SelfDefense Committee '1 . . . in which his precise mathematical brain played an important role. It was said that he could build differential equations of the highest 2 degrees for the use of the Committee, in which he even outdid the work of the artful Modzelewski. 3 (Yes, w e do k n o w that a differential equation usually has an order, not a degree, but the a n o n y m o u s author of the article appears not to have consulted any mathematicians). This sentence was popular a m o n g mathematicians all over Poland. The Chairman of the Differential Equations D e p a r t m e n t at one of the Polish universities joked that he was going to change the n a m e of the d e p a r t m e n t to " D e p a r t m e n t of Differential Equations of Low O r d e r . " Released from i n t e r n m e n t , 4 0 n y s z k i e w i c z d i d n ' t stop his activity for the people's rights in Poland, alt h o u g h Solidarity h a d been officially b a n n e d . Newspapers published slanderous allegations against him; he w o u l d be s t o p p e d b y the police. There was even a joke: " W h a t is a sufficient c o n d i t i o n for b e i n g arrested? To appear in a Warsaw street on the anniversary of the introduction of martial law and be n a m e d Onyszkiewicz." In spring 1989 Onyszkiewicz started to a p p e a r freq u e n t l y on television again. The Polish authorities b e g a n to n e g o t i a t e w i t h Solidarity a n d the official s p o k e s m a n of the trade u n i o n could again present his views on television (which he did with "mathematical precision"). As a result of these negotiations "partially free" elections took place in Poland in June 1989. In the L o w e r H o u s e of the Parliament, 65 percent of the seats were reserved for m e m b e r s of the ruling party and its allies, 35 percent were left to free election. The result surpassed all expectations. All the seats of this 35 percent were taken by candidates from Solidarity; all of t h e m w o n the election! Onyszkiewicz was a candidate in the region of Przemy~l and garnered 82.79 percent of its votes, so he became a m e m b e r of the Polish Parliament. In N o v e m b e r 1989 he accompanied 1 Social Self-DefenseCommittee (KOR) was an underground organisation created in Poland in 1976 for the purpose of defence of the workers persecuted and fired by the government after the strikes in Poland in 1976. 2 Yes, not "high," but "the highest." 3 Karol Modzelewski was the officialspokesman of Solidarity before Onyszkiewicz. 4 We heard from Onyszkiewicz the following story. In the internment camp his mathematical notebook was confiscated, because, as he was told, "it contained illegal numbers." Subsequently Onyszkiewicz sent a secret message to his friends, mathematicians working on number theory, with the advice to hide their notes-who knows which numbers can be regarded as illegal by the secret service?

Janusz Onyszkiewicz with Lech Wal~sa. Lech W a ~ s a during his visit to the United States. T h e s e are the r e a s o n s w h y O n y s z k i e w i c z is so famous in Poland. H o w e v e r , he is well k n o w n also as a mathematician! Almost all students of mathematics k n o w his n a m e for reasons other than political. T h e r e are m a n y m a t h e m a t i c a l b o o k s b u t not so m a n y problem books (especially good problem books i n t e n d e d for students). Janusz Onyszkiewicz is the author (together with Wiktor Marek) of Elementy logiki i teorii mnogodci w zadaniach ("Problems in Logic and Set T h e o r y " ) for students of mathematics. This is one of the best m a t h e m a t i c a l p r o b l e m books p u b l i s h e d in Polish. It contains m a n y easy and basic exercises, as well as m o r e advanced and difficult problems, suitable for those w h o are going to specialize in foundations. The b o o k has had four editions (the most recent one in 1978). In the first edition (1972) there were m a n y misprints a n d small mistakes, which were corrected in the next edition. We heard of an anecdote that w h e n the first edition of the book a p p e a r e d in shops, the authors p u t a note on the door of their office at the university, saying: " A coffee is offered to a n y o n e w h o finds a mistake in the b o o k . " O n the same day the w o r d "coffee" was replaced by "tea. ''5 J a n u s z O n y s z k i e w i c z (born in 1937 in Lw6w, the city of Stefan Banach) w o n an a w a r d in the Polish Mathematical Olympic Games for students 6 in 1955. 5 In Poland tea was in those years much cheaper than coffee. 60rganised each year since 1949, for boys and girls (age 16-19) from secondary schools, not yet specializing in any subject. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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Janusz Onyszkiewicz (center) during a press conference. On the left is Tadeusz Mazowiecki (now the Polish Prime Minister).

The books written by Janusz Onyszkiewicz: Elements of Set Theory and the first and the second editions of Problems in Logic and Set Theory.

Gasherbrum III in the Karakoram, (second on the left), first climbed by the group: W. Rutkiewicz, A. Chadwick-Onyszkiewicz, J. Onyszkiewicz, K. Zdzitowiecki.

After his studies he began work at the Mathematical Institute of Warsaw University. His area of research is logic and foundations (which is rather far from differential equations). His Ph.D. thesis, written under the supervision of Andrzej Mostowski, was presented in 1967. Note that he is also the author of many articles and of another book: Wiadomo~ci z teorii zbior6w ("Elements of Set Theory"), published in 1972, written for secondary-school students who are going to study a subject connected with mathematics. 42 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990

People who have watched Onyszkiewicz on television in recent years probably do not realize that he appeared on it twenty years ago. At that time there was a very popular quiz show, "20 Questions." Two teams had to guess an unknown word (or name) within 20 questions, with only "yes" or "no" answers. If both teams guessed correctly, the quicker one was the winner. When the team of Warsaw mathematicians started playing in the game, the program had to be discontinued after a short time. Why? The quiz made no sense any more, the mathematicians would always win. They introduced the optimal way of asking the questions, called Metys ("M6tis"). The method is very simple, we will show it using an example. Let x be an u n k n o w n word, n be a number of entries in an encyclopedic dictionary. "Is x in the dictionary before word n u m b e r n/2? Yes. Before n u m b e r "/4? No. Before number 3%?- And so on. Of course, it was necessary to p r e p a r e a special list of w o r d s with suitable numbers. Metys is the word that was exactly at the midpoint of one Polish dictionary. It is easy to prove that such a method is really the most effective. In this w a y the team of mathematicians (J. Onyszkiewicz, A. Pedczyfiski, 7 W. Szlenk, R. Bartoszyfiski) could beat anyone, which put an end to the program. Onyszkiewicz is also an excellent mountaineer and speleologist. He has climbed many difficult mountain faces and peaks, even in the Himalayas. In particular, he was one of the four climbers who first ascended Gasherbrum III (7952 meters), the world's fifteenth highest peak, during an expedition in 1975. (From 1964 to the time of this expedition Gasherbrum III was the highest unclimbed peak in the world.) Among his expeditions there was also one (led by him) that conquered Pik Komunisma (The Communism Peak), the highest peak in the Soviet Union, in 1974. Finally, let us note that when we asked people in Krak6w about the names in our list, these who knew of Onyszkiewicz spoke about him in friendly terms. He is not only famous, but also popular, liked, and admired. Postscript. An important development has occurred since this article was submitted for publication: In April 1990 Onyszkiewicz became Vice-Minister of National Defense. This is the first time since World War II that a civilian has attained such a high position in this Ministry in Poland. Could the journalist of "A Soldier of Freedom" have predicted it?

Jagiellonian University Mathematics Institute Reymonta 4, Krak6w, Poland 7 The same mathematician w h o presented a plenary lecture at the International Congress of Mathematicians in 1983.

Mathematical Motifs on Greek Coins Benno Artmann

Sources concerning the history of Greek mathematics from the time before Euclid are sparse; all that we have are some fragments from the History of Mathematics written by Eudaemos, a pupil of Aristotle, which have been handed down by Proclus and others, and various remarks scattered throughout the works of the philosophers Plato and Aristotle. Therefore, any additional information obtained from other areas, however slight, is highly welcome. In this article I want to present some references to mathematics that appear on the coins of Greek cities. Of course these sources are solely pictorial, but I think in such obvious ways that little doubt remains. I would like to thank my colleague Andreas Mehl from the Department of Ancient History of the Technische Hochschule Darmstadt for many helpful discussions on philological and historical matters as well as Professor Maria R.-Alf61di for numismatic advice and for providing the pictures from her excellent collection.

Melos The coinage of the city (and island) of Melos in the Aegean sea shows some obvious mathematical influences. The obverse of the Melian coins shows an apple (Greek: ~ -q X o v), chosen as a pun on the island's name; on the reverse we sometimes find the inscription ,a4/,~ A I O t,/ ('of the Melians'). Two early coins (Kraay 124 from about 480 B.C. and Kraay 128 from 450 B.c.) are struck on the reverse with a diagonally divided square or a similarly divided circle.

General Remarks The obverse side of Greek coins from archaic and classical times generally bears the sign or emblem of the issuing city. The pictures that interest us are invariably on the reverse side. For all general matters on antique coins (techniques, values, etc.) one should consult one of the standard books by R.-Alf61di, Kraay, or Seltman that are listed in the references. Coins are identified by their numbers in the respective publications, i.e., 'Kraay [Number] . . . . ' All dates for coins are clearly meant to be B.C. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4 9 1990 Springer-Verlag New York

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Figure 1. Kraay 124 (left) and Kraay 128 (obverse and reverse).

Figure 3. Old Babylonian mathematical text.

Figure 4. Kraay M3.

A single hoard of more than eighty coins, which has been thoroughly analyzed by Kraay [1964], was discovered on the island in 1907. Kraay concludes that all of these coins were minted in about 426-416. (Melos was destroyed by the Athenians in the year 416 B.C.) In order to distinguish the numbering I will use the letter M to indicate the numbers used in Kraay's paper [1964] on the Melos hoard as opposed to the numbers without letters from Kraay's book [1976]. The coins M21, M33, M34 show a square subdivided into eight triangles (see Figure 2). This is exactly the same geometric diagram described by Plato in his 44

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Figure 2. Kraay M33.

Menon 82b-85b, where he explains the doubling of a square. This component of elementary instruction in geometry seems to have been so popular that it found its way into the decoration of coins. The diagram itself is age-old. It can already be seen on old Babylonian cuneiform tablets like BM [British Museum] 15285 (Obverse, cf. Neugebauer vol. 1, p. 137, and vol. 2, table 3). The picture is taken from Hoyrup [1987, p. 68a]. The generation of a square from four isosceles right triangles is described a second time by Plato in his Timaeus 55b, where he talks about the cube as a regular polyhedron. The precisely formed pentagram on M3 is apparently of mathematical origin. The diagonals of the regular pentagon play a key role in the construction of the pentagon, the important theorem being Euclid XIII, 8. In his earlier construction in the fourth book, Euclid conceals this by proceeding without any motivation. Furthermore, the pentagram seems to have been a symbol of the Pythagoreans as stated by the second century A.D. writer Lucan (De lapsu in salutando 5, and Scholium to Aristophanes' Nubes, 609; for details see van der Waerden [1979], p. 179). As far as I know, this is the only exactly drawn pentagram from the classical Greek period (about 500-300 B.C.). We will come back to this symbol when we talk about the coins from Metapontion. The mathematical inclination of the die cutter(s) of Melos 430-416 manifests itself in another way. In the hoard we find coins with two-, three-, four-, five-, six-, seven-, eight-fold symmetry. Of the 46 different types of coins listed by Kraay, thirty-one present a symmetrically designed reverse. We list them according to their symmetry.

2-Symmetry

Crescent (= half moon) (Kraay M7-M12), executed in various ways. Bilateral symmetry is natural to many motifs and is found on several other coins (M6, M23, M41, M44, M45). It may have been designed without any special mathematical intention (see Figure 5).

Figure 5. Kraay M8.

Figure 6. Kraay M42.

Figure 7. Kraay M36.

Figure 8. Kraay M15.

Figure 9. Kraay M19.

Figure 10. Kraay M20. Figure 11. Kraay M22. Figure 12. Kraay M39. Figure 13. Kraay M17. Figure 14. Kraay M46.

3-Symmetry

5-Symmetry

(a) Three dolphins swimming around the central boss (Kraay M4, M42, M43) (see Figure 6).

Here we have the coin M3 with the pentagram (see Figure 4).

(b) 'Triskeles,' three h u m a n legs (Kraay M29, M35, M36).

6-Symmetry Similar to M22, three diameters divide the circle into six equal sectors (Kraay M39, M40) (see Figure 12).

4-Symmetry 7-Symmetry (a) Wheel with four spokes (Kraay M13, M14, M15, M16). This motif is c o m m o n on Greek coins, see below under Chalcis and Phlius (see Figure 8). (b) Four grains of barley arranged in the form of a star (Kraay M19) (see Figure 9). This design calls to mind a similar coin with five grains of barley from Metapontion, to be discussed below.

Flower with seven petals (Kraay M17) (see Figure 13). 8-Symmetry (a) Flower with eight petals (Kraay M24, no picture). (b) Eight-pointed star (Kraay M46) (see Figure 14).

(c) Ornamented four-star (Kraay M20) (see Figure 10). (d) Circular incuse divided into four segments by thick bands, a simple cross (Kraay M22). This coin should be compared with the six-part symmetric coin M39 below. The die cutter's mathematical intention is especially evident in the reduced design (see Figure 11). (e) The 'Menon-design,' a square divided into four raised and four recessed triangles (Kraay M21, M33, M34) (see Figure 2).

Naturally one wonders how it came to be that just the island of Melos had such strongly mathematically oriented designs on its coinage. The following might be an explanation: According to Kraay [1964], pp. 16 and 19, the coins in the Melos hoard were struck within a short period before 416 in order to finance the defense of Melos against the Athenians. We will see a faint similarity between the Melos coins and some from Metapontion. Perhaps, in a state of emergency, the THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

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Figure 15. Kraay 595 (obverse and reverse) and BMC Italy 45.

Melians hired a die cutter from Metapontion for the rapid minting of their silver and let him follow his mathematical interests with the design of the reverses, which at any rate was in accord with their previous coinage. The fourth book of Euclid's Elements contains the mathematical theory of the division of the circle into equal parts, that is, the construction of regular polygons. According to a scholium, the theorems of Book IV have their origin with the Pythagoreans. (For a detailed discussion see N e u e n s c h w a n d e r [1972].) This directs us once more towards Mentapontion, where Pythagoras is said to have died. Metapontion

The coinages from the Greek mainland and from Sicily have a well-established chronology, achieved by placing the coins in the context of the well-known histories of Athens and Syracuse. A similar scale is lacking in southern Italy (Magna Graecia), so that much of the numismatic chronology tends to be vague and tentative (Kraay [1974], p. 161). Seltman credits Pythagoras personally with the introduction of the art of minting coins in Magna Graecia, especially with the so-called incuse technique, whereby the relief design on the obverse is repeated intaglio on the reverse (Seltman, p. 78). This is not completely unlikely, because Pythagoras's father Mnesarchos is said by Diogenes Laertius (De vita et moribus philosophorum VIII 1, written c. 250 A.D.) to have been a gem engraver, but further arguments are vague, to say the least (cf. R.-Alf61di, Vol. I, p. 93). Pythagoras left the island of Samos about 530 B.C. (date very uncertain) and went to Magna Graecia, first to the city of Croton and later to Metapontion, where he established his school of followers. (For Pythagoras's biography and his school, see Burkert [1962] and van der Waerden [1979].) The influence of his school may show up on some later coins. The pentagram as a symbol of the Pythagoreans has already been mentioned. Figure 15 depicts two coins with a striking five-fold symmetry from Metapontion. 46

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Figure 16. Kraay 705 from Velia c. 320.

