The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
H o w to Write about Teichmiiller The Spring 1995 issue of The Mathematical Intelligencer contains two very different contributions, by M. R. Chowdhury (pp. 12-14) and B. Booss-Bavnbek (pp. 1520), which pick up our article on the life and work of Oswald Teichmfiller in the Jahresbericht der Deutschen Mathematiker-Vereinigung 94 (1992), 1-39. Chowdhury's text, with which we have only minute differences, is written as a service to the reader of the Mathematical Intelligencer, providing a partial translation of and commentary on Teichmiiller's infamous letter to Edmund Landau of November 1933, which had been first published in the appendix to our article on Teichmfiller. In contrast to this, Booss-Bavnbek's article appears in the "Opinion" column of The Mathematical Intelligencer and aims at fundamental criticism. This criticism is directed partly against the general editorial policy of the journal Jahresbericht D M V in matters of obituaries of German mathematicians. We also feel quite uneasy about some of the obituaries that have appeared in the Jahresbericht DMV. For instance, we consider Leichtweil~'s biographical remarks on Karl Strubecker, quoted by Booss-Bavnbek, generally superficial and occasionally offensive. One of our motivations for publishing the Teichmfiller article in the Jahresbericht D M V was in fact to help improve the quality of these obituaries. Bernhelm Booss-Bavnbek expresses an emphatic normative position on how to deal with the difficult and painful history of the Nazi period. We do not want to pass over our basic differences with him in silence. Booss-Bavnbek seems to believe that the historian ought
to be some sort of moral teacher; the more rigidly the teacher declares that certain protagonists of his story were bad guys, the more effectively he educates the readers for a better future. We hold a more sceptical, but certainly not cynical, view. Moral condemnation seems to us to be but a frail protection against a hideous past. We believe instead in close and detailed confrontation with the historical material. Of course we do place ourselves in the universe of values when we approach and present a historical subject. But we try to do so in a tempered, controllable, and criticizable manner. This was openly stated in the introduction to our article on Teichmfiller. Our restriction to documented evidence 1 seemed to 1For instance, it is virtually certain that it was Ernst Witt, not Oswald Teichmiiller, who participated (wearing SA uniform) in Emmy Noether's seminar on Hasse's notes on Class Field Theory, which she held privately at her home in the summer of 1933 because she had already been put on leave by the ministry. Thus footnote 3 in BoossBavnbek's article almost certainly relates a flawed piece of oral history. See Clark Kimberling, "Emmy Noether and her Influence," in: Emmy Noether, A Tribute to her Life and Work, James W. Brewer & Martha K. Smith (ed.), New York, Basel (M. Dekker), 1981, pp. 3-61; here: p. 12, and footnote 13, p. 47. In an early first draft of N. Schappacher, "Das Mathematische Institut der Universitat G6ttingen 1929-1950" in: Becker, Dahms, Wegeler (Hrsg.), Die Universitf~'t G6ttingen unter dem Nationalsozialismus, Mfinchen (K.G. Saur), 1987, 345-373, Schappacher also conjectured that Teichmfiller was the one. This was corrected in a letter by Fenchel; but the mistake has unfortunately survived in: C. Tollmien, "'Sind wir doch der Meinung, dal~ ein weiblicher Kopf nur ganz ausnahmsweise in der Mathematik sch6pferisch t~tig sein k a n n . . . ' - - E m m y Noether 1882-1935. Zugleich ein Beitrag zur Geschichte der Habilitation von Frauen an der Universitat G6ttingen." G6ttinger Jahrbuch 38 (1990), 153-219; here footnote 188.
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us natural when writing about a man whose contradictory personality and whose despicable political ideas and actions have given rise to many anecdotes whose accuracy is impossible to ascertain. However, since we dwelt quite a bit in the article on the previously unp u b l i s h e d - l e a d i n g r61e of Teichm/iller in the Landau boycott, it is incomprehensible to us how any careful reader could conclude that we were attempting to rehabilitate Teichmiiller's politics. Our refusal to make Teichmfiller into a "negative hero"--which seems to have caused the most serious misunderstandings of our article--was simply motivated by a desire to show that Teichmiiller was not a lonely "hero" who single-handedly forced Landau's resignation by organizing the student boycott. In fact, for the boycott to have been successful, the new ministerial policy (in the fall of 1933) was needed, which made it possible to get rid of Jewish professors disregarding the exceptions stipulated in the racist clauses of the law of April 7, 1933. Only a gross misreading of the text could make it possible to interpret our formulation as saying that we wanted to excuse Teichmiiller for his action, and put all the blame on the state administration, as Booss-Bavnbek suggests (pp. 16-17). Had such a reading occurred to us, even in our worst nightmares, our sentence would have probably come out different, and better! Clearly, the biographical section 1. of the article on Teichmiiller, written by Scholz, is not a full-fledged biography which lives up to the principles laid out nicely in the quotation from S6derqvist towards the end of Booss's article. To go further in this direction would have called for a closer investigation of the genesis of mentalities in Weimar Germany (and earlier) which prepared large parts of the German population to support or actively participate in Nazi politics, whereas others kept a sceptical distance, practised disobedience, or, in a few cases, took up active resistance. Such a major work could not be our goal in that article. We are also perfectly aware that sections 2--6 of the article on Teichm/iller, surveying most of Teichmiiller's papers, do not represent a satisfactory history of the mathematical questions that Teichm/,iller addressed. To give more would have required an effort, and an expertise, which seemed beyond our capabilities. In this sense Booss-Bavnbek's remarks (p. 16) concerning the contributions of Gr6tzsch, Lavrent'ev, and Schiffer to the early development of "Teichmfller theory" are welcome. But to insinuate that our failure to mention these mathematicians might have something to do with their being (quoting Booss-Bavnbek) an "anti-Nazi," a "Soviet mathematician," and a "Jew forced to emigrate," verges on slander. We are acutely aware of the limitations of our publication. At the same time, we see no merit in the wholesale condemnation of our partial achievements on the basis of normative ideals. Historical understanding is 6 THEMATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
reached through discourse and controversies--preferably, controversies more to the point than this one. Norbert Schappacher Mathdmatique et Informatique Universitd Louis Pasteur 7, rue Rend Descartes 67084 Strasbourg CEDEX France Erhard Scholz FB7: Mathematik GHS Wuppertal Gaussstrasse 20 5600 Wuppertal 1 Germany
Bernhelm Booss-Bavnbek Replies This exchange has been helpful and enlightening. It is fortunate it could be conducted in The Mathematical Intelligencer; I would have preferred that also the pages of the Jahresbericht would be open to such dialogue. Did N. Schappacher and E. Scholz express themselves in favor of this? Institut for matematik og fysik RUC/IMFUFA Postboks 260 DK-4000 Roskilde, Denmark
I1 Circolo Matematico di Palermo In The Mathematical Intelligencer, vol. 17, no. 2, 1995, in the Letters to the Editor there is a list of the oldest mathematical societies of the world, but it fails to include one of the most famous: II Circolo Matematico di Palermo, founded in Palermo on March 2th, 1884 by the Italian mathematician G.B. Guccia (1855-1914). II Circolo Matematico di Palermo was internationally known, and among its fellows were all the greatest mathematicians of the world.
References Brigaglia, A. & Masotto, G. 1982, II Circolo Matematico di Palermo, Dedalo, Bari. Cardamone, L. 1962, Le scuole matematiche in Sicilia dopo l'Unit~, published in La Sicilia e l'Unit?~ d'Italia, Feltrinelli, Milano, pp. 271-300. Lombardo Radice, L. & Bartolozzi, F. 1977, Matematici siciliani dell'ultimo secolo, published in La presenza della Sicilia nella cultura degli ultimi cento anni, Palumbo, Palermo, pp. 1107-1120. Nastasi, P. 1990, Matematica e matematici in Sicilia, published in Scienza in Sicilia, a special number of "Cronache Parlamentari Siciliane', N.S., a. VII, supplem, al n. VI, pp. 59-79. Atdo Scimone GRIM (Research Group for Mathematical Teaching) Department of Mathematics via Archirafi 34 90123 Palermo, Italy.
