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].

20

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.
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, ~ ) 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