The coin on the left is dated by Kraay at about 440 B.C. It has on its obverse the usual ear of barley of Metapontion together with the inscription META and on the reverse, barley grains arranged in the form of a five-pointed star. A similar design is found on the reverse of the coin BMC [British Museum Collection] Italy, Metapontion 45, which is overstruck on an earlier coin of different origin and where a little crescent is added as a marker. It is dated at about 440 B.C., a little earlier than the coins of Melos (about 420). In Melos the crescent is prominent on some coins; the coin Kraay M19 from Melos has four grains placed in the form of a star on the reverse that is similar to the Metapontion coins. As mentioned above, it is possible that the die engraver travelled from Metapontion to Melos. (That the grains are of barley is irrelevant for this hypothesis; barley has religious significance, cf. Burkert [1977], p. 102.) The P y t h a g o r e a n s lost their political p o w e r in Magna Graecia sometime b e t w e e n 460-440, which agrees with the likewise uncertain date 440 of the coin. (For the political history of the Pythagoreans, see again the books of Burkert and van der Waerden.) Small pentagrams as subsidiary symbols or markers can be found on many later coins. A typical example is Kraay 705 from Velia (= Elea) in Magna Graecia from about 320 (Figure 16). The catalogue of the British Mus e u m Collection lists six other coins from various places in Italy and two from Sicily with little pentagrams on them. There are a few more from other regions, but on the whole we see a clustering in Magna Graecia. These may be late traces of Pythagorean influence, but it might well have been the other way around inasmuch as the Pythagoreans made their acquaintance with the pentagram and dodecahedron through the Etruscans. There are nicely crafted regular d o d e c a h e d r a of Etruscan origin in the archeological museum in Perugia and numerous dodecahedra dating A.D. from all over w e s t e r n Europe, with a certain geographical center in the western Alps. [For a detailed historical analysis see Herz-Fischler [1987], especially Chapter III, pp. 52-62.]

Aegina

Until about 480 B.C. the city (and island) of Aegina was the dominant trading post in the Aegean Sea. The Aeginetans were the first ones in this region who minted coins. After 480 they received more competition from the Athenians and finally were expelled from their island in 431. After the end of the Peloponnesian War in 404, they returned and resumed issuing coins, but the people were unable to regain their former importance. The obverse of the coins from Aegina is always presented with a turtle or tortoise; as usual we are interested in the reverse pattern (Figure 17). Until 480 B.C. we have variously patterned punches which are derived from a square subdivided by its symmetry axes. From about 480-431 we see a formalized skew pattern always done in the same way (Kraay 123 and 127). The n e w coins after 404 B.c. (Kraay 137 and 138) have a mathematical, rectangular design. What we now see is precisely the diagram of Euclid's Theorem II.4; just the diagonal in the smaller square is missing (Figure 18 and 19). Euclid II.4 is the geometrical version of the binomial theorem (a + b)2 = a 2 + 2ab + b2. Similarly subdivided rectangles and parallelograms abound in the Elements; the so-called 'Gnomon' is a familiar tool in geometrical (and arithmetical) proofs. The missing diagonal in the small square could be a means to emphasize the gnomon (Figure 20). We know that Hippocrates of Chios was the first one to write Elements of mathematics, about 430 B.c. (in Athens?). It seems clear to me that the redesigning of the reverse of the Aeginetan coins was done under the influence of the newly established science of mathematics.

Figure 17. From Aegina, c. 580-540 B.C.

Figure 18. c. 500-431 (left) and after 404 (Franke-Hirmer, 335 and 336).

Figure 19. From Euclid.

Figure 20. Gnomon.

Figure 21. Chalcis c. 480 (FrankeHirmer).

Figure 22. Kraay 265 and 266 (top); also Seltman, Plate IV, 16 and 17 (bottom).

Figure 23. From Bo~thius's translation of the Elements.

Chalcis

Chalcis is one of the principal cities on the island of Euboea. According to Kraay [1974], pp. 89-90 and Chantraine [1958], the mathematically interesting coinage of Chalcis was minted in the period 530-470, which is considerably earlier than the coins we have studied so far. The obverse is an eagle or, from about 480 onwards, an eagle carrying a snake in its talons. On the reverse "appears a wheel of either five or four 48

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

or three spokes and the letters ' f ~ A , all this in a shallow incuse triangle or incuse square" (Seltman, p. 54, similarly Kraay [1974], p. 90). The first wheels of Chalcis (c. 515) may have been abbreviations for the earlier quadriga (a four-horse chariot) on the reverses. Kraay 264 clearly shows the physical wheel of a chariot, not a mathematical abstraction. The situation is very different from the next coins from 485 on. As in Figure 21, we often find a circle inscribed in a triangle or square and the subdivision of the circle according to the regular triangle, quadrilateral, or pentagon just as is presented in Euclid's fourth book. The design of the reverse die as an equilateral (Kraay 265) or right (Seltman, Plate IV, 16 and 17) triangle (Figure 22) can hardly be understood without considering mathematical motives. A geometrical diagram like the one on Kraay 265 (equilateral triangle with inscribed circle, the circle subdivided by two perpendicular diameters) surfaces again in a medieval manuscript, namely the so-called pseudo-Boethian "Geometria II" (see Folkerts [1970], p. 224, Fig. 116) (Figure 23). This diagram belongs to the parts of Geometria II which come from Boethius's translation of Euclid's Elements (from Greek to Latin, done about 505 A.D., Folkerts, p. 72), more precisely to Euclid Book IV, Proposition 3: "About a given circle to circumscribe a triangle e q u i a n g u l a r with a given triangle." Euclid himself has a " b e t t e r " (i.e., better suited to the problem and its solution) diagram. Some other diagrams of the Geometria II are exactly the same as in Euclid's Elements, e.g., the rather complicated ones belonging to Euclid II, 9-10 (Folkerts, p. 223, 100 and 101). If we accept that the diagrams of Geometria//--at least for the parts that belong to Boethius's translation of Euclid--go back to late antiquity, then we have a second example for the design on the Chalcidean coin Kraay 265 (Figure 22). But there is a gap of about one thousand years between the coin and Boethius's time. Could it be that in archaic diagrams the circle used to be d r a w n with two perpendicular diameters? This w o u l d correspond to the cuneiform tablet s h o w n above, to the diagrams in Euclid IV, 6-9 and many of the 'wheels' on Greek coins. Viewed from the history of mathematics, the years 530-470 B.C. are the time of Pythagoras and his immediate followers, whose interest in the pentagon has a l r e a d y been m e n t i o n e d . (The d o d e c a h e d r o n of Hippasos belongs to this t h e m e as well, see van der W a e r d e n [1979], Pythagoreer pp. 71-72 after Iamblichus' Vita Pythagoras 88 and 246-47~'; Iamblichus, a third century A.D. philosopher, gives ~i vague hint of the connection of later Pythagoreans with Chalcis, or at least its daughter cities, when he writes about late Pythagoreans: "The most important Pythagoreans have been Phanton, Echecrates, Polymnastes

Figure 24. Seltman, Plate XXVIII, 11. Figure 25. Wheels as circles or ellipses (Franke-Hirmer). and Diocles from Phlius and Xenophiles from the Thracian peninsula of Chalcidice" (Vita Pythagoras 251; the same words are found in Diogenes Laertius VIII, 46). Iamblichus is speaking about late Pythagoreans, surely later than 480. But anyway Chalcis was the mother city of several settlements on the Chalcidice. Below we will see some coins from Abdera (from about 430 B.C.), which give another hint to the presence of Pythagoreans in this part of the country. The city of Phlius on the Peloponnese, which is mentioned by Iamblichus as the home of some other late Pythagoreans, has, like Chalcis, a wheel (with four spokes) on the reverse of its coins from 420 on (Kraay 302-304). There are a few other coins listed by Kraay with wheels on the reverse; all of them seem to be singular (one from Athens c. 540, two Macedonian tribal, one from Tanagra in Boeotia). I have stressed here the connection with the Pythagoreans. It should be clear, however, that there were more mathematicians in early Greece than just the Pythagoreans. This point is made and elaborated by Burkert [1962] in his chapter on Pythagorean mathematics. Abdera The city of Abdera in Thrace and its Ionic mother city Teos have the same civic device of a griffin, which they show on the obverse of their coinage. A characteristic feature of the whole coinage of Abdera is the prominence given to the names of minters or magistrates. From about 470 these names appear in full on the reverses around the incuse square, which in turn is filled with varied deities, inanimate objects, animals, or other things. On one of them "the name Pythagoras is accompanied by a bearded head which can hardly be intended as anything but a representation of a famous philosopher from the preceding century" (Kraay [1974], p. 155, 535 from about 430; similarly Burkert [1962], p. 185, note 70 and May [1966], who praises the fine artistic quality of this coin.) Art historians may be skeptical with respect to the portrait of an historical person on a coin from the fifth

century B.c. Nevertheless, I quote extensively from Seltman, w h o emphasizes the connection with the well-known school of philosophy (Protagoras, Democritus) from Abdera: "Nor are we without links with the celebrated school of philosophy of which Democritus of Abdera, the atomist, was the leading light. His birth is usually placed at about 460 B.C., his name appears on a coin . . ." (Seltman, p. 143; the coin is not listed by Seltman, but there are four coins with the name of Democritus published by May 234-237 from about 415 B.c.). " D e m o c r i t u s , too, w a s a great admirer of Pythagoras [after Diogenes Laertius IX, 3 8 ] . . . and, philosophy being the fashion in Abdera, that one [magistrate] who bore the name Pythagoras should adopt as the emblem for his year in office an idealized portrait of his famous namesake. In no other way can we account for the remarkable coin which bears the fine bust of a bearded man and the name IIY@AFOPH~ around it" (Seltman, p. 144) (Figure 24). "This tetradrachm is a monument of first importance for the history of sculpture; it can be dated to before 432 B.c., and it is the only original surviving named portrait of the fifth century before our era" (Seltman, p. 144). Being executed about seventy years after the death of Pythagoras, this coin can hardly have been intended as a realistic portrait in the sense of today, which is clear as well from the two different coins. That it is not the portrait of a magistrate can be inferred from other Abderite coins; e.g., Kraay 530-542, where we find connections between name and symbol, but, except for this one, no portrait. Circle a n d Ellipse in Syracuse The Roman architect Vitruvius reports in the introduction to the seventh book of his De Architectura on books written by Anaxagoras and Democritus about some sort of perspective design of scenes for the theater. (For more information see van der Waerden [1966], Erw. Wiss., pp. 224-226.) The change in artistic representation from a direct view to an oblique representation can be dated on the coins from Syracuse. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

49

"Until about 425 B.c., Syracuse and all other mints which used the quadriga type had with rare exceptions shown [the horse] walking sedately as though from the winning post: thereafter the horses were always represented at the gallop as in the course of the race itself" (Kraay [1974], p. 221). In fact, on the latter coins the quadriga is shown in the most dramatic point of the race: the horses galloping full speed around the turning post and the wheels of the chariot s h o w n obliquely as ellipses. The r e p r e s e n t a t i o n changes from Kraay 807 (wheel as a circle) to Kraay 808 (as an ellipse) (see Figure 25). The dating of the coins harmonizes with the dates for Anaxagoras (c. 500-428) and Democritus (c. 460-380), so that in this case we see a parallel development of artistic practice and theoretical reflection. It is usually assumed to have been Menaechmos (c. 350 of Plato's A c a d e m y , after a r e m a r k by Eratosthenes) who first studied the conic sections. (For the specifics see van der Waerden [1966], Erw. Wiss., pp. 265-267 and 331-334 and the detailed discussion in Knorr [1986], chapters 3 and 4.) When we see ellipses in works of art some seventy years ahead of Menaechmos, we might assume that the mathematicians might have studied this figure earlier. In fact Democritus speculated about conic sections (van der Waerden, p. 228) in the context of his atomistic theories. But otherwise there are no sources in literature before Euclid, who lived around 300 B.c. In Euclid's Optics we find the following Proposition 36: "'The wheels of chariots appear sometimes circular, sometimes drawn in (awry)" (Euklid-Heiberg, Optica [1885], pp. 80-81). For the proof he uses a preceding lemma and adds that in the case of the oblique view one diameter appears of maximal length, another minimal, and the others are somehow in between. Not a single word about a conic (or cylindrical) section. On the other hand, it is obvious that we have an allusion to the artistic representation that we see on the coins: As the chariot suddenly changes course around the turning pole, we get the distorted view of the wheels. On another, mathematically equivalent occasion, Euclid brings together the sections of a cone and a cylinder: "If a cone or a cylinder be cut by a plane not parallel to the base, this section is a section of an acute-angled cone, which is like a shield" (Heath [1921], p. 439, quoted from Euclid's Phaenomena; the Phaenomena is a treatise on astronomy with many diagrams, where all the circles on a sphere are drawn wrongly as 'lentils,' not as ellipses, cf. Euklid-Heiberg, vol. VIII). Here 'like a shield' means elliptical; the terminology had not yet been established at Euclid's time. However, the wheels of the chariot lie in a plane "'parallel to the base" and they are circular, so the situation is somewhat different. Even in Theon's recension (around 380) of Euclid's Optics no word is said about the elliptical form of the wheels. That no effort 50

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

was made to describe the true form of the wheels is all the more curious, because Euclid himself wrote four books about conics. Contrary to a modern mathematician, who is proud to see a connection between different phenomena, Euclid keeps the sciences of optics and mathematics well apart and seems to be afraid of trespassing established boundaries. This leads to a general concluding remark about our observations. Unfortunately, we do not know very much about mathematics in the general cultural setting of ancient Greece. We see the coins--and generally the mathematics--of the Greek world with the eyes of today, and, as Goethe says, one sees what one knows.