More on M a g n u s I much enjoyed Abe Shenitzer's "In Memory of My Friend Wilhelm Magnus" in the Spring 1995 issue of The Mathematical Intelligencer. I, too, have experienced Magnus as one of the most considerate and polite persons I ever met. I got to know Magnus in my first semester at G6ttingen in 1947 when he taught Differential- und Integralrechnung I to beginning students. At one point during a lengthy proof he had used the same letter t twice for two different variables. When I called this out to him, he thanked me profusely, bowing several times in my general direction (he couldn't identify the source---there were some 200 students in the audience). Then Magnus started to crank down the blackboard, marking all earlier t's with primes. For some high t's that he couldn't reach, he jumped up in the air to apply the proper primes--on the fly, so to speak. To catch his breath, he interrupted the priming once or twice, turning around, bowing, and thanking his unseen benefactor again. After another 10 minutes into the proof the two t's, the primed and the unprimed, occurred in the same equation, whereupon Magnus stopped and turned to the audience one more time, exclaiming, "Es ware eine Katastrophe geworden. Nochmals vielen Dank, vielen Dank!" At the end of the semester, beginning students had to undergo a Fortsetzungsexamen, which determined the right to continue their studies. Magnus's colleague, Arnold Schmidt (known as "Hilbert's last assistant"), who taught Analytische Geometrie und lineare Algebra, announced the results with the words, "Das Ergebnis des Fortsetzungsexamens war niederschmetternd"--The result of the examination was crushing (whereupon the lectern on which he was leaning actually collapsed). But in spite of this ominous announcement, most students actually received passing grades. How did Magnus handle this onerous task for his course? He read about 200 individual names, one after the other, always adding, "Sie haben bestanden"--you passed. Then, at the end of this long recitation, he invited the (few) students whose names he had not called out to come to his office, where he broke the sad news in complete privacy. When I visited Magnus at New York University in 1955, now myself an immigrant to the U.S., the first thing he did was to apologize--apologize for the modest Hundehfitte (kennel) in which he had to receive me (modest compared to his lavish director's office at the Mathematisches Institut in G6ttingen). Magnus loved America but he was also a keen observer of its idiosyncrasies. When he learned that (at age 28) I was still a bachelor, he ventured the prediction that I would either marry within a year or return to E u r o p e "This is not a country for bachelors." Magnus was right--I married, in February 1956, a Bulgarian who
worked for Radio Free Europe and whom I had met three days after m y arrival in America. Some of our joint hikes in the "mountains" north of New York City were along routes Magnus had recommended to us.
Manfred Schroeder Drittes Physikalisches Institut Universitil"t Go'ttingen D-37073 Go'ttingen, Germany Visit to Hua Loo-Keng Thanks to Shuzhong Zhang and Caspar Schwiegman for their excellent article on Hua Loo-Keng in The Mathematical Intelligencer, vol. 16, no. 3. He was a friend of mine, and I published several of his books. I first met him in June 1974, during the "Cultural Revolution." When I asked to see the great mathematician, my host in Beijing said he was "out of town," and only after strong insistence by me did the-y bring Hua Loo-Keng to my hotel. Later, after receiving an honorary degree at Nancy, Hua Loo-Keng visited me in Heidelberg, and'told me of his life while banished to the countryside. He joked that as a mathematician he could at least do some work in the latrine, writing on toilet paper. The main feature of his character was deep love for his country.
Heinz GStze Springer-Verlag Tiergartenstrasse 17 D-69121 Heidelberg Germany
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The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author,
and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in-chief, Chandler Davis.
The Case Against Computers in K-13 Math Education (Kindergarten through Calculus) Neal Koblitz
In Peru, as in many Third World countries, the system of public education is in crisis. Teachers' pay--traditionally low--is falling rapidly because of inflation. The schools are dilapidated, and there is no money for basic supplies. The government has not been responsive to the teachers' protests. Under pressure from the International Monetary Fund, 1 it insists that the state sector--includes public schools--must be cut back. Yet President Fujimori has said that he wants to get computers into the schools as soon as possible. The government's priority is to "modernize" the economy and the educational system, and computerized learning is supposedly one way to do this. For teachers who are trying to cope with financial hardship and abysmal working conditions, what could be more demoralizing than the message that machines come before people? To the Peruvian teachers, Fujimori's advocacy of computers adds insult to injury. On the other hand, not everyone loses. The U.S. computer industry has an interest in creating new markets in the Third World. Thanks to such strategies as interconnected products and planned obsolescence, greater and greater payments will flow to the north as countries like Peru become dependent on U.S. technology in more areas of national life. Throughout the Third World, for about a decade pressure has been mounting to import computer learning from the wealthy countries. In February 1986, a major conference called "Informatics and the Teaching of Mathematics in Developing Countries" was held in
Tunisia. Participants were predominantly from Northern Africa, but many came from the universities and educational establishments of other regions of the Third World. All of the mathematicians and math educators sang the praises of computers and pled for the rapid introduction of computers into their school systems. Not
1The I n t e r n a t i o n a l M i s e r y F u n d , a s E g y p t i a n P r e s i d e n t M u b a r a k h a s said. THE MATHEMATICAL INTELLIGENCERVOL 18, NO. 1 9 1996 Springer-Verlag New York
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a single note of skepticism was raised; not a single question was explored in depth. One could have asked, for example: Are computers truly what schools in Africa need? In practice, would computers be introduced on a general level, or only in a few of the elite schools for the wealthy? Could resources be better spent in other ways--to raise teachers' salaries, purchase classroom supplies, expand libraries? It seems bizarre that the Tunisia conference adopted as an axiom the notion that the introduction of computers should be a priority for elementary education in Africa. It should be noted that much of the support for this conference came from the large French computer companies. Thus, even though the meeting was of little value for people seriously interested in educational issues, the funding companies must have viewed it as a great success, holding open the possibility of lucrative new markets, particularly in the former French colonies of Northern Africa.
Some Caveats Despite my skepticism about computers in education, I do not take an extreme position--I do not want to throw out the baby with the bathwater. There are some appropriate uses for computers in math education. Thus, I will not argue 1. that it is unwise to use calculators in class. (I have m y calculus students use them.) 2. that computer-based math courses are always a bad idea. (My university has an optional computer lab course, taught in conjunction with third-quarter calculus, that seems to be functioning well.) 3. that other technology in the classroom should be avoided. (Nice films on area, volume, and so on have been around for decades.) 4. that special learning programs based on calculators or computers are never successful. (Almost any teaching method can work well under suitable conditions and with a dedicated and enthusiastic teacher.) I will argue, however, that there has been too much hype about technology in math education, and it is time to consider the downside. In my opinion, computers should not be a major component in math education reform. The downside can be divided into several broad areas: 1. 2. 3. 4.
drain on resources (money, time, energy) bad pedagogy anti-intellectual appeal corruption of educators
I shall discuss each of these objections in turn, after a few preliminary remarks. 10
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Popular Culture Any discussion of math education reform should take into account the cultural environment in which we live. American popular culture, which has come to dominate the entertainment industry and the mass media in many parts of the world, can easily cause distortions of the movement for reform. Youngsters who are immersed in this popular culture are accustomed to large doses of passive, visual entertainment. They tend to develop a short attention span and expect immediate gratification. They are usually ill equipped to study mathematics, because they lack patience, self-discipline, the ability to concentrate for long periods, and reading comprehension and writing skills. In the United States, attempts to popularize mathematics tend to be influenced by this ambience. The public television program "Square One" is an example of a well-intentioned effort that went astray because of too much gimmickry and too little attention to content. The politics of funding adds to the pressure on reformers to water down the content. Funding agencies are impressed if grantees can demonstrate short-term success. This may explain why one often comes across pilot projects in American schools where the math content is not appropriate for the target age group--it is too trivial. Of course, the students find it easy, and the project organizers can report great success, thereby satisfying the funding agencies. Such an approach to curricular development contributes to the "dumbing down" of the curriculum. The most important example of gimmickry in math education reform is computermania.
A Drain on Resources We already saw an extreme example of misallocated resources in Peru. But even in a "wealthy" country one has to be concerned about this problem. Many educators in North America share the feeling of betrayal of the teacher who said, They can give us the axe, but they can spend thousands on computers. We have to fire our music coordinator, we have to fire our music teachers, we have shitty libraries. --Lynn, a Canadian schoolteacher, quoted in [14], p. 41 At the University of Washington, we also have resource limitations. After considering various ways to reform the calculus course, we selected a low-tech approach using some applications-oriented lecture notes that I had written. Our calculus reform was implemented relatively quickly, painlessly, and inexpensively, largely because it was not based on computers or graphics calculators. Cost is an issue not only for schools but also for individual students. Ironically, it is sometimes the colleges
Ann Hibner Koblitz discusses a graph theory example with seventh grade girls in Harare, Zimbabwe.
with the highest proportion of working-class students that become most enamored of expensive new gadgetry for teaching mathematics. At a reception for students planning to transfer to m y university from community colleges, the students I talked with were complaining about having to spend more than $80 for a graphics calculator, in addition to $60 for a textbook. (By contrast, the required material for m y 20-week calculus sequence costs a total of about $14.) Another resource issue is that techniques that work in an experimental program with an enthusiastic instructor will not necessarily work under less ideal conditions. One of the lessons of the 1960s "new math" fiasco is that we must look beyond the pilot programs and demonstration classes and selected testimonials, and ask ourselves what is going to happen in a typical classroom with a typical teacher. Every week, in conjunction with a course I teach for math education majors, I spend a morning visiting just such a school. My students and I work with regular sixth-grade classes at Washington Middle School (WMS), an inner-city school in Seattle. The teacher has 5 math classes every day, with a total of over 100 students, many of whom have .severe personal as well as learning problems. When I started visiting her classes in 1992, she had several computers in the back of the room. But they were just gathering dust, and so have been removed. What, if not computers, does an overworked teacher like her need in order to be more effective? She herself told me that she very much regrets not having any time in the
day to talk as colleagues with the other teachers--about pupils they have in common, and about teaching methods and materials. A fundamental problem in education is the failure to treat teachers like professionals. Teachers at schools like WMS need opportunities to upgrade their qualifications, to learn about different teaching materials, and to interact with other teachers as colleagues. This requires release time, light teaching years, or Sabbaticals. If money used to buy computers and software were instead devoted to improving conditions for teachers, it would be money well spent. If pay and working conditions improved for teachers, then maybe more of our best students would go into teaching. All of the fuss about computers serves to divert attention away from the central human needs of the school system--better conditions for teachers and better teacher training.
Bad P e d a g o g y
In response to the grandiose claims of such computers-in-the-schools gurus as Seymour Papert, many experts in child development have pointed out that those claims rest on a flimsy pedagogical foundation, especially where young children (K-5) are concerned. The point is that children benefit most from material that stimulates them to exercise their imagination. For example, simple, unstructured play material like clay, sand, blocks, rag dolls, and finger-painting sets are more THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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w h o l e s o m e entertainment than TV (even "educational" TV) and Power Ranger toys. Harriet K. Cuffaro of the Bank Street College of Education comments on computer painting versus ordinary painting: 9 in "painting" via the computer, the experience is reduced and limited by eliminating the fluid, liquid nature of paint9 In this painting there are n o . . . opportunities to become involved in the process of learning how to create shades of colors; gauging the amount of paint to be mixed; experimenting with and discovering the effects of overlaying colors; understanding the relationship of brush, paint, and paper, the effects achieved by rotating the brush and varying pressure, or how one gains control of or incorporates those unexpected, unintended drips . . . . There is an absence of texture, of smell, a lessening of qualitative associations with the experience of painting ... computer graphics have a "stamped-out," standardized, "painting-by-numbers" quality. Though the design or arrangement of colors, lines, and forms will vary with each child, there is a quality of sameness in appearance ... individuality is flattened by the parameters of the program9 [15, p. 25] M o r e generally, according Columbia Teachers College:
to
Douglas
Sloan
of
For the healthy development of growing children especially, the importance of an environment rich in sensory experience color, sound, smell, movement, texture, a direct acquaintance with nature, and so forth--cannot be too strongly emphasized . . . . At what points and in what ways will the computer in education only further impoverish and stunt the sensory experience so necessary to the health and full rationality of the human individual and society?.. 9 What is the effect of the flat, two-dimensional, visual, and externally supplied image, and of the lifeless though florid colors of the viewing screen, on the development of the young child's own inner capacity to bring to birth living, mobile, creative images of his[/her] own? [15, pp. 5 and 8] Some have also questioned the effect of c o m p u t e r s on the teacher-student interaction 9 Larry Cuban, w h o has m a d e a detailed s t u d y of the history of attempts since 1920 to introduce technology into American schools, writes: In a culture in love with swift change and big profit margins, yet reluctant to contain powerful social mechanisms that strongly influence children (e.g., television), no other public institution [besides schools[ offers these basic but taken-for-granted occasions for continuous, measured intellectual and emotional growth of children . . . . The complex relationships between teachers and students become uncertain in the face of microcomputers.., introducing to each classroom enough computers to tutor and drill children can dry up that emotional life, resulting in withered and uncertain relationships9 [2, pp. 89 and 91] Several educators have criticized the public's unquestioning acceptance of so-called " c o m p u t e r literacy" as a goal for education 9 C o m p u t e r scientist Joseph W e i z e n b a u m has said, "There is, as far as I know, no m o r e evidence [that] p r o g r a m m i n g is good for the mind 12
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The author checks the solution of two children in Cuzco, Peru.
than [that] Latin is, as is sometimes claimed" (quoted in [2, p. 94]). According to John M. Broughton of Columbia Teachers College, Inherent in that term ['computer literacy'] is the promise of generalizability comparable with [that] of reading and writing. However, there is no evidence that programming skills transfer to other areas of psychological development, even cognitive ones. In fact, a recent comprehensive review of the literature by Pea and Kurland suggests that virtually all the claims made about the beneficial educational effects of learning to program are not only inflated, but probably incorrect. Moreover, Pea and Kurland reveal that there is not even support for the . 9 notion that learning to program aids children's mathematical thinking9 Their own research study on transfer revealed that Logo programming experiences had no effect on the planning skills that are deemed central to problem-solving skills9 The tradition of grossly inflated claims identified in the artificial intelligence literature ... appears to have carried over into the ... area of electronic learning. [15, p. 109] The inability to develop good translation software has been one of the most embarrassing failures of Artificial Intelligence 9 If the best computers in the world are unable to translate from French into English, then they certainly cannot help m y calculus students do what is the main point of the course: translating w o r d problems into mathematics. C o m p u t e r s in this course w o u l d only be a distraction and a waste of time and resources. If the focus in beginning calculus is p u t w h e r e I believe it bel o n g s - - o n analyzing real-world problems and choosing the appropriate mathematical t e c h n i q u e s - - t h e n the course cannot be centered a r o u n d computers or graphics calculators. At m y university we have an optional one-credit computer lab for students in their third quarter of calculus 9 At this point it makes sense to offer the technology, because (1) the emphasis shifts from w o r d - p r o b l e m applications to geometric applications of calculus in three
The author presents math enrichment topics in an after-school program in a poor barrio in San Juan, Puerto Rico.
dimensions, where computer graphics serve a useful purpose and (2) students at that level are sophisticated enough to benefit from computers. At all levels of schooling we have to ask: Do the students learn to punch buttons, or do they learn mathematics? One day at Washington Middle School, we had the sixth graders play a math game that involves dividing by 7 and rounding off to the nearest integer. When they had to find 60 - 7, they punched it correctly into their calculators, which displayed 8.5714 . . . . But most of them could not read or interpret the answer: they did not understand the significance of the decimal point. Similar dangers exist among older students. For this reason, in the calculus final exams at my university we usually ask for exact (not decimal) answers. For example, sin(60 ~ = 1//2 V 3 , not 0.866; the circumference of a circle is 2~-r, not 6.283r. Anti-Intellectual Appeal
Computers reinforce the fascination with gadgetry, as opposed to intellect, that is endemic in American popular culture. As pointed out by Brian Simpson, former education advisor to IBM in the United Kingdom, Technological solutions to educational problems often have a seductive appeal, promising to make education easier and more enjoyable than ever before. In the past, extravagant claims have been made for teaching machines, educational television, language laboratr)ries, and even such improbable, esoteric phenomena as sleep learning and learning under music-induced hypnosis. [15, p. 84] There is a long history of people wanting and expecting technology to transform education. It has been over 70 years since the following prediction was made by a famous American:
I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of fextbooks. --Thomas Edison, 1922 (quoted in Ref. 2, p. 9) Perhaps the most frequent argument for computers in the schools--and also the most illogical--is the inevitability argument: "Calculators/computers are going to be everywhere, so we might as well incorporate them into math class. One can't fight against the tide." Using the same reasoning, one could say that, since automobiles are everywhere in our society and play a crucial role in all of our lives, therefore cars should be used as much as possible in education, and driver education should be regarded as a centrally important subject in school, much more so than such relatively useless subjects as music and art. The inevitability argument for computers in the schools is exactly the same sort of antiintellectual non sequitur. Computers are usually used in the classroom in a way that fosters a Golly-Gee-Whiz attitude that sees science as a magical black box, rather than as an area of critical thinking. Much computer use is "teaching by demo" with the student as spectator. There is then little difference between so-called electronic learning and simply watching television. Most software is based on immediate gratification and does not encourage sustained mental effort. While physically playing an active role, the pupil is intellectually passive, and has little opportunity to be creative; that is, the pupil is programmed to follow a path already laid out in detail by others. Even when software designers try to get the children into a creative mode, in many cases the same could be better accomplished without the technology. Educators tend to put the cart before the horse: instead of asking whether or not technology can support the curriculum, THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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they try to find ways to squeeze the curriculum into a mold so that computers and calculators can be used. Like a quack cure in medicine, perhaps the most harmful effect of the computer craze is that it diverts people from other, more solidly grounded approaches to treating the ailments of mathematics education. Might not the Golly-Gee-Whiz-Look-What-ComputersCan-Do school of mathematical pedagogy eventually come to be regarded as a disaster of the same magnitude as the "new math" rage of the 1960s?