References BMC (British Museum Collection): A Catalogue of the Greek Coins in the British Museum. Vol.: Italy, Vol: Sicily. W. Burkert, Weisheit und Wissenschaft. Studien zu Pythagoras, Philolaos und Platon, N~irnberg: Hans Carl (1962). (English translation: Lore and Science in Ancient Pythagoreanism, Cambridge, MA (1972).) , Griechische Religion, Stuttgart: Kohlhammer (1977). H. Chantraine. Zur M~inzpr~igung in Chalkis im 6./5. Jh., Jahrbuch fiir Numismatik und Geldgeschichte 9 (1958), 7-17. Euklid-Heiberg, Euktides Opera Omnia (Vol. 7: Optica, Vol. 8: Phaenomena, Leipzig: Teubner (1883f). M. Folkerts, "Boethius'" Geometrie II, Wiesbaden: Steiner (1970). P. R. Franke, and M. Hirmer, Die griechische M~inze, M(inchen: Hirmer (1964). Th. Heath, A History of Greek Mathematics, Oxford: Clarendon Press (1921). R. Herz-Fischler, A Mathematical History of Division in Extreme and Mean Ratio, Waterloo: Wilfred Laurier Univ. Press (1987). J. Hoyrup, Algebra and naive geometry. An investigation of some basic aspects of old ,babylonian mathematical thought, (194 pages) Roskilde Universitetscenter Preprint Nr. 2 (1987). W. R. Knorr, The Ancient Tradition of Geometric Problems, Basel, Boston: Birkh/iuser (1986). Colin M. Kraay, Archaic and Classical Greek Coins, Berkeley: University of Calif. Press (1976). , The Melos Hoard of 1907 Re-examined. The Numismatic Chronicle, VII series, vol. 4, 1964, p. 1-20, Plates I, II, III. I. M. F. May, The Coinage of Abdera. London: Roy. Numism. Soc. (1966). E. Neuenschwander, Die ersten vier Bficher der Elemente Euklids, Arch. Hist. Ex. Sci. 9 (1972), 325-380. O. Neugebauer, Mathematische Keilschrift-Texte, 2 Vols., Heidelberg: Springer (1935); Reprint 1973. Maria R.-Alf61di, Antike Numismatik, Teile I, II, Mainz: Ph. v. Zabem (1978). Charles Seltman, Greek Coins, London: Spink (1977; 1st. Edition 1933). B. L. van der Waerden, Erwachende Wissenschaft, Basel: Birkh/iuser (1966).(English translation: Science Awakening, Dordrecht: Kluwer (1975).) , Die Pythagoreer, Z(irich: Artemis (1979). Fachbereich Mathematik Technische Hochschule Darmstadt D-6100 Darmstadt, Federal Republic of Germany

How to Build Minimal Polyhedral Models of the Boy Surface Ulrich Brehm

Introduction and History In the middle of the last century A. M6bius gave a combinatorial description of a closed, one-sided, polyhedral surface, which was soon recognized as a topological model of the real projective plane. It also turned out to be the surface that J. Steiner had defined geometrically. Soon, algebraic definitions followed which were used to construct plaster models of the cross-cap and the Roman surfaces. Until the Klein bottle, none of these non-orientable closed surfaces, whether smooth or polyhedral, was known to be "immersible" in R 3. A topological immersion i : M ~ R 3 is a locally injective continuous mapping. An immersion i:M--* R 3 of a c o m p a c t 2 - m a n i f o l d ( w i t h o u t boundary) is called polyhedral if the image of i is contained in the union of finitely many planes. In 1903 D. Hilbert's student W. Boy proved in [3] that the real projective plane RP 2 allows an immersion in R 3 (with an axis of symmetry of order 3). Several efforts have been made to give an explicit description of such an immersion. A survey of explicit combinatorial, analytic, and algebraic descriptions of such immersions is included in F. Ap6ry's recent book on the subject [1]. In [5] polyhedral immersions of RP 2 with eighteen vertices were described. The polyhedral immersions given in [1] have even more vertices. In this paper the existence of symmetric polyhedral versions of the Boy surface with only nine vertices and ten facets is shown and an easy recipe for building cardboard models of these objects is given. Formal definitions of the terms "'vertex," "'facet," and "'edge" will be given near the end of the introduction. T. Banchoff showed in [2] that an immersion of RP 2 in R 3 in general position must have a triple point. A

polyhedral immersion can always be perturbat6d so that the vertices are in "very general" positioff(for example, with the coordinates of the vertices being algebraically independent). Thus we get the generic case with at least one triple point in the relative interior of three triangular facets. The intersection of any two of these triangles is a line segment which cannot contain a common vertex because a polyhedral immersion is injective in some neighborhood of any vertex. Thus the three triangles containing the triple point have together nine different vertices, so nine is a lower bound for the number of vertices of a polyhedral immersion of RP 2. We will show that this lower bound can indeed be attained.

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4 9 1990 Springer-Verlag New York

51

The nicest way to prove this is to construct a 3-dimensional model of the object wanted. This paper contains an easy recipe for building your own cardboard models of minimal polyhedral versions of the Boy surface. We also give the coordinates of the vertices together with the combinatorial structure. Using these data you can also construct a computer model.

i,

e

g h

a

d e

f

Figure 1. A net of a polyhedral immersion P1 of RP2. In Figure 2 and Figure 3 we show orthogonal projections of P1 in the direction of the axis of symmetry from "above" and from "below." We have indicated the self-intersection lines by dotted lines. Visible lines, dotted or solid, are drawn much thicker than invisible lines. For any two edges with intersecting projections we indicate which of the two edges is above the other one. e

b /Z

f a

1 Y

D e f i n i t i o n s . Let M be a compact 2-manifold (without boundary) and i : M ~ R 3 a polyhedral immersion; (a) a facet is a connected component of a non-empty set of the form int(i-l[H]) where H C R3 is a plane and int denotes the interior of a set; (b) a vertex is a point of M that is in the intersection of the closures of (at least) three facets; (c) the connected components of the set of points of M that are neither vertices nor contained in some facet are called edges. Thus if two vertices happen to be mapped onto the same point in R 3, they are still counted as different vertices. On the other hand, the intersection of (the relative interior of) the images of a facet and an edge is not regarded as an additional vertex. If each facet is a t o p o l o g i c a l o p e n disc, t h e n E u l e r ' s f o r m u l a f2 - fl + fo = x(M) holds, where f2, fl, f0 denote the numbers of facets, edges, vertices, respectively, and x(M) denotes the Euler characteristic of M. If no misunderstandings can occur we call the image of a vertex, edge, or facet also a vertex, edge, or facet, respectively. In particular, by coordinates of a vertex we mean always the coordinates of the image point in R 3. If M is triangulated such that i is piecewise linear, then the local injectivity of i has to be checked only in a neighborhood of each vertex of (the simplicial complex) M.

a

Figure 2. The orthogonal projection in the direction of the axis of symmetry from above.

Polyhedral Immersions of RP 2 with Nine Vertices and Ten Facets

We describe three combinatorially different symmetric polyhedral immersions P1, P2, P3 of RP 2 with nine vertices and ten facets. P1 has six quadrangular and four triangular facets, whereas each of P2 and P3 has three pentagonal and seven triangular facets. The coordinates of the vertices of P1 are

Y

a (-2,0,0) d (-1,2,1) g (-1,1,0) x w

"

~

b

i

~'*Z e

Figure 3. The orthogonal projection in the direction of the axis of symmetry from below. 52

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

b (0,-2,0) e (1,-1,2) h (0,-1,1)

c (0,0,-2) f (2,1,-1) i (1,0,-1)

In Figure I we give a net of our immersed polyhedron P1. The dotted lines indicate the self-intersection lines. The list of coordinates of the vertices of P1 shows that the mapping (x,y,z) --~ (z,x,y) is a rotation by 2"rr/3with axis R(1,1,1) inducing the permutation (a,b,c) (d,e,f) (g,h,i) of the vertices. Because this permutation induces an automorphism of the net (see Figure 1), P1

f

d

e

h

d

a

b

h

e

e

e

Y

i

f

d

b Z,,,,.

f

c

.,,..X

r

Figure 6. Orthogonal projection of the MObius strip from above,

Figure 7. Orthogonal projection of the MObius strip from below.

has an axis of symmetry of order 3. The cell-complex defined by Figure I (vertices and edges being identified in the obvious way) clearly is an RP 2 with f0 = 9, fl = 18, f2 = 10. In Figure 10 you can see some pictures of a cardboard model of P1. It is easy to check that Figure 2 is correct and that a,c,d,g are affinely d e p e n d e n t and g - a = f - i. With the symmetry this implies that P1 is indeed an immersed polyhedron with the combinatorial structure given in Figure 1. So we get the following result.

Next we describe two minimal polyhedral versions of the Boy surface containing three pentagonal facets. The coordinates of the vertices of P2 and of P3 are

is a symmetric polyhedral immersion of nine vertices, eighteen edges, and ten facets, six of which are quadrangles. THEOREM 1:P1 R P 2 into R 3 with

a (0,1,-1) d (2,2,0) g (1,1,0)

b (-1,0,1) e (0,2,2) h (0,1,1)

c (1,-1,0) f (2,0,2) i (1,0,1)

In Figure 4 and Figure 5 we give nets of our immersed polyhedra P2 and P3. The dotted lines indicate the self-intersection lines. The list of coordinates of the vertices shows that the mapping (x,y,z) ~ (z,x,y) is a rotation by 2"rr/3 with axis R(1,1,1) inducing the permutation (a,b,c) (d,e,f) (g,h,i) of the vertices. Because this permutation induces an automorphism of the nets THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

53

f Figure 8. Part of P1.

a

P

Figure 9. Part of P3. 54

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

Figure 10. A model of P1.

Figure 11. A model of P3.

(see Figure 4 and Figure 5), P2 and P3 have an axis of symmetry of order 3. The cell-complexes defined by Figure 4 and Figure 5 (vertices and edges being identified in the obvious way) clearly are Rp2's with f0 = 9, f~ = 18, f2 = 10. P2 and P3 are combinatorially different because P3 contains 6-valent vertices, whereas P2 does not contain such vertices. Note that the M6bius strips arising from P2 and P3 when omitting the four triangles that do not contain the triple point are identical. In Figure 6 and Figure 7 we show orthogonal projections of this M6bius strip in the direction of the axis of symmetry from "above" and from "below." We have indicated the self-intersection lines by dotted lines. Visible lines, dotted or solid, are drawn much thicker than invisible lines. Note that the three pentagons form a symmetric poly-

Figure 12. A model of P2.

hedral M6bius strip (without self-intersections). In Figure 11 and Figure 12 you can see some pictures of P3 and P2, respectively. It is easy to check that Figure 6 is correct, that a,d,g,f,i are affinely dependent, and that g lies in the interior of the convex hull of the vertices. With the symmetry this implies that P2 and P3 are indeed immersed polyhedra with the combinatorial structure given in Figure 4 and Figure 5, respectively. So we get the following result. THEOREM 2 : P 2 and P3 are combinatorially different polyhedral immersions of RP 2 into R 3 with nine vertices,

eighteen edges, and ten facets, three of which are pentagons. N o w let us modify P1 by adding a new vertex j = ( - 2, - 2 , - 2) and replacing the triangle abc by the triangles abj, acj, bcj. Because abj and abh are coplanar, we can omit the edge ab and get a non-convex pentagon THE MATHEMATICAL INTELL1GENCER VOL. 12, NO. 4, 1990 5 5

aehbj. Symmetrically we omit the edges ac and bc. Thus

we get a polyhedral immersion P I ' with f0 = 10, h = 18, f2 = 9. Similarly, we can modify P2 and P3, getting combinatorially different polyhedral immersions P2' and P3' with f0 = 10, h = 18, f2 = 9. Thus we have shown: THEOREM 3: There exist symmetric polyhedral immersions of R P 2 into R 3 with ten vertices, eighteen edges, and nine facets. REMARKS: 1) One gets a symmetric modification of P1 such that the quadrangle a c d g is convex if a,b,c,d,e,f are chosen as the vertices of a regular octahedron with diagonals af, bg, ch and g,h,e are chosen as the midpoints of the edges ad, be, cf, respectively, and the quadrangle a g f i is split into the triangles a g i and f g i (splitting c i e h, d g b h similarly). 2) A triangulation of the M6bius strip with 9 vertices, whose boundary forms a triangle and whose automorphism group has order 6, but which cannot be immersed in R 3 was first described by the author in [41. 3) In order to build nice models I suggest cutting a circular " w i n d o w " into the regular triangle and cutting curved " w i n d o w s " into the triangles containing the triple point, such that no material self-intersections occur, but such that the full self-intersection figure, and in particular the triple point, are still visible (and marked by lines on the model on both sides). In Figure 8 we show the central part of the net of P1 and a third of the net of the self-intersecting part of P1 (cf. Figure 1) indicating the self-intersection lines and

the suggested windows. The lengths of the line segments areab = ac = bc = 2 V ~ , a g = dg = bh = eh = ci = fi = X/2, ad = be =,__s = gi = X/-6, fg = ai = ei = V ' - ~ , ae = bf = cd = V14, gp = ir = qt = st = 2 X / ~ , tr = tu = iq = ~ X / - ~ .

In Figure 9 we show the central part of the net of P3 and a third of the net of the MObius strip which P2 and P3 have in common (cf. Figure 5) indicating the selfintersection lines and the suggested windows. The lengths of the line segments are be = cf = ad = ai = di = ei=fg= V~,ag=dg=gi=fi= V~,ae= bf=cd = V~,de=df=eef=2V2, gp=ir=qt=st= 89 tr = tu = iq = 2V6, (ab = ac = bc = V 6 for P2). Acknowledgment: I wish to thank D. Ferus for taking

the photos of the models (Figures 10, 11, 12) and B. Morin, U. Pinkall, D. Ferus, and E. Tjaden for helpful discussions.