Computer Science Is Not the Same As Computers One can strongly advocate increased teaching of computer science (and related areas, such as discrete math) while opposing the use of computers. Computer scientist Michael Fellows, who is an active campaigner for computer science in grade school (see [ 3]), has said: "Most schools would probably be better off if they threw their computers into the dumpster." Fellows uses the term "Cargo Cult" to refer to the fetishization of computers by the media and educational establishment.
Definition of Classical Cargo Cult: An isolated civilization comes into initial contact with European technology. Ignorant of modern science, they interpret the benefits of technology in terms of their familiar world and their familiar mode of operation. They pray and perform sacrifices, or do whatever they think might be necessary to induce the deities to bring them the Cargo. Modem Cargo Cult- In the United States and many other countries, most of the general public is prescientific, in the sense of having no rational understanding of the intellectual processes that go into scientific advances or their application to the real world. On the other hand, like the classical Cargo Cultists, they realize that technology is associated with economic well-being, and that something must be done so that youngsters will later be able to reap the benefits of the "computer age." The natural response, then, is to fetishize computers and fit them into the familiar world of traditional mindless school math. The public needs to understand that math and computer science are not about computers, in much the same w a y that cooking is not about stoves, and chemistry is not about glassware; that is, COMPUTER SCIENCE q~ COMPUTERS The meaning of this inequality is: What children need in order to become mathematically literate citizens in the computer age is not early exposure to manipulating a keyboard, but rather wide-ranging experience working in a creative and exciting way with algorithms, problem-solving techniques and logical modes of thought. 14
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Money Corrupts Much of the effort to introduce technology in the classroom is profit-driven. As Douglas Sloan says, It does not take a flaming Bolshevik, nor even a benighted neo-Luddite, to wonder whether all those computer companies, and their related textbook publishers, that are mounting media campaigns for computer literacy and supplying hundreds of thousands of computers to schools and colleges really have the interests of children and young people as their primary concern. [15, p. 3] In the words of Joseph Menosky, an American writer and former science editor at National Public Radio, Certainly those who have a great deal to gain from a universal acceptance of computer literacy--microcomputer firms selling hardware, textbook companies selling educational software, organizations selling worker and teacher retraining courses, and writers and publishers selling books and instructional guides--have done a brilliant, if morally indefensible, job of commercial promotion. [15, p. 77] Corporate domination of math education reform has corrosive effects. The profit motive has an excessive influence; the intrinsic value of a pedagogical idea is not considered to be as important as its saleability. Educational ideas that are not based on expensive gadgetry or new textbooks are not likely to be supported strongly. There is an excess of faddism and hype. It is regrettable that computers have been so aggressively marketed to teachers and school systems. In speaking to parents, teachers, and school boards, many company sales representatives take the hard-sell approach: "If you don't buy our latest products, you will be neglecting to prepare your children for the 21st century." Foundations and government agencies, such as the National Science Foundation (NSF) in the United States, compound the problem, because they share the biases of the commercial interests. Money from corporate and foundation sources seduces educators, whose objectivity and independent judgment become compromised. Grants can corrupt. The technology-in-education movement has some of the characteristics of a religious evangelical campaign, fueled by corporate and foundation money. One sees the same reliance on testimonials. A technology enthusiast might proclaim, "How graphics calculators have changed my life!"--just as a born-again Christian talks about rediscovering Jesus. Like their religious counterparts, technology boosters tend to adopt a Manichaean view, dividing educators into two camps: those who have seen the light, and the fractious infidels. I recently had a personal encounter with this closedmindedness. About 2 years ago, the NSF asked me to help evaluate calculus reform proposals. But when they learned that I am skeptical about computers and graphics calculators in calculus, they changed their minds and
Asian students using abacuses. (Photo used by permission of Superstock.)
decided not to send me any proposals to review. They did not want any input from a nonbeliever.
Low-Tech Is Better To end on a positive note, the good news is that the vast majority of enrichment topics do not require high technology, but only pencil and paper and inexpensive manipulatives, such as geoboards and abacuses? All material that my math education students and I present to the Washington Middle School youngsters is low-tech. Of the four examples given below, the first two have worked well with the WMS sixth graders. The cryptography exercise is easy. The answer is "iHasta la vista!"-a phrase that non-Spanish-speaking kids usually know from the movies. The second exercise is more challenging, since we ask the children to solve it by first constructing a mathematical model: a graph in which two vertices are joined by an edge whenever the corresponding girls can room together. The third example is accessible to slightly older children. The fourth problem is taken from a calculus final exam at the University of Washington. A calculator might be useful (but is not necessary) for the third problem, and should certainly be-used in part (c) of the fourth problem. 2Accordingto the Wall StreetJournal (November22, 1994),the abacus is making a comeback in Asian schools, replacing calculators. The sound and tactile sense give children a feel for the algorithmic dynamics-kids say things like "to subtract 7 you add 3 and then subtract 10"--whereasthe use of calculatorsmade the childrenfeelalienated from the arithmetic.
Examples Example 1: Cryptography. You intercept the coded message EXPQXIXSFPQX, which you know was formed by a shift in the 26-letter alphabet (Caesar encryption). The message is in Spanish. Using the fact that "a" is the most frequently occurring letter in Spanish, break the code and read the message. Example 2: Mathematical Modeling. Ten girls from different parts of the world are coming to spend 4 weeks at International Summer Camp. The girls will live two to a room. As Head Counselor, you have the job of selecting roommates. Each girl must room with a girl with whom she has a common language. The list of campers, together with their countries of origin and the languages they speak, is given below. Try to find a perfect matching. Conchita (Guatemala)--Spanish Eliane (Brazil)--Portuguese, Spanish Farkhonda (Egypt)--Arabic, English Hilweh (Jordan)--Arabic, French Keiko (Japan)--Japanese, French Mei Li (Macao)--Cantonese, Portuguese Pilar (Spain)--Spanish, French Selani (Zimbabwe)--Shona, Ndebele, English Sisai (Botswana)--Shona, English Zeineb (Tunisia)--Arabic, French
Example 3: Statistics. In a certain Third World country, in 1985 the annual salaries of THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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7,000,000 people were approximately $800 2,000,000 people were approximately $1200 900,000 people were approximately $2000 100,000 people were approximately $20000 After several years of a new economic policy called "Structural Adjustment Program" (SAP), a statistical study in 1995 found that 10,000,000 people earn approximately $600 per year 1,800,000 people earn approximately $1000 per year 550,000 people earn approximately $6000 per year 150,000 people earn approximately $40000 per year
time was heavily French-influenced, imported the new pedagogy despite the strenuous opposition of the leading Peruvian mathematicians (particularly Professors C. Carranza and M. Helfgott). Of course, the introduction of Bourbaki-style mathematics in the schools was a disaster in Peru, as elsewhere. The Peruvian mathematicians had been right, and the French had been wrong. Will history repeat itself? Will countries around the world again import a poorly thought out and unworkable pedagogy from the United States and France? This time the cost will be higher. The "new math," for all its foolishness, at least was relatively inexpensive.
References Find the mean, median, and mode of the annual incomes in 1985 and in 1995. (a) Which average would be used by someone who wanted to argue that the SAP had been a success? (b) Which average would be used by someone who wanted to argue that the SAP had been a failure?
Example 4: Calculus. At time t = 0 you borrow $5000 to buy a car. You pay back the loan with monthly payments of $50. The annual interest rate is 8% comp o u nd ed continuously. Make the simplifying assumption that the payment of $50/month is also made continuously, rather than in lump sums once a month. (a) Write a differential equation for the amount y(t) you still owe after t years. (b) Find a formula for y(t) by solving this differential equation. Please show all steps clearly. (c) After how many years will you have paid off the debt? Give a decimal answer. (d) Make up a problem about a unicorn farm* which leads to the identical mathematics as the above problem about loans (i.e., the same differential equation and the same initial value of the unknow n function). Your problem should involve bi r t h / deat h rates and something about how many unicorns are being sold as pets. Use complete English sentences. What question about the unicorn farm is analogous to part (c) above? *A unicorn is a mythical horselike animal which does not exist outside of ancient legends and modem calculus exams.
Epilogue: First Time it's Tragedy, Second Time it's Farce At the beginning of the article I mentioned that President Fujimori of Peru has become a fervent advocate of computers in the schools. It is interesting to note that in the late 1960s, Peruvian education officials came under influences that were in some ways a foretaste of the computer rage. The "new math" was then the reigning paradigm in math education in the wealthy countries. The Peruvian education ministry, which at that 16
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1. Consortium for Mathematics and Its Applications, For All
Practical Purposes: Introduction to Contemporary Mathematics, New York: W. H. Freeman (1988). 2. L. Cuban, Teachers and Machines: The Classroom Use of Technology Since 1920, New York: Teachers College Press (1986). 3. M. R. Fellows, Computer science and mathematics in the elementary schools, in N. D. Fisher, H. B. Keynes, and P. D. Wagreich, editors, Mathematicians and Education Reform 1990-1991, Providence, RI: American Mathematical Society (1993), pp. 143-163. 4. M. R. Fellows, A. H. Koblitz, and N. Koblitz, Cultural aspects of mathematics education reform, Notices of the American Mathematical Society, 41 (1994), 5-9. 5. M. R. Fellows and N. Koblitz, Math Enrichment Topics for Middle School Teachers, Seattle, WA: Math Liberation Front (1994). 6. N. Koblitz, Math 124/125 (Calculus Lecture Notes), University of Washington Mathematics Department (1995). 7. N. Koblitz, The profit motive: the bane of mathematics education, Humanistic Mathematics Network Journal, No. 7 (1992), 89-92. 8. Mathematical Sciences Education Board and National Research Council, Measuring Up: Prototypes for Mathematics Assessment, Washington, DC: National Academy Press (1993). 9. National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics, Reston, VA: NCTM (1989). 10. National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics, Reston, VA: NCTM (1991). 11. S. Papert, Mindstorms: Children, Computers, and Powerful Ideas, New York: Basic Books (1980). 12. R. D. Pea and D. M. Kurland, On the cognitive effects of learning computer programming, New Ideas in Psychology 2 (1984), 137-168. 13. R. D. Pea and D. M. Kurland, Logo Programming and the Development of Planning Skills, Technical Report No. 16, New York: Bank Street College of Education (1983). 14. K. Riel, Factors That Influence Teachers" Use and Non-Use of Computers, M.A. in Education thesis, University of Victoria (1994). 15. D. Sloan (ed.), The Computer in Education: A Critical Perspective, New York: Teachers College Press (1985). 16. C. Stoll, Silicon Snake Oil: Second Thoughts on the Information Highway, New York: Doubleday (1995). Department of Mathematics University of Washington Seattle, WA 98195, USA
Some Kinds of Computers for Some Kinds of Learning: A Reply to Koblitz Ed Dubinsky and Richard Noss
Introduction It is surely time to examine the computer's role in mathematical teaching and learning. The initial flush of enthusiasm over "new technologies" is beginning to pass, and we should now begin to think carefully about the ways in which computers may help students. A dispassionate, well-informed examination of costs, benefits, and difficulties is needed. Unfortunately, Neal Koblitz has failed to make such a start. Moreover, by confounding different aspects, he has ironically ceded decision power to the computer and software giants which he and we--deplore. Koblitz begins by casting doubt on the need of the third world for computers in schools. The question is up to them to decide for themselves, but he is free to advise them. We read on, awaiting his analysis. The view of computers in education which he offers is, alas, so narrow as to border on distortion. Following Koblitz's organization, we will begin with the economics of educational commodities, which come from the industrialized countries and are sold wherever the market can be found--in the first, second, or nth world. Then we will respond to his discussion of the use of computers in education (in any ith world), and argue for a broader view, encompassing phenomena of which Koblitz seems unaware and leading perhaps to very different conclusions. We believe a computer environment to help students learn mathematics can be designed in ways very different from those which Koblitz (quite properly, for the most part) rejects; we believe these methods can be extremely effective.
medicine. The pharmaceutical giants certainly" put profits ahead of third-world welfare: artificial milk products are killing babies throughout the "undeveloped" world. Do we conclude that the introduction o,f new medicines should be stopped? Hardly. Rather we distinguish between therapeutic alternatives, evaluate them on the basis of research, and try to see how cures and prevention of illness can be integrated into people's lifestyle. Just so for computers. They are used for both missile guidance and microsurgery. It is the culture that determines how the technology enters, not the technology itself. As one of us has written elsewhere, "Pieces of knowledge are appropriated (or not) depending upon pupils' own agendas, how they feel abouf their partici-
Use of Computers in Third-World Education Koblitz suggests that computers add "insult to injury" in the third world, and their spread is aimed primarily at expanding the market for the first world manufacturing industry. True enough. True for the first world as well: for example, the intl:oduction of computers into UK schools in the 1980s was handled by a government department whose brief was to expand the UK's computer industry, not to help education. True for other industries, too: everything Koblitz says about technology holds for the billion-dollar textbook publishing industry (including sales of left-over stock to the third world). The conclusions are not so clear. Take the example of THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 9 1996 Springer-Verlag New York
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pation, teacher intervention, and above all, the setting in which the [computer] activities are undertaken." (Hoyles and Noss, 1992).
for the potential of an environment that includes all of these and also appropriately used computers.
The Charge of Anti-Intellectualism Resources and Cultures Koblitz attacks computers as the epitome of "popular culture," which is based on passive "entertainment" and which, in the educational domain, entails a "dumbing down" of content. Of course. Modern economic and social life is based on the commodification of essential physical, emotional, and intellectual life: on casting people as consumers rather than as whole human beings (see Noss, 1994). This is the central role assigned in modern capitalist economies to all technology. But here again Koblitz has a simplistic view. Here again it is not just the technology, but how it is used. Koblitz is right to observe that technology can be used to lull a child into a passive inability to "concentrate for long periods," but the same technology can hold a child engaged for hours; the machines that offer immediate gratification for minimal reading and writing skills can also afford young children ways to craft complex and meaningful pieces of writing, and introduce them to the skills of editing and redrafting; the same computer that runs software in which mathematics problems are solved simply by pushing the right buttons can also support systems that students use to construct mathematical tools on the computer and simultaneously construct mathematical concepts in their minds. We can no more blame "computers" for a passive popular culture than we can blame "medicine" for encouraging Indian mothers to stop breast-feeding. Our big disagreement with Koblitz is that he bases his argument on one way of using computers--admittedly a widespread w a y - - a n d ignores other uses to which his complaints do not apply. We are also not much enamored of Koblitz's alternative, which is, apparently, that a mathematician gets up a set of notes for a course and if the notes are good enough, calculus will be fixed. Where have we heard that before? He would have us do things in the same old way, but better. There are already many talented and dedicated practitioners of traditional pedagogies, and still, even in the industrialized countries, children are neither learning nor liking mathematics. Koblitz asserts that computers are a "drain on resources." We are certainly opposed to any attempt (and such attempts are underway) to replace teachers by computers. Computers alone cannot make good teachers, and we are not encouraged by attempts to get computers to do the same things that teachers do (see Noss, 1995, for evidence of this assertion in undergraduate mathematics teaching). We agree that computers are a drain on resources if used as an excuse to fire teachers. Teachers are a higher priority, as are books, desks, roofs over classrooms, warmth, and running water. We argue 18
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Koblitz argues that computers foster anti-intellectualism. This overlaps his earlier argument concerning passivity. "Most software is based on immediate gratification, and does not encourage sustained mental effort," he says. Software of that sort we reject also. But a small (yet substantial) fraction does not supplant mental activity; indeed it does not do very much at all. The wordprocessors with which we are writing, or the high-level programming languages in which we express mathematical ideas do not "do" or "encourage" or "gratify" in any way. Each is a medium of expression, like a piano or a pen. Whether creativity occurs depends on social responses of people doing things with the cultural tools they have at their disposal. If a particular musical composition is boring, don't blame the violin. In the case of learning mathematics, we could not agree more with Koblitz that "teaching by demo" is bad pedagogy. But we cannot avoid the observation that amongst the worst examples of this pedagogical strategy are having students listen to lectures, follow worked-out problems in a text, or read notes written by a mathematician who is not paying attention to how learning actually occurs.