References 1. F. ApOry, Models of the real projective plane, Braunschweig: Vieweg (1987). 2. T. Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974), 407-413. 3. W. Boy, Ober die Curvatura integra und die Topologie geschlossener F1/ichen, Math. Ann. 57 (1903), 151-184. 4. U. Brehm, A non-polyhedral MObius strip, Proc. Amer. Math. Soc. 89 (1983), 519-522. 5. K. Merz, P. Humbert, Einseitige Polyeder nach Boy, Comm. Math. Helv. 14 (1941-42), 134-140. FB 3--Mathematik Technische Universitdt Berlin StraJJe des 17. Juni 136 1000 Berlin 12, Federal Republic of Germany

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Electrically Active Cells and Singular Perturbation Theory Jane Cronin

Electrically active biological cells must function with great precision, and consequently their study requires careful quantitative scrutiny. The first cell thus considered was the squid giant axon, which was investigated experimentally and modeled mathematically in the Nobel prize-winning work of A. L. Hodgkin and A. F. Huxley [9]. We will describe two models of electrically active cells: the Hodgkin-Huxley model of the squid axon and a model of the cardiac Purkinje fiber due to D. Noble [15]. For descriptions of other such models, see J. Cronin [3], Chapter 4. We summarize some arguments for regarding these models as singularly perturbed equations, describe the singular perturbation theory that can be used, and indicate some further developments of the singular perturbation theory. Singular perturbation theory is also applicable elsewhere: e.g., in other biological models, models in chemistry, and control theory. First we describe briefly the kind of singularly perturbed equations that occur in this work. We consider the initial value problem for systems of the form: dx dt

The Hodgkin-Huxley model of the squid axon consists of the four equations: dV

1 -

dt

[I -

gNa m3 h ( V -

VNa )

CM -- XK n4( v -

-

-- V~) V

)l

dm dt

-

O~m(1 -- m) -- ~mm (H-H)

dh dt -

~

-

h) -

~h h

dyl dt -

cxn(1 - n) - ~,n

1 -

f(x,y)

(s) dy_

d-7 - g(x,y)

where x and f are m-vectors, y and g are n-vectors, f and g are "well-behaved" (e.g., continuous second derivatives in all variables), and e is a small positive parameter. The basic problem is to utilize the smallness of e to obtain information about solutions of (S). THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4 9 1990 Springer-Verlag New York

57

where V is the potential difference across the surface of the axon; m and h measure the flow of sodium ions across the axon surface, n measures the flow of potassium ions across the axon surface, and t represents time. The terms C M, I, gnu, VNa, gK, VK, gl, V1 a r e constants, and the expressions o~m, Bm, ~h, fBh, OLn,Bh are well-behaved, messy functions of V. A typical such function is: (0.1) (V + 25) exp[(V + 25)/10] - 1"

C~m

(For a complete description of the Hodgkin-Huxley equations and a description of how they were obtained from the experimental data, see Cronin [3], Chapter 2.) The numerical analyses of (H-H) carried out by Hodgkin and Huxley [9] yielded results that agreed very well with a number of quantitative laboratory observations (see Cronin [3], pages 57-65.) These results caused the Hodgkin-Huxley theory to be characterized as "a spectacularly successful example of such a [phenomenological] model" (Rubinow [16], page vii). The success of this model calls for a qualitative study of the system (H-H). The first such study, undertaken by R. FitzHugh [7], consisted of replacing (H-H) with a 2-dimensional "caricature," now wellknown as the FitzHugh-Nagumo equation. FitzHugh recognized that the caricature equation was singularly perturbed. Indeed Figure 1 of his paper is the usual picture drawn for a prototype singularly perturbed system. Later, FitzHugh [8], page 27, also pointed out that the system (H-H) can be regarded as a singularly perturbed system. That is, taking into account the values of the physiological constants that appear in (H-H), one can, with a simple change of variables, recast (H-H) in the form: dV

--

dt

-

dm dt

1

- F ( V , m , h, n )

9

1 -

9

M(V,

m)

(H~)

dh d---t = H(V, h) dn

dt - N ( V , n)

where F, M, H, N are well-behaved functions of the same order of magnitude and ~ is a small positive number. Unfortunately, FitzHugh did not use the term "singularly perturbed," and consequently the mathematical community missed a valuable clue. A second model of an electrically active cell is the 58 THE MATHEMATICAL INTELLIGENCERVOL. 12, NO. 4, 1990

Noble [15] model of the cardiac Purkinje fiber. Before describing the model, we indicate a little of what the Purkinje fiber does. The electrical impulses that govern the heart rate originate in the sino-atrial node (the pacemaker region) and are then transmitted to the muscles in the walls of the heart, where the impulses order the muscles to contract, thus producing the heartbeat. The last part of the journey of the electrical impulse is along the Purkinje fiber, which brings the impulse to the walls of the heart. Although transmission of this impulse is the primary function of the Purkinje fiber, laboratory studies show that the Purkinje fiber displays another activity: if the Purkinje fiber is not subject to stimulus, it fires spontaneously and regularly. That is, it generates electrical impulses regularly. The Noble model of the Purkinje fiber is the following system of equations: dV dt

1

{(400m3h + 0.14)(V - 40) Cm + (1.2 e x p [ ( - V - 90)~0 ] + .015 exp[(V + 90)~0] + 1.2n4)(V + 100)}

dm

dt - o%(1 - m) - f3mm

(N) dh

d'-t- = %(1 - h) - f3hh dn

dt-

o~,(1 - n) - ~nn

where V, m, h, n have the same meanings as in (H-H), Cm is a constant, and the terms oLm, ~m, o% an, ~, are, again, well-behaved, messy functions of V (but quite different from the c~'s and ~'s that appear in (H-H)). For a more detailed description of (N), see Cronin [3], pages 85-89. The model (N) was d e r i v e d by m o d i f y i n g the Hodgkin-Huxley equations in accordance with laboratory data obtained for the Purkinje fiber. Thus it is not surprising that (N) formally resembles (H-H). However, the behavior of the solutions of (N) is very different from the behavior of the solutions of (H-H). This too is unsurprising because the nerve axon and the Purkinje fiber perform very different functions. Following stimulus, the nerve axon has certain reactions and then returns to a quiescent condition. The Purkinje fiber, on the other hand, oscillates regularly. This suggests that the Hodgkin-Huxley equations should have a globally asymptotically stable equilibrium point and the Noble equations should have a periodic solution. As might be expected, the Noble model has been

supplanted by more accurate models based on the extensive laboratory data obtained since 1962. See McAIlister et al. [13] and DiFrancesco and Noble [5]. (Computer analysis of unpublished extensions of the DiFrancesco-Noble model is given in OXSOFT HEART program manual, Version 2.1, dated 1 Mar. 1988, copyright 1987 by OXSOFT Ltd., Oxford, England.) However, the more recent models are obtained by using the same viewpoint used to derive (N) and consequently turn out to be more complicated versions of the Noble equations. Thus, to reach the desired goal, which is a qualitative analysis of the more recent models, it seems reasonable to study first a prototype model, i.e., the Noble equations.

A question about w h y certain research did not take place at a particular time generally has a complicated answer with historical or sociological overtones. The first purpose of a model of the Purkinje fiber is to describe the activities mentioned earlier: transmission of impulses and spontaneous regular firing. A description of the firing would consist of a periodic solution of (N), preferably with some fairly strong stability properties. A description of the transmission is more complicated because a nonautonomous term representing the influence of the impulse must be added and one is led to use of the theory of entrainment of frequency (see Cronin [3], pages 240-243). We will consider here only the problem of finding a description of the regular firing, that is, a periodic solution of (N). By simple transformations, system (N) may be recast as: dV dt

1 9

-

[F(V,

-

dm dt

_ 1 [m.(V) -- m] ~L

ah _ dt dn dt -

m,

h,

n)]

--

TIn(V)

(N3

- h.] [_ Th(V) [n.(V) _- n] n [

T,,(V)

where e and ~q are small fixed positive numbers and the functions in the square brackets are well-behaved and are of the same order of magnitude. The functions m| h~(V), m,(V), Tm(V), Th(V), Tn(V ) are all messy but have simple behavior (see Cronin [3] pages 216-222). The role of the e suggests that (N 3 should be

regarded as a singularly perturbed system with singular perturbation parameter 9 The role of ~ suggests that -~ should be regarded as a regular perturbation parameter. Viewing ~ as such a parameter, however, leads us astray. The reason is this: if ~qis a regular perturbation parameter, then the usual procedure is to set "q = 0, solve the resulting simpler problem, and use the results to approximate solutions of (Nl) if ~q is small. But it is straightforward (although a little tedious) to show that if -q = 0, system (Nl) has no periodic solution (see Cronin [3], pages 222-233). By regarding ~qas a regular parameter we "lose" the desired solution. A more detailed analysis of (N1) suggests that a better strategy is to recast (N) (by a change of variable) as:

l{1L,V,o,h,n)} 1{1

dV dt

,11

dm dt

,rI

dt

-m| -.~m(l,O m]}

(1',I2)

_ l - hi hi_ Th(V)

dn _ [n.(V) - n 1

L Tn(V) J System (N2) is also, of course, a singularly perturbed system. Thus, for simple quantitative reasons, we are led to viewing the Noble model and the Hodgkin-Hodgkin model as singularly perturbed systems. More important, however, there are physiological reasons for regarding these models as singularly perturbed systems. For example, laboratory data show that the potential V and the function m in the nerve axon and the Purkinje fiber undergo very rapid smooth changes. (Television addicts often see these changes during dramas in which some character requires intensive care. The oscilloscope pictures everyone watches so anxiously are representations of the cardiac potential.) Also, numerical studies of (H-H) and (N) show that the V and m components display these very rapid changes. Such rapid smooth changes are characteristic of certain components of solutions of singularly perturbed systems. Thus we have a strong physiological argument for reg a r d i n g (H-H) a n d (N) as singularly p e r t u r b e d systems. For further arguments favoring the singular perturbation viewpoint, see Cronin [3], pages 182-183. By coincidence, part of the general study of singularly perturbed systems was developed about the same time as the Hodgkin-Huxley theory. Those aspects of the theory that will turn out to be most useful in the study of models of electrically active ceils were initially developed by A. Tychonov [21], N. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990 5 9

Levinson [12], and Y. Sibuya [18] (Sibuya's paper is an extension of Levinson's work). In view of these facts, it seems, at first, surprising that singular perturbation theory was not widely applied, starting in the 1960s, to models of electrically active cells. Although there are studies employing singular perturbation theory (e.g., Casten, et al. [2]) and Carpenter [1]), there has not been the systematic, widespread application of the singular perturbation viewpoint that might be expected. A question about why certain research did not take place at a particular time generally has a complicated answer with historical or sociological overtones (if, indeed, the question can be answered at all). But the question of w h y singular perturbation theory was not widely applied in the 1960s to models of electrically active cells has one very simple answer. In order to explain this answer, we need first to look a little bit at the n a t u r e of solutions of singularly p e r t u r b e d systems. We can do this by studying the following version of the van der Pol equation dx dt

1 ( y - 3 x + x 3) 9

dy_ dt

-

-

- -

X

- -

3

X

1 2"

- -

- -

The usual way to approach a problem with a small parameter 9 is to set 9 equal to zero, solve the resulting simpler problem, and then show that solutions of the original problem are close to the solutions of the simpler problem. So in this case, we would first solve (D). To study (D), we can take the analytic viewpoint: solve the first equation for x as a function of y, say h(y), and then solve the equation dy_ d--t - h(y)

1 2

Or we may take the geometric viewpoint and speak of seeking the solutions of dy dt

-

60

-

I

~

1

I

[

1

2

~

T

S: ( - 1 , - 2 )

Figure

(D) -

R

1.

- -

0=y-3x+x dy_ dt

2)

(v)

where 9 is a small positive number. The degenerate system associated with (V), which is defined to be the system obtained from (V) by setting 9 equal to zero (first multiply through by 9 is:

-

j: (1,

d~

1 2

- -

..-.

~

X

B

1 2

_

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

which are on the manifold d~ defined by 0 = y-

3x + x3.

(M)

(This manifold is called the slow manifold.) See Figure 1. For our purposes, it is convenient to take the geometric viewpoint; then the orbits of the resulting solutions are subsets of ~t, that is, subsets which do not include either of the points ( - 1 , - 2 ) or (1, 2). (These points must be excluded because (M) cannot be solved uniquely for x as a function of y in neighborhoods of these points.) By general theory it follows that if 9 is sufficiently small, there exist solutions of (V) near these orbits. The "general theory" just mentioned originates with a paper of A. N. Tychonov [20] and has been developed by numerous mathematicians. For ongoing research and references, see H. W. Knobloch [10]. There is a serious drawback to the procedure just described: An important class of solutions entirely eludes this analysis. It is well known from numerical analysis and also from laboratory study of the physical systems modeled by equation (V) that (V) has a periodic solution. But the analysis just described does not suggest the existence of a periodic solution, because orbits of the solutions obtained must stay close to segments of ~t that certainly do not resemble closed curves. Consequently, to find a periodic solution, we must take a different viewpoint. We select a point P0

= (x0,Y0) not on the manifold ~t and sketch the orbit of (V) that passes through P0. Since y - 3x + x 3 < 0 at P0, if 9 is sufficiently small, then dx/dt at P0 is very large; since dy/dt is not large, it follows that the tangent to the orbit at P0 is practically horizontal and is directed to the right. Using the same kind of arguments as we move along the orbit to the right, we may conclude that the orbit is reasonably well approximated by the dashed horizontal segment PoR drawn in Figure 1.

The usual way to approach a problem with a small parameter ~ is to set ~ equal to zero, solve the resulting simpler problem, and then show that solutions of the original problem are close to the solutions of the simpler problem. This argument breaks down, of course, when we reach d~ (at the point R) because on ~ we have: dx/dt = 0. The question then arises: where does our "approximate orbit" proceed from R? We know that it cannot turn back toward P0 because dx/dt ~ 0 at the left of R. But the "approximate orbit" cannot cross ~t and proceed to the right because on the right of R, dx/dt < O. Hence our "approximate orbit" must proceed either up or down on ~ . Since dy/dt > 0 at R, the "approximate orbit" proceeds upward (as sketched) until it reaches J, the point (1,2), which is the maximum point on ~ . At the point J, the circumstances of thee "approximate orbit" are radically changed. As indicated by arrows in Figure 1, the directions of the nearby orbits no longer force the approximate orbit to stay on ~ . In fact there is every inducement to lure the "approximate orbit" away from ~t. So we assume that the approximate orbit leaves ~t as soon as possible, that is, at the point J. Since dx/dt < 0 above d~, if 9 is very small so that ]dx/dt I is very large, the "approximate orbit" becomes the dashed horizontal line segment JQ directed to the left. The terminal end of this line segment is the point Q on ~t. By arguments analogous to those used at R and J, respectively, we conclude that the "approximate orbit" (sketched as a d a s h e d curve) proceeds along d~ to the minimum point S on ~t and then along the horizontal line segment from S to T. Then the "approximate orbit" proceeds up ~t to the point R, and after that, the "approximate orbit" simply repeats itself. No decent person (we will not give a formal definition of the term decent) can read this description of an "approximate orbit" without flinching. Indeed, many with righteous indignation will refuse to proceed past the point R or even to reach the point R. There are so many weaknesses in the argument that one scarcely

knows where to begin a critical analysis. Yet the remarkable fact is that this slipshod description is the basis for an important and deep part of singular perturbation theory: the theory of discontinuous solutions. The curve we termed an "approximate orbit" is called a discontinuous solution of (V). (Note that the discontinuous solution is neither a solution of (V) nor is it discontinuous. The word "discontinuous" refers not to the curve itself but to the tangent to the curve. The tangent is discontinuous at the points Q and T.) The first step in the theory of discontinuous solutions is to define precisely the concept of discontinuous solution for a general system dx dt

-

1 fix,y, 9 9

(c)

dy_ d--t - g(x,y, 9 where x is an m-vector, y is an n-vector, 9 is a small positive parameter, and the functions f and g are well behaved. The idea of the definition is, in essence, a straightforward generalization of the "approximate orbit" described for (V). Very roughly speaking, the discontinuous solution is defined to be a continuous curve that consists of a finite collection of solutions: first a solution of a "fast system" dx dt