Computers and Pedagogy Now we come to the meat of Koblitz's paper. He claims that computers generate "bad pedagogy." He attacks the "grandiose claims" of, among others, Seymour Papert, drawing evidence from several references. Some of these (e.g., Cuffaro, Sloan, Cuban) are ten-year-old expressions of opinion, from an interesting but bygone era when it was fashionable to discuss whether computers were as good as paintbrushes for artistic expression, or whether computer sounds were as pleasant as those produced by a Stradivarius. Such discussions had appeal to some, but "evidence" they are not, and it is somewhat disingenuous of Koblitz to cite them as such. We have two points regarding computers and pedagogy. One is that, as we have already indicated, there is a multitude of ways in which computers might be used in education. These include "microworlds" where mathematical phenomena are an integral part of the learner's environment; computer algebra systems that students use to explore ideas and solve sophisticated problems; mathematical programming languages used to construct mathematical concepts on the computer; graphing facilities that allow the learner to see complex phenomena; multiple-representation software with which a student can see the effect that changing a feature of one representation has on the others; playful and experimental "games" in which learners try to accom-
plish entertaining tasks; and spreadsheets in which the structure of the software provides a powerful w a y to model some mathematical situations. We do not endorse everything done with all of these pedagogical tools; we are not agnostic as to their relative merits (nor do we suggest they are well-ordered). We do claim that there is a lot more to this multitude of software than appears in Koblitz's simplistic report-and much of it can be used to make a significant difference in how much is learned by how many people. The second point we wish to make is this. If Koblitz rejects "grandiose"--unjustified--claims, well, of course. But there is a growing literature describing effective strategies and providing evidence of their effectiveness. All we can do here is mention some examples. For effects of having students write programs to learn mathematical induction, see Dubinsky, 1989; to learn predicate calculus, see Dubinsky, in press; to learn functions, see Breidenbach, et al, 1992. The contributions in Hoyles and Noss (1992) testify to the breadth of the encounter of computer programming in Logo with mathematical topics including three-dimensional geometry, group theory, and dynamical systems. For an overview of experiments showing both success and failure of ways of using calculators to learn various mathematical concepts, see Dunham, 1993; for spreadsheets, see Smith, 1992. The diverse contributions in diSessa, Hoyles, and Noss (1995) amply illustrate the wide range of mathematical ideas which the computer (e.g., programming languages such as Logo and Boxer, dynamic geometry tools such as Cabri Geometry, and the trusty spreadsheet) can open to learners and teachers alike. We feel that this literature is quite different from Koblitz's choice of the Holy Grail of research, the paper which has become ubiquitous in arguments that computers have no (or detrimental) effect on mathematical learning. That paper is by Roy Pea and Midian Kurland, two researchers from Bank Street College who undertook a series of studies of the computer programming language Logo and its effect on learning mathematics. We are sorry to see Koblitz join the ranks of writers who turn to Pea and Kurland as evidence without comment. For the benefit of readers unfamiliar with Pea and Kurland's research, we outline here their "findings." Pea and Kurland mounted a series of studies in the early 1980s (a typical and oft-quoted example is Pea and Kurland, 1984). Their aim was to explore the "effects" of young children's learning to program a computer in the programming language Logo. As psychologists, not mathematicians, they were interested in "transfer" of "human cognition." In a typical experiment, they took (not randomly) a sample of 24 children from a private school, half of whom learned Logo, and half of whom did not (we are not told what these children did instead). At the end of a four-month period, the children were asked to devise a plan to carry out six classroom chores on a transparent plexiglass map of a fictitious classroom.
Success on this task (as measured by a fairly arcane set of criteria) was found to be not significantly different between the Logo and "control" groups. N o w many charges might be laid against this experiment, concerning misplaced objectives, faulty statistics, and inadequate methodology (see Hoyles and Noss, in press). We will restrict ourselves here to the point most relevant for Intelligencer readers: the "finding" tells us nothing about the learning of mathematics. It does not show that "all the claims made about the beneficial educational effects of learning to program are not only inflated, but probably incorrect." Neither does it show that "there is not even support for t h e . . , notion that learning to program aids mathematical thanking." It is simply an irrelevance. Worse, an irrelevance based on a finding of "no significant difference." If it is supposed to be a counterexample, what is it a counterexample to? Most telling is the failure of Pea and Kurland to document what the children did with Logo. Were they given specific activities? If so, what kind? Did the activities relate to the idea of planning which would subsequently be tested? We simply do not know. Pea and Kurland showed that young children who had been given a copy of Logo were no better at solving a cute puzzle involving pushing in chairs and cleaning tables than those who had not. So what? The contrast between Pea and Kurland's actual research and the way it is invoked by Koblitz points to the bottom line on the literature in this field. There are many reports, some of which may provide evidence of something; but if the subject we are addressing is how to make the learning of mathematics more effective and enjoyable, we must weigh the studies which bear on the subject, and leave aside the studies which do not.
What Are We Trying to Achieve? We find this curious sentence in Koblitz's article: "If the best computers in the world are unable to translate from French into English, then they certainly cannot help my calculus students do what is the main point of the course: translating word problems into mathematics." We share Koblitz's goal that students should learn to translate word problems, although we would rather think of it as using mathematical structures to model situations described in word problems. His argument here seems misdirected. It is superficial to insist that the computer be capable of doing what we wish the student ultimately to accomplish. The computer may play a role different from the human's, as a piano may accompany an oratorio. But there is a deeper issue. Does Koblitz really believe that the main point of learning calculus is translation of word problems in the w a y that he could imagine a computer doing? To us, the point is to help students learn to respond to a situation by mathematizing it, by constructing functions and relationships which can make THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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sense of it, and by using tools such as calculus to answer questions about it. This is a very different matter from mindless translation of the sort that computers might (but, in general, cannot) do. It may be that Koblitz is in agreement with us on this substantive point, but he blames the computer for a failure to think through an appropriate pedagogy. Let's agree that translation (in the sense of modeling) is important: why can't we use appropriate computational tools to help us, and reject those that are inappropriate? Certainly the work we have referred to suggests that appropriate uses do exist. Turning to computers themselves, Koblitz offers the following banal observation: "What children need in order to become mathematically literate citizens in the computer age is not early exposure to manipulating a keyboard, but rather wide-ranging experience working in a creative and exciting way with algorithms, problem-solving techniques, and logical modes of thought." If anybody thinks otherwise, it must be an advertising agent of a computer company, whom we join Koblitz in despising. But why is there that word "rather"? Is exciting work for children possible in some technologies but not in others? Are pens and paper acceptable? Presumably. Light pens and word processors, presumably not. Dynamic geometry tools and programming languages, definitely not. We ask Koblitz this question: Where does he draw the divide? Does the inclusion of a chip rule out the tool? And if so, why should the tools used by the most creative professionals (including creators and users of mathematics) be closed to children?
Cash, Computers, and Choices Koblitz says that "money corrupts." He reiterates the negative effect of corporate profit on education. We concur. He ridicules "believers" who say, "Graphics calculators have changed m y life." We take his point. Our (horror: electronic) thesaurus says, "Belief: creed, conviction, faith, tenet, doctrine." Koblitz is objecting to those who make high-tech a religious prejudice. It is a shame that he reduces his objection to the same level. If Koblitz is really concerned about the corruptive effect of cash, then he should go after the economic system in which the only way that educational materials can be produced and widely disseminated is through someone's making a profit. In such a system, it is natural that profit-makers want more profit, and there arises a conflict between what makes sense educationally and what will sell. Many calculus reformers in the US are just now discovering how difficult this makes it to introduce innovations into their materials. This is a very big problem, but Koblitz is wrong in suggesting that it has to do with computers. It goes much deeper. In his understandable effort to resolve the dilemma, Koblitz concludes that "low tech is better." He gives four examples, one of which is a "real world" calculus prob20 THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996
lem based on continuously compounded interest payable on a car loan. Apparently the technology of the internal combustion engine, the business practice of banks, and the role of the car in consumerism are on the right side of Koblitz's technological divide. Whose "real world" does Koblitz have in mind? And can he cite a single study that shows that using his kind of example has any positive effect on student learning? We wouldn't even dare to ask for such information about the effectiveness of the "low-tech approach using some applications-oriented lecture notes that [Koblitz] had written." Koblitz must know that such an approach has been tried by a multitude of mathematics teachers. Some, after looking at the results, have concluded that much more serious thinking about learning is needed. Some of us, mathematicians and educators, feel that we are not in a position to be complacent, and that we cannot afford to reject out of hand new technologies that might--just might--help us in our task.
References Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992), Development of the process conception of function, Educational Studies in Mathematics 23, 247-285. diSessa, A., Hoyles, C., & Noss, R., eds. (1995), Computers and Exploratory Learning. Berlin: Springer-Verlag. Dubinsky, E. (1989), Teaching mathematical induction II, The Journal of Mathematical Behavior 8, 285-304. Dubinsky, E. (in press), On learning quantification, Journal for Computers in Mathematics and Science Education. Dunham, P. (1993), Does using calculators work? The jury is almost in, UME Trends 5.2, 8-9. Hoyles, C. & Noss, R., eds. (1992), Learning Mathematics and Logo. Cambridge MA: MIT Press. Hoyles, C. & Noss, R. (1992), A Pedagogy for mathematical microworlds, Educational Studies in Mathematics 23, 31-57. Hoyles, C. & Noss, R. (in press), Windows on Mathematical Meanings. Dordrecht: Kluwer. Noss, R. (1994), Structure and ideology in the mathematics curriculum, For the Learning of Mathematics 14, 1. Noss, R. (1995), Reading the Sines. Unpublished report of the evaluation of the Teaching and Learning with Technology Project #15, Institute of Education, London. Pea, R. & Kurland, M. (1984), On the cognitive effects of learning computer programming, New Ideas in Psychology 2, 137-168. Smith, R. (1992), Spreadsheets at Joint Meetings in Baltimore, UME Trends 4.2, 1-3. Ed Dubinsky Department of Mathematics Purdue University West Lafayette, IN 47907-1968 Richard Noss Department of Mathematical Sciences University of London, Institute of Education 20 Bedford Way London WCIH OAL England
Hilbert's Problems and Their Sequels Jean-Michel Kantor II n'y a pas de probl~mes rdsolus, il n'y a que des probl~mes plus ou moins rdsolus. 1 Henri Poincar6
"To speak of the future of mathematics is an exercise in fantasy which can not be discouraged too strongly; it is really a p u r e absurdity . . . . [The state of mathematics in the year 2000] will be just fine if the atomic physicists or some peace conferences don't s u d d e n l y interrupt the course of progress" (R. Godement, 1948, Gen. Ref. [LeL]2). T o d a y the historical d e v e l o p m e n t of the subject can no longer be treated more than partially, and even then it takes a team such as was assembled at the conference in the United States in 1974 on the problems David Hilbert had posed in 1900. Poincar6 and Hilbert were no d o u b t the last mathematicians to have an overall conception of their science; it is no longer within the scope of one mathematician to handle the mathematical legacy of Hilbert. This m a y serve to excuse in part such omissions and errors as m a y be found in the following, which are m y responsibility alone, despite the kind help of V.I. Arnold, M. Berger, M. Berry, H. Br6zis, P. Cartier, J.-L. Colliot-Th616ne, J. Dixmier, M. H i n d r y , J.-P. Kahane, V. Kharlamov, B. Malgrange, Y. Matijasevich, B. Mazur, J. Milnor, C. Sabbagh, J.-P. Serre, G. Tenenbaum, and M. Waldschmidt. M y w a r m thanks to them, and to the staff of the interuniversity science library of Jussieu, and to Alberto Arabia.
Who would not lift the veil that hides the future from us, to have a look at the progress of our Science and the secrets of its further development in future centuries? [H, p. 58] H e lectured on 10 of the 23 problems later published in the Proceedings of the Congress: On foundations--P.r.obtems 1, 2, and 6. Four problems on arithmetic and a l g e b r a - - P r o b l e m 7, Irrationality and transcendence; P r o b l e m 8, The Riemann hypothesis; P r o b l e m 13, Superposition of two functions; and P r o b l e m 16, Ovals and limit cycles. Three m o r e on theory of f u n c t i o n s - - P r o b l e m 19, Calculus of variations; P r o b l e m 21, Fuchsian differential equations and m o n o d r o m y ; a n d Problem 22, Uniformization. To Hilbert, the choice of problems was supposed to show the d e e p unity of mathematics and support his declaration of faith in a glorious future (in mathematics there is no " i g n o r a b i m u s ' - - D u b o i s - R e y m o n d ) . The
The Context 3
Poincar6 m a r k e d the first International Congress of Mathematicians in 1897 b y a lecture on the relations between mathematics and physics. When Hilbert was invited to speak to the next International Congress (Paris, 6-10 August 1900), he hesitated between an answer to Poincar6 and a list of problems to stimulate the research of the new century. He asked Minkowski, "Might I point to probable directions of the mathematics of the new century . . . a look to the future?" Minkowski was enthusiastic. At Paris, Hilbert had a rapt audience:
Jean-Michel Kantor
Jean-Michel Kantor of the Universit~ de Paris VII has been
"From Plato to Queneau" on mathematics for students in
1,,There are no solved problems, there are only more-or-less solved problems." 2Referencesare indicated by authors' initials. Some are general (Gen. Ref.) and some pertain to particular problems. 3For the entire section, see Gen. Refs. [HI and [MI.
the French adaptation of the beautiful book Rasskazy o matematikakh i fizikakh by Simon G i n ~ , under the title
Pendules, horloges et mdcanique cdleste (Diderot Editeurs, Paris, I995). We await future installments.
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 9 1996 Springer-Verlag New York
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Hilbert problems went on to have the success we all know; their history is rich in instructive anecdotes, but also in blind alleys. There are also some surprises waiting in their present status. Let me try to bring the problems up to date; I emphasize the progress made between 1975 and 1992, because a serious work was produced by the American Mathematical Society seminar in 1974. For an introduction to the situation in number theory (before the "Wiles bomb"), see [Ma].
T h e Present Status Problems 1 and 2: The continuum hypothesis; consistency of arithmetic. Associated with the first two problems are the names of Kurt G6del and Paul Cohen. The continuum hypothesis, the subject of Problem 1, may be stated as Every uncountable subset of R has the same cardinality as R,
of exhaustion, whereasfor plane figures the calculation of the area of any polygon is reducible by decomposition ("scissors geometry") to that of the square. To show that such a reduction is impossible for volumes. Actually the problem was already solved by Bricard (1896) and Hilbert's student Dehn (1900), based on the notion of invariant; two solids which can be cut up into congruent pieces necessarily have the same Dehn invariant. Call "scissors congruence invariant" an element D(P) (of a group) associated to every polytope P in such a way that D(P N P') + D(P 0 P') = D(P) + D(P') if P, P', and P U P' are polytopes. Further, D(P) = 0 if the polytope is planar; and for g a motion of space, D(g(P)) = D(P). The Dehn invariant is defined as follows. The group of angles between lines in the plane is identified with R/2"rrZ, and one sets D(P) = Z
ILil Q
8i, D(P) E R C)z(R/2rtZ),
i or
2~~ = R1. The consistency of arithmetic was treated by G6del, who showed (1931) the incompleteness and the undecidability of arithmetic. G6del showed in 1938 that if set theory (according to the Zermelo-Frankel axioms) is consistent, then it remains consistent if augmented by adjoining the axiom of choice a n d / o r the continuum hypothesis. Finally, in 1963 Paul Cohen showed that if the Zermelo-Frankel axioms are consistent, then the negation of the axiom of choice or even the negation of the continuum hypothesis4 can be adjoined and the theory will remain consistent! G6del's impact was felt especially outside of mathematical circles: it did not change the way of working of mathematicians--with the obvious exception of logic i a n s - a n d it gave rise to a huge trove of citations eligible for the next edition of the Dictionary of Stupidity. 5 J.Y. Girard sums up G6del's discovery by this image: There are things in mathematical thought which are not part of the mechanism. [G] Problem 3: Congruence of polyhedra. "On the equality of the volumes of two tetrahedra with equal base and altitude." Two pyramids with triangular base and the same altitude are in the same ratio as their bases. The proof (Euclid) uses the method
where Li is an edge and 3/the dihedral angle at Li. Then Dehn shows that the cube and the regular tetrahedron of the same volume do not have the same Dehn invariant. The problem was considered as solved. However, Sydler in 1965 showed that two polytopes are equivalent if and only if they have the same volume and the same Dehn invariant (see [B], [C], and [S]). On the other hand, no analogous results are known in higher dimensions, or in non-euclidean geometry in three dimensions or more. More recently, other invariants, the Hadwiger invariants, were introduced in arbitrary dimensions, with respect to the group of translations, and it was shown that for two polytopes to be equivalent when only translations are allowed, it is necessary and sufficient that they have the same Hadwiger invariants. We refer to [C] for the connection to the calculation of Eilenberg-Mac Lane homology groups and for current developments. Problem 5: Topological groups and Lie groups. Is a locally euclidean group a Lie group? This problem was in fashion in the 1950s. It was solved in 1953 by Gleason and by Montgomery and Zippin. The following question still remains open: Given a locally compact topological group acting faithfully on a topological manifold, is it a Lie group? For example (actually, equivalently), can the p-adic integers act faithfully on a compact topological manifold? We observe that the Montgomery-Zippin theorem plays a crucial role in a recent important work of M. Gromov [G].