-

1 f(x,yo, 9 9

where Y0 is fixed; then a solution of the degenerate system o = f(x,y,o)

dy_ a-7 - g(x,y,0), then another solution of a fast system, and so on. (We speak sloppily here and identify " s o l u t i o n " and "orbit.") The discontinuous solution in our example (V) consisted of a solution of the fast system dx 1 ----(Y0dt 9

3x + x 3)

from P0 to R followed by a solution of the degenerate system from R to J, and so on. Conceptually, the definition of discontinuous solutions arises directly from that curve P0 R J Q S T. But in order to give a precise definition in the (m + n)-dimensional case, we must give a careful description of the curve, especially in neighborhoods of "corners" such as R, J, Q, S, and T. To make the definition of THE MATHEMATICAL INTELL1GENCER VOL. 12, NO. 4, 1990

61

discontinuous solution useful, it is necessary to im- problem in dealing with a specific application is to depose some analytic conditions in neighborhoods of the termine the discontinuous solutions. It is exactly at corners. So it turns out that the definition of discon- this point that we encounter a substantial difficulty: a tinuous solution, although conceptually simple, is difficulty which was insurmountable until the advent lengthy (e.g., several pages long in Mishchenko and of computers. The discontinuous solution of (G) conRozov [14]). Once the discontinuous solution has been sists partly of solutions of the degenerate system, that defined, then it can be proved that if e is small, the is, solutions of the equation discontinuous solution is a good approximation (in a sense that will not be described in detail) to solutions dy_ a-/- g(x,y,0) of (G) that have an initial value near the discontinuous solution. Also the part of the solution of (G) which is approximated by a solution of a "fast system" exhibits in the slow manifold defined by rapid smooth changes. Moreover, if the discontinuous solution is a closed curve (e.g., the discontinuous sof(x,y,O) = O. lution T J Q S T of system (V)) and if ~ is sufficiently small, there is a periodic solution of (G) near the dis- So the first step in finding the discontinuous solutions continuous solution. These results are the import of is to delineate the slow manifold. In the study of the work of Tychonov [21], Levinson [12], Sibuya [18], s y s t e m (V), finding the slow manifold consisted and the work of Soviet mathematicians, among them simply in graphing the equation Pontryagin and his colleagues, which is summarized in the book of E. F. Mishchenko and N. Kh. Rozov 0=y-3x+x 3, [14]. (The work of Levinson and Sibuya concerns systems somewhat more general than (G), but we will the kind of task assigned to a beginning calculus stunot venture into the details of this generality.) dent. But the general problem of determining the form We have described two methods for studying singu- of the slow manifold is practically impossible unless larly perturbed equations: examining solutions that the components of f have a particularly simple form, are near solutions of the degenerate system and scru- and such is certainly not the case with either the tinizing solutions that are near discontinuous solu- Hodgkin-Huxley equations or the Noble equations. tions. Actually the first method is a special case of the This is consequently a reason, if not the main reason, second method for which the discontinuous solution w h y this singular perturbation theory was not applied is simply a solution of the degenerate system. The first to models of electrically active cells in the 1960s: It was method is useful in many applications, indeed any ap- not possible to determine the form of the slow maniplication in which the solution of the singularly per- fold and hence the existence of discontinuous soluturbed equation quickly approaches and then stays tions could not be investigated. near the slow manifold. This is the case, e.g., for certain problems in enzyme kinetics. See Rubinow [16], pages 38-45. No decent person (we will not give a formal In electrically active cells, however, we see at once definition of the term decent) can read this that the situation is more complicated. For example, in description of an "'approximate o r b i t " the Purkinje fiber, the V-component of the solution without flinching. undergoes a rapid smooth change followed by an interval in which slow changes occur and then followed by another rapid smooth change and so on. This sugHowever, with the advent of computers the situagests that the solution is close to a solution of a fast tion has become very different. The calculations resystem during one short time interval and then, at a quired to graph, with reasonable accuracy, the slow later time interval, is again close to a solution of a fast manifold for (N2), i.e., the manifold defined by system. Thus the solution does not stay close to the slow manifold. Hence we conclude that we must use F(V, m, h, n) = 0 the theory of discontinuous solutions. We have seen that the Hodgkin-Huxley equations m = m.(V) and the Noble model can be regarded as singularly perturbed systems and that these solutions should be h = h~(V), studied by using discontinuous solutions. It remains actually to apply the discontinuous solution theory de- are quickly and easily carried out with a computer. veloped by Tychonov, Levinson, Sibuya, and Mish- (Also, despite the messiness of the function F, the corchenko and Rozov. We have already indicated how responding slow manifold has a very simple form, as the discontinuous solution is defined, and the first will be seen below.) 62

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

metric, has been proposed (Cronin [4]). This approach is an extension of classical geometric methods discussed, e.g., in Lefschetz [11, Chapter XI] and Stoker [19]. N o w we describe this procedure in very brief, informal terms. The underlying idea of the geometric approach is to construct a globally attracting set for the orbits of (N2). We regard "q as a singular perturbation parameter. (Despite the suggestive notation used in (N2), it turns out to be unnecessary to regard 9 as a singular perturbation parameter. However, in visualizing orbits of a fast system, it is convenient to think of 9 as a singular perturbation parameter.) A computational s t u d y s h o w s that the slow manifold described by F(V,m,h,n) = 0 m = mo~(V)

h = h|

Figure 2.

Actually by using a computer to graph the slow manifold, we are considering not the slow manifold itself but an approximation to it. This is a serious theoretical point which deserves more study: the exact nature of the assumptions made when the approximation to the slow manifold is used should be stated explicitly. Using this approximation to the slow manifold makes it possible to apply the Mishchenko-Rozov theory to these singularly perturbed systems. A graduate student at Rutgers, Teresa Dib, has obtained results in this direction for the Hodgkin-Huxley equations. However, a detailed investigation of the Noble model reveals some drawbacks to the MishchenkoRozov theory. First, the formal definition of discontinuous solution is lengthy and includes a number of analytic conditions w h o s e verification may not be easy. Indeed, in the more complicated models, of which the Noble model is a prototype, these verifications might be extremely difficult, if not impossible, to carry out. Second and perhaps more serious, the M i s h c h e n k o - R o z o v t h e o r y yields no information about stability properties of the solutions. In a physiological model, only solutions with fairly strong stability properties can be expected to be good predictors of behavior of the system. For these reasons, another approach, more geo-

can be described roughly as a curve % whose projection in the (V,n)-plane is S-shaped and whose m and h coordinates are monotonic increasing and decreasing, respectively, as functions of V. Thus the projection of c~ in (V,m,n)-space has the appearance sketched in the dashed curve A B D C in Figure 2. This suggests the presence of a closed discontinuous solution sketched as the solid curve ABCDA. (The curves AB and CD are subsets of the slow manifold. The curves BC and D A are orbits of fast systems.) But instead of trying to determine the closed curve A B C D A exactly and trying then to prove that the closed curve satisfies the conditions in the definition of a discontinuous solution, we construct an attracting set that contains the curve ABCDA. First we show that a "sleeve" (sketched in black) can be constructed around AB so that any orbit of (N2) that intersects the sleeve enters it. Denote the sleeve and its interior by ~ . Next let E be the "end" of the sleeve at B. (The end of the sleeve at B must be constructed with some care and is not indicated in detail in Figure 2. The point B is a junction point (see Mishchenko and Rozov [14], p. 173) and orbits of (N2) near a junction point tend to have complicated behavior.) It can be proved that if "q is bounded there is a number To > 0 such that if S(t) is a solution of (N2) with S(0) E E, then for all t ~> T0, solution S(t) stays very close to curve CD. Let ~ be the set = U {S(t)lt ~ [0,T0]},

where the union is taken over all solutions S(t) of (N2) such that S(0) E E. A second sleeve X2 is constructed around CD, and finally a second tube of solutions ~-2 that starts from the end of ~2 and ends very near the curve AB is constructed. It is then proved that the set lI defined by THE MATHEMATICAL INTELLIGENCER VOL. I2, NO. 4, 1990

63

H

=

Y-,lU ~-lU Y.2u ~-2

has the following attracting property: given a compact set K in (V,m,h,n)-space such that K does not contain the equilibrium point of (N2), there exist ~q0 > 0 and To > 0 such that if 0 < "q < "q0, then a solution of (N2) that passes t h r o u g h a point of K at t = 0 has the property that for t > T0, the solution is contained in 1I. Also the Poincar6 m a p p i n g that acts on

10.

11. 12. 13.

~1 n {(V,m,h,n)/n = no} (where n o is a n u m b e r between the n-coordinates of the points A a n d B) has a fixed point and hence It contains a periodic solution. The attracting property of 1I is a kind of global stability for this periodic solution. Only ordinary differential equations have been discussed here. However, it should be pointed out that Hodgkin and Huxley [9] extended system (H-H), essentially by a d d i n g a diffusion term, to a system ( ~ - ~ ) of partial differential equations. Their numerical analysis of this latter system resulted in a theoretical prediction of the nerve impulse. The agreement of this theoretical prediction with laboratory observations is the most spectacular triumph of the Hodgkin-Huxley theory. For references to and discussion of the work on (~f-~f) and similar partial differential equations, see Scott [17] and Fife [6]. It is questionable w h e t h e r other models of electrically active cells can be extended to partial differential equations that are physiologically valid. See, e.g., McAllister, Noble, and Tsien [13], pp. 53-54.

References 1. G. Carpenter, Periodic solutions of nerve impulse equations, ]. Math. Anal. Appl. 58 (1977), 152-173. 2. R. Casten, H. Cohen, and P. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quarterly of Applied Math. 32 (1974-75), 365-402. 3. Jane Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge: Cambridge University Press (1987). 4. Jane Cronin, Qualitative analysis of a cardiac fiber model, to appear. 5. D. DiFrancesco and D. Noble, A model of cardiac electrical activity incorporating ionic pumps and concentration changes, Philos. Trans. Roy. Soc. Lond. B 307 (1985), 353-398. 6. Paul C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, Vol. 28, New York: Springer-Verlag (1979). 7. R. FitzHugh, Impulses and physiological states in models of nerve membrane, Biophys. J. 1 (1961), 445-466. 8. R. FitzHugh, Mathematical models of excitation and propagation in nerve, Biological Engineering. H. P. Schwan, ed., Chapter 1. New York: McGraw-Hill (1969). 9. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to 64

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

14.

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

conduction and excitation in nerve. J. Physiol. 117 (1952), 500- 544. H. W. Knobloch, "Invariant manifolds and singular perturbation," Proceedings of the Eleventh International Conference on Nonlinear Oscillation d Budapest: Janos Bolyai Mathematical Society (1987). S. Lefschetz, Differential Equations: Geometric Theory, Second Edition, New York: John Wiley & Sons (1962). N. Levinson, Perturbations of discontinuous solutions of non-linear systems of differential equations, Acta Math. 82 (1951), 71-106. R. E. McAllister, D. Noble, and R. W. Tsien, Reconstruction of the electrical activity of cardiac Purkinje fibers, J. Physiol. 251 (1975), 1-58. E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (translated from Russian) New York: Plenum Press (1980). D. Noble, A modification of the Hodgkin-Huxley equations applicable to Purkinje fiber action and pace-maker potentials, J. Physiol. 160 (1962), 317-352. S. Rubinow, Mathematical Problems in the Biological Sciences, Philadelphia: Society for Industrial and Applied Mathematics (1973). Alwyn C. Scott, The electrophysics of a nerve fiber, Reviews of Modern Physics 47 (1975), 487-533. Y. Sibuya, On perturbations of discontinuous solutions of ordinary differential equations, Natur. Sci. Rep. Ochanomizu Univ. 11 (1960), 1-18. J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems. New York: Interscience Publishers (1950). A. N. Tychonov, Systems of differential equations containing small parameters with derivatives, Mat. Sbornik N.S. 31 (1952), 575-596. A. N. Tychonov, On the dependence of the solutions of differential equations on a small parameter, Mat. Sbornik N.S. 22 (1948), 193-204.

Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA

Steven H. Weintraub* For the general philosophy of this section see Vol. 9, No. 1 (1987). A bullet (e) placed beside a problem indicates a submission without solution; a dagger (~) 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 February 1991. This column will have a new editor as of the next issue. Problem solutions and other correspondence should be directed to David Gale, Department of Mathematics, University of California, Berkeley, CA 94720 USA.

In taking my leave of the editorship of this column, I note that in the time I have had it (the past four years), the Mathematical Entertainments column has amply demonstrated the international character of the Mathematical lntelligencer. Counting only problems and solutions that have appeared in print, this column has had contributors from the following twenty-one countries: Australia, Austria, Belgium, Bulgaria, Canada, Colombia, Czechoslovakia, England, the Federal Republic of Germany, Finland, France, Hungary, Israel, Italy, Japan, Mexico, the Netherlands, Poland, Romania, Switzerland, and the USA.

such that for any sequence of vectors z 1. . . . . z t in Rn satisfying Izil ~ 1 for each i and z 1 + . . . + zt = 0, there exists a rearrangement zp(1). . . . . Zp(t) such that

•

zp( 0 <~ K,

s=

1. . . . .

t.

i=1

If we let K(n) be the infimum of K as above, K(n) is k n o w n as the Steinitz c o n s t a n t in d i m e n s i o n n. Clearly, K(1) = 1. Prove that K(2) = V5/2. (The exact value of K(n) is unknown for n > 2.)

Problems The modified Fermat problem: Quickie 90-7 by Flejberk Jaroslav (Pardubice, Czechoslovakia) For any two relatively prime positive integers n and k, show that the equation

x.+y~=zk has a solution in positive integers x, y, and z.

The Steinitz constant in dimension two: Problem 90-8t" by the Column Editor In 1913 Steinitz proved the following result: Let n be any positive integer. Then there exists a number K

* 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.

THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4 9 1990Springer-VerlagNew York 65

66

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

From Springer-Verlag Stephen Wiggins, California Institute of Technology, Pasadena, CA

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67

Inertial Manifolds R. T e m a m

Inertial manifolds are new objects that have been recently introduced in relation to the study of the longterm behavior of solutions of dissipative evolution equations. When they exist, inertial manifolds are finite dimensional invariant smooth manifolds that contain the global attractor and attract all the orbits at an exponential rate. Most of the dynamics for the system under consideration takes place on this manifold, producing a considerable simplification in the study of the dynamics. Also, the system obtained by restriction to the inertial manifold is a finite dimensional system, even if the initial system was infinite dimensional. This system, called the inertial system, reproduces most of the dynamical properties of the initial system. In addition to their mathematical significance, inertial manifolds m a y have some computational and physical relevance. As is shown below, inertial manifolds produce an interaction law between the small and large wavelength components of a flow; alternatively, using a t e r m i n o l o g y n o w c o m m o n in the physics literature, we can say that the small wavelengths are enslaved by the large ones. This will be discussed in more detail below; we shall also relate inertial manifolds to slow manifolds, which are well known in meteorology for short-term weather forecasting. From the computational point of view, inertial manifolds and approximate inertial manifolds produce finite dimensional objects that approximate the attractor (perhaps a fractal set). They reproduce the coarse structure of the attractor that we want to approximate, while they neglect the fine details. Simple approximate inertial manifolds have been constructed and, although this is still very recent, promising numerical algorithms have been derived using these manifolds. This too will be briefly discussed below. 68

Dissipative Evolution Systems We are interested in evolutionary dissipative systems. Such s y s t e m s arise in p h y s i c s , c h e m i s t r y , mechanics . . . . Usually the state of the system is described by an element ~ of a Hilbert space H of finite or infinite dimension (the phase space) and its evolution on an interval of time I C ~ is defined by a function u : I - - * H. An example of this is a chemical system for which H is Rn and u(t) is an n-tuple (cl(t) . . . . . cn(t)), where cl(t), . . . . cn(t) are the concentrations at time t of the reactants. In fluid mechanics or meteorology, H is an infinite dimensional function space and u(t) ~ H is a function J

u(t) : f~--~ ~,

where f~ C ~3 is the domain filled by the fluid; here u(t) might be the velocity field (at time t) throughout fL or the field of velocities and temperatures.

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4 9 1990 Springer-Verlag New York

In many cases of interest the evolution of u is determined by the solution of an initial value problem for a differential equation in H, say du(t) dt - F(u(t)),

t > 0,

(1)

u(0) = u0,

(2)

where F is a function from H (or part of H) into itself. When the dynamics of the system is trivial, the system evolves towards rest or towards a steady state. In (1), (2) this is reflected by the fact that the solution u(t) of (1), (2) converges as t ~ oo to an equilibrium point u,, i.e. a solution of F(u,) = 0.

(3)

For turbulent systems, the dynamics is not trivial. The solutions of (1), (2) remain always time dependent; their evolution looks essentially unpredictable at our present level of understanding of nonlinear dynamics. Some of the relevant questions are then 9 What can we predict concerning the evolution of u(t)? 9 H o w can one effectively compute the solutions of (1), (2) on a large interval of time? 9 Assuming that we know the solutions of (1), (2), what can we learn from this (abundant) information that is physically relevant? H o w can we economically extract from a large system a specific item of important information? We shall try to show how inertial manifolds could perhaps provide a new insight to these problems in nonlinear dynamics.

Figure 1. Absorbing Set. sending u 0 to u(t). Concerning the operators S(t) it may also be appropriate, in some situations, to consider only discrete values of t; for instance, when we consider a Poincar6 map. From the physical point of view a dissipative system is a system dissipating energy; hence some real function of u(t) will decrease as t increases. From the mathematical point of view a well-accepted definition of dissipativity for a semigroup S(') or equation (1) is the existence of a bounded absorbing set. This is a bounded set ~0 C H such that: For every bounded set ~ C H, S ( t ) ~ C ~o for all sufficiently large t.

Thus, by contrast with Hamiltonian (nondissipative) systems, the orbits do not wander in the whole space H and they do not fill any open nonempty set in H. Instead they concentrate in the region ~0. The time at which a given orbit enters ~0 depends on its initial point u 0, but all orbits eventually enter ~0; and the entering time is uniformly bounded for u 0 in a bounded set ~. (See Figure 1.)

Absorbing Sets We assume that the evolution of the system under consideration is governed by a semi-group of operators S(t):H---~H,

The Global Attractor Another aspect of dissipativity is the existence of a

t~0,

global attractor. This is a compact set ~ C H enjoying

where

the following properties: S(O) = L S(t + "r = S(t) " S('t),

t,,r ~ O.

(4)

Knowing the state u(';) of the system at time % its state at time ~ + t,t > 0 is given by u(t + "0 = S(t)u('r)(= S('r

S(t + "r)u(0)if ,r > 0).

S(t) ,~ = ~l, V t ~ O,

(5i)

attracts all bounded sets, i.e., for every bounded set ~ C H, d i s t ( S ( t ) ~ , ~ ) := sup inf IIS(t)x - yiJn

(5ii)

9

For example, under suitable hypotheses on F, the initial value problem (1), (2) is well posed, i.e., for any u 0 E H there exists a unique function u from [0, oo) to H satisfying (1), (2): in this case S(t) is the mapping

.

,

^

xE~yE.~

renas ro u as t ~

+ oo.

In particular, each orbit converges to ~/as t ~ oo 9 dist(S(t)u o, ~ ) --~ 0 as t ~ % V u o. THE MATHEMATICAL

INTELLIGENCER

V O L . 12, N O . 4, 1990

(6) 69

Because an abundant literature is already available on attractors, we shall just emphasize here that a global attractor is maximal for the inclusion relation among all attractors and that conditions (5i) and (5ii) make the set ,~ unique, if it exists. If the dynamics is trivial, reduces to a single equilibrium point (F(u,) = 0). In the nontrivial case ~ may contain or consist of the set of equilibrium points with manifolds connecting them (the unstable manifolds); orbits of time-periodic solutions; orbits of time-quasiperiodic solutions lying on a toms; or even more complicated sets of nonintegral dimension, i.e., fractals. For ordinary differential equations, i.e., when H has finite dimension, the existence of the global attractor has been known for many years and some of these attractors have attracted particular attention (H6non, Lorenz . . . . ). In the infinite dimensional case the existence of the attractor has only recently been proved, for some classes of dissipative evolution equations and for some specific dissipative equations such as the Navier-Stokes equations. When a nonstationary turbulent flow takes place, the attractor ~ (or part of it) is the natural mathematical object for the description of the permanent regime. Ruelle and Takens attempt to explain the temporal chaos corresponding to a turbulent flow by conjecturing that ~ is a complicated, fractal set (strange attractor), and that the temporal chaos is due to the orbits wandering along such a set. For this reason it would be, of course, interesting to understand better the geometry of such sets, but actually little is known at present. One of the geometrical aspects of attractors that has been extensively studied, in particular in the infinite dimensional situation, is the dimension of the a t t r a c t o r - - e i t h e r the Hausdorff dimension or the fractal dimension. For many dissipative systems it has been shown that even if the phase space H has infinite dimension, the attractor itself has finite dimension, and hence permanent turbulent regimes actually have a finite dimensional structure; this reduction of infinite dimension to finite dimension will be further discussed below. In some cases, the estimates on the dimension of the attractors in terms of the physical data are physically relevant. For example, in fluid mechanics these estimates are in good agreement with and give a rigorous mathematical proof of some fundamental aspects of the Kolmogorov and Kraichnan theories of turbulence; in particular, the maximum number of degrees of freedom of a turbulent flow predicted by these theories is the same as the dimension of the global attractor attached to the flow.

Inertial Manifolds Infinite dimensional dissipative systems seem to display a finite dimensional behavior. The relation be70

THE MATHEMATICAL INTELUGENCER VOL. 12, NO. 4, 1990

t w e e n finite and infinite dimensional dynamical systems, already transparent in the results on dimension of attractors, deserves to be further investigated; this was one of the motivations for inertial manifolds. Another, more general motivation, is the imbedding of the global attractor in a smooth manifold. This is a natural problem in geometry and topology which has been investigated by several authors under hyperbolicity assumptions which may not be easy to verify for specific equations (see in particular M. Shub's book and the references therein). An inertial manifold for the semigroup {S(t)}t>~o (or for equation (1)) is a smooth manifold ~ of finite dimension such that S(tfiY~ c ~ ,

V t/> 0,

attracts all the solutions of (1), (2) at an exponential rate.

(7i) (7ii)

By (7ii) we mean that for every u 0 ~ H there exists K1, K2 > 0, such that dist(S(t)uo, ~ ) <<-K1 exp (-K2t ). The comparison of ~ and ~J~ is transparent; ~ may not be regular and may even be a fractal set, whereas is required to be a smooth manifold, Lipschitz at least. The equality sign in (50 is replaced by an inclusion sign in (7i), and while the orbits may converge to at an arbitrarily slow rate, they converge to ~ at an exponential rate. Finally (7ii) obviously implies that C ~l~. If u 0 E ~ , then by (7i) the solution u(t) = S(t)u o of (1), (2) is included in ~ for all t/> 0. The restriction of (1) to ~ is therefore a well defined finite dimensional system called the inertial system. Under appropriate hypotheses the inertial system possesses the property of asymptotic completeness: by this we mean that to every orbit solution of (1), (2), we can associate a companion orbit lying in ~ and producing the same behavior as t ~ ~: Vu 0 E H , 3 v 0 E . ~ a n d ~ > 0 s u c h t h a t dist(S(t)u o, S(t + T)v0) --~ 0 as t ~ ~. Thus, in such cases the inertial manifold produces a reduction of dimension without any loss of asymptotic information. Existence theorems of inertial manifolds have been proved for many dissipative differential equations. Other results include the asymptotic completeness and further regularity results, e.g., ~ is a Cl-mani fold. Equations for which such results were proved include reaction-diffusion equations, the GinsburgLandau equation, and pattern-formation equations such as the Kuramoto-Sivashinsky and the Cahn-Hilliard equations. However, for many dissipative equations possessing a global finite dimensional attractor,

the existence of an inertial manifold is still an open problem. In particular this is the case for the NavierStokes equations, even in space-dimension 2. It is not certain that an inertial manifold necessarily exists for any dissipative evolution equation; nonexistence results have even been proved for reaction-diffusion equations and for damped wave equations similar to the Sine-Gordon equation. All the existence results that have been proved give an inertial manifold that is a graph. Equation (1) is rewritten more specifically in the form

du(t) d-----~+ Au(t) + R(u(t))

= 0.

(8)

folds S(t)PmH, as t increases; under appropriate hypotheses these manifolds are graphs above PmH converging to a limit as t -+ % and the limit is an inertial manifold. eThe Sacker method reduces the determination of 9 to a hyperbolic equation in infinitely many dimensions. The e q u a t i o n is easy to derive formally: setting P = Pm = Pmu , q = qm = Qmu , w e p r o j e c t ( 8 ) onPm H and QmH and obtain the system

dp --~ + Ap + PR(p + q) = O, dq

Here A is an u n b o u n d e d self-adjoint closed positive operator with domain D(A) C H and R is a nonlinear operator from D(A) into H. Assuming that A-1 is compact, we find by elementary spectral theory the existence of an orthonormal basis of H consisting of eigenvectors wj of A:

(11) + Aq + QR(p + q) = O.

For a trajectory lying on ~ldt, we have q(t) = cI)(p(t)), V t i> 0, and by elimination we readily find

{ Awj = )*jwj, j = 112 . . . . . 0 < ).1 <~ )*2 . . . . Xj--+ ~ a s j - - + oo.

- ~ ' ( p ) ' ( P R ( p + ~(p)) + Ap) + Ac~(p) + QR(p + C~(p)) = O,

For a given integer m we denote by Pm the orthogonal projection of H onto the space spanned by w 1. . . . . w m. Let Qm = I + Pm" We look for ~ as the graph of a function c~: pmH --+ QmH.

(9)

In this case, besides the technical hypotheses, the major and most restrictive hypothesis is a spectral gap condition )*m+~ -- Xm ~> K3()*m~+l + )*m~),

(10)

where K3 is a constant depending on the equation: because )*m --+ ~ as m --+ % (10) implies that the spectrum of A must contain large gaps. The methods that are available for the determination of 9 are the following: eThe Perron-Lyapunov method, which reduces the determination of cI) to the solution of a fixed-point problem ~ - 9 = 9 of a certain mapping ~-. This method resembles that used by Perron and Lyapunov for the construction of the central manifold, of which inertial manifolds are a global analog. eThe Cauchy method is based on the construction of integral curves. We choose an appropriate large spherelike set F in PmH and consider the family of integral curves

(12)

where ~ ' is the Fr6chet differential of ~. We can obtain a good idea of the meaning of the spectral gap condition (10) by considering the case of reaction-diffusion equations with space-periodicity b o u n d a r y conditions. In this case the condition is slightly weaker than (10), namely hm+l

- - )*m ~

K4,

(13)

and in space-dimension n the )*m are the numbers n

(ki)2,

kl . . . . .

k, E ~.

(14)

i=1

By well-known results in number theory any positive integer is of the form (14) if n/> 4, and condition (13) is not likely to be satisfied; while if n = 2, there are arbitrarily large gaps in the set of numbers (14) and therefore (13) is satisfied for arbitrarily large values of QmH

E = tO S(t)F. t~0

It can be shown that the closure of s is a graph above PmH and that s is an inertial manifold. eThe Hadamard method consists of studying the mani-

Figure 2. Convergence of orbits to the attractor and to the inertial manifold. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990 7 1

m. For reaction-diffusion equations the principle of spatial averaging permits a weakening of (13) and leads to existence results in space-dimension >2. In order to resolve the unsolved problems of existence of inertial manifolds one should look perhaps for more general forms of inertial manifolds: less regular manifolds (e.g., H,Older continuous) and manifolds that are not graphs (Figure 2).

Turbulence At this point, it is not clear to what extent inertial manifolds can help understand turbulence, but we would like to describe here the connection existing between these theories. For the sake of simplicity we consider an evolution equation (8) where A is the abstract linear operator corresponding to an elliptic differential operator associated with the space-periodicity boundary conditions. In this case the eigenvectors wi are simply proper combinations of sines and cosines from the Fourier series:

on the manifold. Because the inertial manifold ~2~ is approached at an exponential rate, we see that after a certain transient period, the orbits lie on ~ during the permanent regime: during the permanent regime small and large wavelengths phenomena are connected by the quasistatic law (15). Alternatively, at each time the small wavelengths are solely and fully determined by the large ones. We can say also in the terminology of Haken that the small eddies are enslaved by the large ones. The existence of an inertial manifold implies the existence of an interaction law for the system under s t u d y . For fluid m e c h a n i c s , t h e r m o h y d r a u l i c s , damped waves . . . . it is not clear whether such laws should exist. We shall see however that approximate forms of such laws do exist.

Meteorology

Before and independently of the concept of inertial manifold, a related concept was developed in meterology, under the name of slow manifold. This concept sin k. x, cos k" x has been developed on heuristical grounds for compuk = (k1. . . . . kn) ~ ~n, x = (x 1. . . . . x~) ~ ~ ; tational purposes without any attempt at mathematical justification. we are assuming that the period-cube is (0, 2~r)n. The idea is that the meteorological variables u correWe write the series expansion of u sponding to wind velocity, temperature, and possibly oo some concentrations, are governed by an equation of u(t) = ~ uj(t)wj form (1). The permanent regime has been reached, so j=l that the orbit u(t) lies on the attractor ~ and in prinor ciple a n y m e a s u r e m e n t u 0 should correspond to a point on ~. Of course this information is of no use u(t,x) = ~ {ak(t ) sin k . x + bk(t ) cos k. x,} since little is known of this attractor. kEN n In meteorology there is a "natural" decomposition and consider a d e c o m p o s i t i o n u = Pm + qm of U of H into PmH•Qm H, where QmH corresponds to small wavelengths, called the gravity waves, while PmH corslightly different from that above; namely for m E ~, responds to the large wavelengths called the Rossby Pm = Pro(t,x) = ~ {ak(t) sin k " x + bk(t ) cos k" x} waves. The conjecture leading to the slow manifold is Ikil~m that we have i n d e e d a relation of the form (10), for some i q,, = q)(Pm), the graph of ~ - w h o s e construction is not explained here-being the slow manifold 8. Furthermore, the slow manifold 8 contains (0,0) and is tanqm = qm(t,x) = ~ {ak(t) sin k" x + bk(t ) cos k- x}. gent to the PmH space at (0,0). The gravity and Rossby Ikil>m f o r all i waves are on a totally different scale, a fact that should be compared to the spectral gap condition (10). Hence Pm is periodic of period 2~r in each direction and The slow manifold is used in meteorology for shortqm is the superposition of periodic functions of period term forecasting. The initial data u0 (either measured 2~r/m in each direction. Although the cut-off number or computed) should be on 8 (and even on ~!). This m is arbitrary here, we can say that Pm corresponds to is not the case, due to errors in measurement and/or large wavelengths and qm to small wavelengths. computation. For short-term weather forecasting it is It appears that this phenomenon of interaction of important to have a good initialization (u0 ~ ~ ), so the small and large wavelengths is an important aspect of measured u 0 is approximated via appropriate algoturbulence in general (fluids or waves) and of statis- rithms by a close point 1/0 (a kind of projection) on 8. tical mechanics. N o w for a system governed by an As mentioned before, no theorem has been proved equation of form (8), if an inertial manifold of type (9) and there is not even a precise definition of 8 in the exists, then meteorology literature. There are also some doubts qm(t) = r (15) about the existence of an exact slow manifold (same 72

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

u 0 a n d on the equation, the n o r m of qm(t) in H (and in other spaces) becomes and remains small:

[qm(t)[n ~ 8,

(16)

where 8 is an appropriate power of )~l/Km+~. A first consequence of (16) is that the attractor ~ lies in the strip of width 28, [qm[n ~ 8. Then with an asymptotic expansion it can be s h o w n that the second equation (11) can be approximated to first order by

Aqm + QmR(pm)= O.

Figure 3. The slow manifold in meteorology.

More precisely, after a certain time the orbits enter the region

[Aqm(t) + QmR(pm(t))[n ~ 82.

(17)

Hence the manifold ~ 1 of equation

AQmq~ + QmR(Pmq~) = 0

Figure 4. Localization of the attractor: ~ lies in the darker shaded region. problem as for inertial manifolds?). However, it is generally accepted that a sort of approximate slow manifold does exist a n d the concept, broadly used in shortterm forecasting, is called the nonlinear initialization (the linear initialization being a mere projection on PmH) (Figure 3). A p p r o x i m a t e Inertial M a n i f o l d s In the absence of a definitive result (positive or negative) on the existence of inertial manifolds, an interesting alternative is that of approximate inertial manifold. This topic is the object of current research a n d we shall give only a brief description of the concept. A first idea is to perturb equation (8) and consider a modified equation containing A s, s > 1:

du~ - - + ~ ASu~ + Au~ + R(u) = 0, dt where ~ > 0 is small. The eigenvalues )~m are replaced by ~ K~,. For a suitable s (s ~ so > 1) and for m sufficiently large (m/-- m0(~, so)), the spectral gap condition is fulfilled and the approximate inertial manifold ~ is obtained. However, as e ~ 0, ~ deteriorates, its dim e n s i o n m becoming larger a n d larger; we do not k n o w if ~J~, converges to a limit. Another more natural concept of approximate inertial manifold results from the following observation: after a certain time d e p e n d i n g only on the initial data

(18)

appears as an approximate inertial manifold in the sense that it attracts all orbits in a thin neighborhood of it, at an exponential rate (Figure 4). Of course the attractor lies in this neighborhood of ~rd~l; this is one more item of information on the localization of ~. Also these localization results of ~ permit the construction of new, more efficient algorithms for the numerical approximation of the equation. For example, the usual Galerkin a p p r o x i m a t i o n w i t h m m o d e s consists of projecting equation (8) on PmH; one can also define a nonlinear Galerkin approximation with m modes, which consists of some sort of projection of the equation (8) onto ~ 1 . Other approximations based on m-dimensional analytic manifolds are also being investigated. In finite dimensions the analogs of (17) a n d (18) are respectively semi-algebraic sets a n d algebraic sets of large dimension; in the infinite dimensional situation they are semi-analytic and analytic sets. Further reading

Nonlinear Dynamics and Turbulence (general presentation) J. Gleick, Chaos, New York: Viking Press (1987). D. Ruelle, The Mathematical Intelligencer 2(3) (1980), 126-137.

Attractors A. V. Babin and M. I. Vishik, J. Soviet Math, 28 (1982), 619-627. A. V. Babin and M. I. Vishik, J. Math. Pures Appl. 62 (1983), 441-491. P. Constantin and C. Foias, Comm. Pure Appl. Math. 38 (1985), 1-27. P. Constantin, C. Foias and R. Temam, Memoirs Amer. Math. Soc. 314 (1985), and Physica D 30 (1988), 284-296. P. Constantin, C. Foias, O. Manley, and R. Temam, J. Fluid Mech. 150 (1985), 427-440. C. Foias and R. Temam, J. Math. Pures Appl. 58 (1979), 339- 368. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

73

J. M. Ghidaglia and J. C. Saut, Equations aux d~rivdes partielles non lindaires dissipatives et syst~mes dynamiques, Paris: Hermann (1988). J. K. Hale, Math. Surveys and Monographs, vol. 25, Providence, RI: Amer. Math. Soc. (1988). E. Lieb, Comm. Math. Phys. 92 (1984), 473-480. R. Marl6, Lecture Notes in Math., vol. 898, New York: Springer-Verlag (1981). J. Mallet-Paret, J. Differential Equations 22 (1976), 331. M. Shub, Global Stability of Dynamical Systems, New York: Springer-Verlag (1987). R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer-Verlag (1988).

Inertial Manifolds P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Appl. Math. Sciences, no. 70, New York: Springer-Verlag (1988). E. Conway, D. Hoff, and J. Smoller, SIAM J. Appl. Math. 35 (1978), 1-16. C. R. Dering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, vol. 99, Contemporary Mathematics, Providence, RI: Amer. Math. Soc. (1989). C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam, C.R. Acad. Sci. Paris, S6rie 1, 301 (1985), 285-288, and J. Math. Pures Appl. 67 (1988), 197-226. C. Foias, G. R. Sell, and R. Temam, C.R. Acad. Sci. Paris, S6rie 1, 301 (1985) 139-141, and J. Differential Equations 73 (1988), 309-353. J. Mallet-Paret and G. Sell, ]. Amer. Math. Soc. 1 (1983). B. Nicolaenko, vol. 99, Contemporary Mathematics, Providence, RI: Amer. Math. Soc. (1989). J. C. Saut and R. Temam, Proceedings of the Luminy Conference, Math. Modelling Numerical Anal. 23 (1989). Approximation of Inertial Manifolds F. Baer and J. J. Tribbia, Mon. Wea. Rev. 105 (1977), 1536-1539. E. Fabes, M. Luskin and G. R. Sell, J. Differential Equations, to appear. C. Foias, M. S. Jolly, I. G. Kevrekides, G. R. Sell, and E. S. Titi, Physics Letters A 131 (1988), 433-436. C. Foias, O. Manley, and R. Temam, Math. Modelling Numerical Anal. 22 (1988), 93-118. C. Foias, G. R. Sell, and E. S. Titi, ]. Dynamics and Differential Equations (1989, to appear). C. Foias and R. Temam, C.R. Acad. Sci. Paris, S6rie I, 307 (1988), 5-8, and 67-70. C. Foias and R. Temam, The algebraic approximation of attractors: the finite dimensional case, Physica D 32 (1988), 163-182. R. Marl6, Lecture Notes in Math., vol. 597, New York: Springer-Verlag (1977) 361-378. M. Marion, Thesis, University Paris-Sud, 1988. M. Marion and R. Temam, Nonlinear Galerkin Methods, SIAM J. Numerical Anal. (1988), to appear. R. Temam, C.R. Acad. Sci. Paris, S4rie II, 306 (1988), 399-402. R. Ternam, Indiana Univ., Preprint no. 8809, 1988; Jour. Fac. Sci. of Tokyo, to appear. R. Temam, Indiana Univ., Preprint no. 8812, 1988; SlAM Jour. Numer. Anal., to appear. E. S. Titi, MSI Preprint 88-119, 1988; Jour. Math. Anal. and Appl., to appear. J. J. Tribbia, Mon. Wea. Rev., 107 (1979), 704-713 and 112 (1984), 278-284. Laboratoire d'Analyse Num~rique C.N.R.S. et Universit~ Paris-Sud Bdtiment 425, 91405 Orsay, France 74

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

Chandler Davis*

its traditional fields its meaning and utility have greatly expanded. It may be sufficient to refer to antisymmetry, dynamical symmetry, generalized crystallography, and the symmetry analysis of music and of artistic design.

Symmetry: Unifying Human Understanding edited by Istvan Hargittai New York: Pergamon, 1986 xiii + 1045 pp. US$125.

In the introduction to the other volume, Symmetry through the Eyes of a Chemist, Istvan and Magdolna

Symmetry Through the Eyes of a Chemist by Istvan Hargittai and Magdolna Hargittai

Hargittai add the following:

New York: VCH Publishers, 1986 xii + 458 pp.

Reviewed by Michele Emmer "Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect." So said the mathematician Hermann Weyl ([22], page 145) in 1952, in what he called "this swan song on the eve of my retirement from the Institute for Advanced Study" ([221 preface). In his preface to the book, Symmetry: Unifying Human Understanding, the editor of the volume, Hargittai, quotes from the letter of invitation he sent to prospective contributors: It is, of course, commonplace today that not only is symmetry one of the fundamental concepts in science, but it is also possibly the best bridging idea crossing various branches of sciences, the arts and many other human acfivities. Whereas symmetry has been considered important for centuries, primarily for its aesthetic appeal, this century has witnessed a dramatic enhancement in its recognition as a cornerstone scientific principle. In addition to traditionally symmetry-oriented fields such as spectroscopy or crystallography, the concept has made headway in fields as varied as reaction chemistry, nuclear physics and the study of the origin of the Universe. At the same time, in

* C o l u m n editor's address: M a t h e m a t i c s D e p a r t m e n t , U n i v e r s i t y of Toronto, Toronto, O n t a r i o M5S 1A1 C a n a d a

Fundamental phenomena and laws of nature are related to symmetry and, accordingly, symmetry is one of science's basic concepts. Perhaps it is so important in human creations because it is omnipresent in the natural world. Symmetry is beautiful although alone it may not be enough for beauty, and absolute perfection may be even irritating. Usefulness and function and aesthetic appeal are the origins of symmetry in the worlds of technology and the arts. Plainly such a subject calls for at least 1000 pages; and indeed Hargittai has recently announced a sequel volume, Symmetry 2. Books, conferences, seminars, etc. are very often dedicated to the theme of s y m m e t r y from various points of view. In the last few years I may cite 9 The interdisciplinary congress on the work of the Dutch graphic artist M. C. Escher at the University of Rome La Sapienza, and the exhibition of his works at the Dutch Institute also in Rome (March 1985) [5, 7, 191F O The exhibition Symmetrie in Kunst, Natur und Wissenschaft in the town of Darmstadt ( J u n e - A u g u s t 1986). The catalogue of the exhibition consists of two volumes of almost 1000 pages in total [13]. 9 The volume TiIings and Patterns by Branko Grfinbaum and G. C. Shephard [11], a comprehensive and systematic treatment of the subject, focusing on classification and enumeration of tilings, including detailed surveys of coloring problems and aperiodic tilings. I have mentioned only a few examples; it is interesting to note that many mathematicians who have contributed to the volume edited by Hargittai have

THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4 9 1990Springer-VedagNew York 75

been involved in the projects and books that ! have recalled. But w h a t is s y m m e t r y ? We m a y r e t u r n to Weyl for a r e m i n d e r of the ambiguity of the word: If I am not mistaken, the word symmetry is used in our everyday language in two meanings. In the one sense symmetric means something like well proportioned, well balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry . . . . In this sense the idea is by no means restricted to spatial objects; the synonym harmony points more toward its acoustical and musical than its geometric applications ([22], p. 31). This b r o a d aesthetic m e a n i n g is d e s c r i b e d by the c r y s t a l l o g r a p h e r A l a n L. M a c k a y in the Hargittai volume: "Symmetry is the classical Greek w o r d ~YMMETPIA, the same measure, due proportion. Proportion m e a n s equal division and due implies that there is some higher moral criterion. In Greek culture due proportion in e v e r y t h i n g was the ideal." It is well k n o w n that the Greek p h i l o s o p h e r Plato (c. 427-347 B.C.) related the five regular solids of threespace to the elements of physical space. He describes t h e m as "the four most beautiful bodies which are unlike one a n o t h e r . . . . A n d t h e n we shall not be willing to allow that t h e r e are a n y distinct kinds of visible bodies fairer t h a n t h e s e " ([17], 53b), see also [8]. This vague notion of s y m m e t r y as h a r m o n y or proportion is still p r e s e n t in our culture a n d will probably remain forever ([22], preface). It is in this sense that H a r o l d O s b o r n e , P r e s i d e n t of the British Society of Aesthetics, deals w i t h " S y m m e t r y as an aesthetic factor" in the Hargittai volume: From classical antiquity the idea of symmetry in close conjunction with that of proportion dominated the studio practice of artists and the thinking of theorists. Symmetry was asserted to be the key to perfection in nature as in art. But the traditional concept was radically different from what we understand by symmetry today--so different that symmetry can no longer be regarded as a correct translation of the Greek word symmetria from which it derives . . . . To the Greeks symmetry meant commensurability, and two magnitudes were said to be commensurable if there exists a third magnitude which divides into both without remainder . . . . This preference for intellectually apprehensible rather than sensuous beauty corresponded to the basic cast of the Greek temperament, their demand for order and comprehensibility everywhere and their abhorrence of all that is vague and formless . . . . The two fundamental assumptions of classical aesthetics continued virtually unquestioned at the time of the Renaissance: first, the belief that the same principles of perfection apply in nature and in art and, second, contrary as it very often was to practice, that the representational realism asserted to be the goal of the artist should depict not the imperfect models available in nature but an ideal symmetry of the type, which can be discovered mathematically. . . . Renaissance artists and scholars were intoxicated with mathematics and it was at this time that the doctrine of the Golden Section or, as it was generally known, the Divine Proportion came to prominence. 76 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990

I personally do not agree with the j u d g m e n t that d u r i n g the Renaissance, painters, artists, and architects like Paolo Uccello, Piero della Francesca, Leon a r d o da Vinci, Albrecht; Dfirer, Filippo Brunelleschi, a n d L e o n Battista Alberti w e r e "intoxicated with mathe m a t i c s , " nor d o I believe that their use of m a t h e matics was a simple use of rules a n d p r o p o r t i o n s . There is more to mathematics (of any period) than Osb o r n e p e r h a p s realizes. N o t that I am denying that Ren a i s s a n c e artists had a m a t h e m a t i c a l bent. (See the two chapters "Painting a n d Perspective," and "Science Born of Art: Projective G e o m e t r y " in the v o l u m e Mathematics in Western Culture b y Morris Kline [12]; the chapter dedicated to the Renaissance in Carl Boyer's History of Mathematics [2]; and the book by Julian Lowell Coolidge [4]. For a history of the theory of proportion during the Renaissance see the chapter b y that title in Panofsky [16].) T h e m o d e r n c o n c e p t i o n of s y m m e t r y is v e r y diff e r e n t f r o m that i n h e r i t e d f r o m classical times, a n d given n e w p r o m i n e n c e at the Renaissance . . . . O u r idea of b e a u t y in art has b e c o m e emotional a n d expressive; intellectual perspicuity counts for less t h a n it did in the past. M o d e r n t h i n k i n g is less t h a n convinced that the same aesthetic principles apply in nature and art. We repudiate the basic assumptions of Renaissance aesthetics, the belief that there is an ideal s y m m e t r y of natural objects, w h e t h e r based o n the golden section, o n dynamic symmetry or o n any o t h e r mathematical formula. P e r h a p s not on a mathematical formula, but w h y not o n a mathematical idea? In his p a p e r "The Mathematical A p p r o a c h in C o n t e m p o r a r y A r t " the Swiss artist Max Bill wrote [1, page 105]: By a mathematical approach to art it is hardly necessary to say I do not mean any fanciful ideas for turning out art by some ingenious system of ready reckoning with the aid of mathematical formulas . . . . It must not be supposed that an art based on the principles of mathematics is in a sense the same thing as a plastic or pictorial interpretation of the latter. Indeed, it employs virtually none of the resources implicit in the term pure mathematics. . . . It presents some analogy to mathematics itself.... Despite the fact that the basis of this mathematical approach to art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination. 1 As Weyl said, and as G y 6 r g y Doczi reminds us in the Hargittai collection, " T h e w o r d s y m m e t r y has a b r o a d e r a n d a n a r r o w e r m e a n i n g . The n a r r o w e r ,

1 For the important role of topology in the art of Max Bill see: Max Bill, Endless Ribbon, in Mathematics Calendar, Berlin: Springer (1979), June; Michele Emmer, "Visual art a n d mathematics: The Moebius b a n d , " Leonardo 13(2) (1980), 108-111; Michele Emmer, Moebius Band, film of the series Art and Mathematics, with Max Bill, Rome: FILM 7 (1980).

M. C. Escher, Without Title (Symmetry drawing), E21 (1938), M. Sachs Collection, 9 1988 M. C. Escher, Cordon Art, Baarn, Holland.

M. C. Escher, Without Title (Symmetry drawing), E34 (1941), M. Sachs Collection, 9 1988 M. C. Escher, Cordon Art, Baarn, Holland.

M. C. Escher, Without Title (Symmetry drawing), El17 (1963), M. Sachs Collection, 9 1988 M. C. Escher, Cordon Art, Baarn, Holland.

R. Farinato, L. Loreto, M. Tonetti, Interactive Software

for Studying Crystallographic Groups, University of Rome 1 (1987).

medium, controlled by the creature which I am conjuring up. It is as if they themselves decide on the shape in which they choose to appear. 2

newer [?] meaning indicates equality between the two opposite sides of a plane or line, while the broader, older meaning refers to beauty or harmony of all parts." The relationship of symmetry and art is the topic of a large and interesting part of the volume, whether approached from the broad or the narrow understanding of the term, and whether the writer is an art historian or a scientist. In the sixty-five papers, each author has tried to give her or his own contribution to the general theme. There appears to have been no discussion among them, and there is no homogeneity whatever. Such extreme variety, juxtaposing "Magic rectangles in the table of the 32 classes of symmetry" by I. I. Shafranovskii and " A n extension of the Neumann-Minnigerode-Curie principle" by J. Brandm/iller with "Symmetry and its 'love-hate' role in music" by D. Wilson may be too much of a good thing. Many of the mathematicians, crystallographers, and physicists write not on symmetry in general but on some specific topic. The paper of Fejes T6th, "Symmetry induced by e c o n o m y , " p r e s e n t s various e x t r e m u m p r o b l e m s which lead to highly symmetric geometrical configurations. Two papers are inspired by the work of the Dutch artist M. C. Escher: one by the crystallographer Caroline Macgillavry, "The Symmetry of M. C. Escher's 'Impossible images'," and the one by the mathematician Douglas Dunham, "Hyperbolic symmetry." They both wrote papers on similar topics in the Proceedings of the Escher conference ([14], [6]). Several years ago Macgillavry a u t h o r e d a n o w famous book illustrated by Escher, Fantasy & Symmetry. In the preface Escher himself wrote ([15], page VIII):

The most interesting periodic drawings were made by Escher in his notebooks. The mathematician Doris Schattschneider started years ago to study them in order to explain the system that the Dutch artist used to create and classify his tessellations of the plane ([18],[15]; see also the catalogue of an exhibition in Japan in 19833). In the Hargittai volume, Schattschneider has a paper entitled "In black and white: How to create perfectly colored symmetric patterns," where perfect coloring means that every symmetry of the uncolored tiling is also a two-color s y m m e t r y of the colored tiling. The mathematicians Branko and Zdenka Grfinbaum as well as G. C. Shephard investigate the Moorish ornaments of the Alhambra in Granada, Spain, by using very beautiful illustrations, including some in color. The volume would not be complete without a contribution from H. S. M. Coxeter, with w h o m it was my great pleasure to work on the movie and conference on M. C. Escher [5]. His article is "The generalized Petersen graph G(24,5)." I am particularly attracted by the subject of soap films and was fortunate enough to work with Fred Almgren and Jean Taylor on a film on this subject [9], so I was glad to see the article "Symmetries of soap films" by A. T. Fomenko. One misses in Fomenko's references the famous paper of Jean Taylor on singularities of soap-film-like and soap-bubble-like surfaces [21]; also completely ignored is the Italian school of Perimeter Theory for the Plateau problem going back to Ennio De Giorgi in the sixties ([3], [10]).

Though the text of scientific publication is mostly beyond my means of comprehension, the figures with which they are illustrated bring me occasionally on the track of new possibilities for my work. It was in this way that a fruitful contact could be established between mathematicians and myself. The dynamic action of making a symmetric tessellation is done more or less unconsciously. While drawing I sometimes feel as if I were a spiritualist

2 For t h e c o n t a c t of E s c h e r w i t h m a t h e m a t i c i a n s a n d crystallographers see [5], [7]; also the two films, M. C. Escher: Symmetry and Space a n d M. C. Escher: Geometries and Impossible Worlds, of the series Art and Mathematics, Rome: FILM 7 (1982, 1984). 3 M. C. Escher's Universe of Mind Play, catalogue of the exhibition, Tokyo: Fuji Television Gallery (1983). THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4, 1990 77

In c o n c l u s i o n , t h e v o l u m e of H a r g i t t a i contains m a n y excellent, t h o u g h perhaps too disparate, papers. Its interdisciplinarity affords the reader information from a vast domain. If any approach is under-represented, it is the use of computers in p r o d u c i n g and analyzing periodic patterns. A n e x c e l l e n t s t i m u l u s for f u r t h e r w o r k on symmetry. The o t h e r v o l u m e by Istvan a n d M a g d o l n a Hargittai, Symmetry through the Eyes of a Chemist, is more traditional. The intention of the authors is to present a relatively short s u r v e y of the entire field of chemistry from the point of view of symmetry. The authors declare in the preface: Despite its breadth, our book was not intended to be comprehensive or to be a specialized treatise in any specific area. Rather, after acquiring a broad perspective, the reader may then refer to the excellent monographs which provide detailed treatments of specialized topics . . . . We would like especially to note here the two classics in the literature of symmetry which have strongly influenced us: Weyl's Symmetry and Shubnikov and Koptsik's Symmetry in Science and Art [20]. An overview of the contents: Simple and Combined Symmetries; Molecules: Shapes and Geometry; Helpful Mathematical Tools; Molecular Vibrations; Electronic Structure of Atoms and Molecules; Chemical Reactions; Space-Group Symmetries; Symmetries in Crystals. The authors write: Chemical symmetry has been noted and investigated for centuries in crystallography, which is the border between chemistry and physics . . . . Later, discussion of molecular vibrations, the selection rules and other basic principles in all kinds of spectroscopy also led to a uniquely important place for the concept in chemistry with equally important implications. Moreover, the discovery of the h a n d e d n e s s or chirality of crystals a n d t h e n of molecules led the symm e t r y concept nearer to the chemical laboratory. T h e y add: It is not yet two decades from the days when, in discussing the importance of symmetry in chemistry, some sort of an apology was felt necessary for giving so much attention to this concept. Symmetry was then considered to lose its significance as soon as the molecules entered the most usual chemical change, the chemical reaction. Orbital theory and the discovery of the conservation of orbital symmetry have removed this last blindfold. A central role in the book is played b y the theory of groups. In particular, group-theoretical m e t h o d s are applied to s y m m e t r i e s of molecular vibrations, electronic structure, a n d chemical reactions. A descriptive discussion of space-group symmetries a n d the symmetries of crystals is also included. The v o l u m e gives the reader a b r o a d perspective, including a detailed list of references. Naturally the v o l u m e is full of illustrations, including examples taken from the arts. It can 78

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 4, 1990

be a v e r y useful book for students in chemistry a n d for a n y o n e w h o w a n t s to u n d e r s t a n d the role of symm e t r y in chemistry.

References 1. Max Bill, "Die mathematische Denkweise in der Kunst unserer Zeit," Werk (1949) No. 3; reprinted in English in Eduard Huttinger, Max Bill, Zurich: ABC Edition (1978), 61-116. 2. Carl B. Boyer, A History of Mathematics, New York: John Wiley & Sons, Inc. (1968). 3. F. Colombini, E. De Giorgi, L. C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Pisa: Scuola Normale Superiore (1972). 4. Julian L. Coolidge, The Mathematics of Great Amateurs, New York: Dover Publications, Inc. (1963). 5. H. S. M. Coxeter, M. Emmer, R. Penrose, M. Teuber, eds., M. C. Escher: Art and Science, Amsterdam: NorthHolland (1986). 6. Douglas J. Dunham, "Creating hyperbolic Escher patterns," in [5], 241-248. 7. M. Emmer, C. van Vlanderen M. C. Escher, Catalogue of the Exhibition, Rome: Istituto Olandese (1985). 8. Michele Emmer, "Art and mathematics: The Platonic solids," Leonardo 15(4) (1982), 277-282; see also the film Platonic Solids of the series Art and Mathematics, Rome: FILM 7 (1980). 9. , Soap Bubbles, film of the series Art and Mathematics, Rome: FILM 7 (1980). 10. Enrico Giusti, Minimal Surfaces and Functions of Bounded Variation, Boston: Birkh/iuser (1984). 11. Branko Grunbaum, G. C. Shephard, Tilings and Patterns, New York: W. H. Freeman and Co. (1987). 12. Morris Kline, Mathematics in Western Culture, Oxford: Oxford University Press (1953); reprinted London: Pelican Books (1979). 13. Bernd Krimmel, ed., Symmetrie in Kunst, Natur und Wissenschaft, Darmstadt: Mathildenh6he (1986). 14. Caroline H. Macgillavry, "Hidden symmetry," in [5], 69-80. 15. - - , Fantasy & Symmetry: the Periodic Drawings of M. C. Escher, New York: Harry N. Abrams, Inc. (1976). 16. Erwin Panofsky, Meaning in the Visual Arts, New York: Doubleday (1955); reprinted London: Penguin Books (1971). 17. Plato, Timaeus, in The Great Books of the Western World, London: Encyclopaedia Britannica 7 (1952), Plato, 442-477. 18. Doris Schattschneider, "M. C. Escher's classification system for his colored periodic drawings," in [5], 82-96. 19. J. J. Seidel, Review of the volume "M. C. Escher: Art and Science," The Mathematical Intelligencer 10, no. 1 (1988), 69-70. 20. A. V. Shubnikov and V. A. Koptsik, Simmetriya v naukakh i iskusstve, Moskva: Nauka (1972); English translation Symmetry in Science and Art, New York: Plenum Press (1974). 21. Jean E. Taylor, The structure of singularities in soapbubble-like and soap-film-like minimal surfaces, Annals of Mathematics 103 (1976) 489-539. 22. Hermann Weyl, Symmetry, Princeton, NJ: Princeton University Press (1952). Facoltd~di Scienze M.F.N. Universit~ della Tuscia 01100 Viterbo, Italy

Robin Wilson*

Czech and Slovak Mathematics The stamps shown here were issued as a set in 1987 to commemorate the 125th anniversary of the Union of Czechoslovak Mathematicians and Physicists. The top stamp features part of the Prague Town Hail clock and a computer graphic from the theory of functions superimposed on a result from real analysis. The middle stamp features three mathematicians and physicists superimposed on a result from number theory. They are (from left to right): Jozef Maximilian Petzval (1807-1891), a Slovak mathematician and physicist who was professor at the Universities of Budapest and Vienna; his research interests lay in differential equations, optics, acoustics, and ballistics. In 1840 he constructed a new type of camera lens that cut exposure time from 30 minutes down to 30 seconds. (~en~k Strouhal (1850-1922), professor at Charles University in Prague, was the founder of Czech experimental physics; he worked on acoustics and the magnetic and electrical properties of steel, and wrote textbooks on optics, acoustics, and mechanics. Vojt~ch Jarnik (1897-1970), a Czech mathematician who also was a professor at Charles University; his research work was in analytic number theory, the geometry of numbers, and mathematical analysis, and he wrote influential textbooks on the differential and integral calculus. The bottom stamp shows a drawing of field measurement from a book by A. M. Mallet (1672), an earth fold diagram, and the trajectory of a particle in Brownian motion.

I am grateful to Ivan N e t u k a , J. Ne~effil, and Krzysztof Ciesielski for their help in preparing this column.

* C o l u m n editor's address: Faculty of Mathematics, The O p e n University, Milton Keynes MK7 6AA England THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 4 9 1990Springer-VerlagNew York 79