4The generalized continuum hypothesis implies the axiom of choice. 5Example: R. Debray (Le Scribe, 1980): "From the day that G6del s h o w e d that there is no consistency of Peano arithmetic formalizable within that theory (1931), political scientists were in a position to understand w h y Lenin had to be m u m m i f i e d and d i s p l a y e d . . , in a mausoleum." 2 2 THE MATHEMATICALINTELLIGENCERVOL.18, NO. 1, 1996
Problem 6" Axiomatization of physics. There are two main reasons w h y all relation has disappeared between this problem as Hilbert stated it and as it would be stated today:
1. The revolution in modern physics: relativity, general relativity, quantum physics. 2. The introduction of new mathematical tools, first of all Hilbert space. Hilbert was starting work on it in winter 1900, in the following spirit: It appears to me of outstanding interest to undertake an investigation of the convergence conditions which serve for the erection of a given analytic discipline so that we can set up a system of the simplest fundamental facts which require for their proofs a specific convergence condition. Then by the use of that one convergence condition alone--without the addition of any other convergence condition whatsoever--the totality of the theorems of the particular discipline can be established. (Gen. Ref. [R, p. 85])
Stimulated by his reading of Fredholm's work, Hilbert introduced the space which bears his name. One notes with astonishment that he makes no mention of this research in the problems stated at the Paris Congress. The notion of Hilbert space completely transformed the mathematical formalism in use in theoretical physics. Skipping forward a century, if we try to indicate the most active directions of the present, these seem to stand out: General relativity and global differential geometry. Construction of manifolds which could be effective cosmological models (S. Hawking, R. Penrose). Mathematics of quantum field theory; gauge theory; introduction of quantum groups in the formalism of quantum physics. Problem 7: Irrationality and transcendence of od3 for c~ algebraic and j8 algebraic irrational (for example, 2x/2). The problem was solved by Gel'fond (1935) and Th. Schneider. Euler's constant remains as mysterious as ever. Let us note some recent progress in this area: A. Baker (1966, Fields Medal in 1970): If the ai are nonzero algebraic numbers whose logarithms are linearly independent over Q, then
1, log al . . . . . log an are linearly independent over the field of algebraic numbers. In the last 20 years tools have been brought into play from various mathematical domains. The theory of functions of several complex variables enabled Enrico Bombieri (Fields Medal 1974) to resolve a conjecture of Nagata and enabled W.D. Brownawell in 1985 to give the first effective version of the Hilbert Nullstellensatz. The latter theorem is useful in problems of algebraic independence; the method initiated by A.O. Gel'fond in 1949 and developed by G.V. Chudnovsky in the 1970s enabled him to prove the transcendence of numbers like F(1/4) or F(1/3) (where F is the Euler gamma function). Commutative algebra, with Chow forms, was introduced in this context with great success by Yu.V.
Nesterenko. Results on transcendence involving the usual exponential function or elliptic functions (Th. Schneider) were extended to (commutative) algebraic groups by S. Lang in the 1960s; this theme was abundantly developed especially by D.W. Masser, first alone and then in collaboration with G. W/istholz, in connection with the work of Faltings on the Mordell conjecture. This result of G. Faltings implies that, for a fixed p -> 3, the equation xP + yP = zP
has a finite number of solutions without a common factor. What remained of the Fermat problem was to show that this number is zero. K. Ribet showed that the Fermat conjecture is implied by a deep conjecture (Taniyama-Weil) related to number theory and also to algebraic geometry and group theory. This got the Fermat problem out of its isolation.6 Problem 8: The Riemann hypothesis. Surely the most celebrated problem in the history of mathematics; according to Hilbert, the most important problem of mathematics, and he even Went so far as to say in conversation that it was the most important problem for humanity! (loc. cit.) It is now known that more than 40% of the zeros are on the fateful line (Gen. Ref. [C] 1989), and the fluctuations among the zeros have been studied. It seems, as was suggested very early, that the distribution of zeros resembles that of the eigenvalues of a self-adjoint operator (Hilbert and P61ya, about 1915). About 1973 Dyson suggested that the operator could be a random hermitian operator. For several decades, progress on the hypothesis has been essentially technical, and the most powerful computers have been tried on it. Some of the achievements have been real tours de force ([B-I], cf. [G-K]). On the other hand, there have been attempts, following a suggestion by Hilbert, to generalize the Riemann hypothesis by replacing the field of rationals by a field of functions on a projective curve defined over Fq. This was a motivation of the founders of modern algebraic
6Having already cited Gen. Ref. [Ma] for number theory through the 1980s, let me draw on the introduction to the 1993 English edition for the recent developments: " . . . the famous problem 'Fermat's Last Theorem,' together with the Taniyama-Weil conjecture for semi-stable elliptic curves, seem to have been completely proved by A. Wiles. Wiles used various sophisticated techniques and ideas due to himself and a number of other mathematicians (K. Ribet, G. Frey, Y. Hellegouarch, J.-M. Fontaine, H. Hida, J.-P. Serre, J. Tunnel . . . . ). This genuinely historic event concludes a whole epoch in number theory, and opens at the same time a new period which could be closely involved with implementing the general Langlands program . . . . One of the characteristic features of the new methods and ideas is the intensive use of p-adic L-functions and Galois representations. Another striking example of this feature is K. Rubin's construction of rational points on elliptic curves using special values of p-adic L-functions and their derivatives." THE MATHEMATICALINTELLIGENCERVOL.18, NO. 1, 1996 23
geometry, A. Weil and O. Zariski. The "Weil conjectures," stated in 1949, which extended the earlier conjectures to higher-dimensional varieties, were proved in 1973 by P. Deligne (Fields Medal 1978) (Gen. Ref. [K]). Problem 10:
"On the possibility of solving a diophantine
equation." The problem is to give an algorithm which tests the solubility of diophantine equations (polynomial equations with integer coefficients whose integer solutions are sought). As the statement indicates, the problem is at the border between logic--recursion theory--and number theory. It was settled negatively by Yu. Matijasevich in 1970. This negative result was obtained by deepening our knowledge of recursive sets (recursively enumerable or "listable') and diophantine sets; that is, sets of integer parameters ai for which
P(al, a2 . . . . .
an,
Zl, Z2.....
Z m) = 0
has solutions (z) in integers. Matijasevich's theorem asserts that these two families of sets of integers are the same. Astonishing application: there exists an effectively computable polynomial (in 10 integer variables) with integer coefficients whose positive values are precisely the set of prime numbers! Other problems of the same sort remain unsolved; for example, does there exist a system fiX1, X 2. . . . .
Xn,
t) -- 0
numerous mathematicians around the world are laboring (whereas Langlands himself has turned to percolation!). See Gen. Ref. [Ma]. In the abelian case, one would like to describe explicitly the abelian extensions of a number field K which is a finite extension of Q, in terms of the values of certain special functions (exponential, elliptic) and the action of the Galois group. For Q, the Kronecker-Weber theorem gives an explicit description of the abelian extensions by means of the action of the Galois group on the roots of unity. The work of Shimura and Taniyama [S-T] concerning abelian varieties with complex multiplication provides an almost complete answer for "CM fields" (totally imaginary quadratic extensions of a totally real field).
"Impossibilityof solving the general equation of degree 7 in terms solely of functions of two arguments": Problem 13:
superposition of functions. The equation of third degree can be reduced by translation to
X3+pX+q=O, which has the solution (Scipione del Ferro, 16th century)
X= ( - q
+
(1)
of equations with integer coefficients which has rational solutions (x~. . . . . Xn) if and only if the parameter t is an integer? Otherwise stated, does there exist a morphism of varieties over Q V --> Affine line such that the fiber at a is nonempty if and only if a is an integer? Problem 11: Classification of quadratic forms with coefficients in rings of algebraic integers.
( _ q _ ~ A p 3 + a Y q 2 ) 1/3 2
4(27)
to
P(X) = 0
Problem 12: This concerns generalizationsof Gauss's quadratic reciprocitylaw, the law from which has flowed the modern algebraic theory of numbers. Consider the equation (in integers) x2 + 1 ~ 0 (mod p) This equation has solutions if and only if the prime p is congruent to 1 modulo 4. This rule is at the origin both of group theory and of modern number theory. To sum up the ramifications of this problem, one would cite class field theory (Hilbert, and around 1930 Furtwangler, Takagi, E. Artin); the introduction of methods of cohomology of groups; the study of L-series; and the vast "Langlands program" aiming to extend the quadratic reciprocity law to the non-abelian case, on which THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
"
The equation of fourth degree can be solved by superposition (composition) of addition, multiplication, square roots, cube roots, and fourth roots. To try to solve algebraic equations of higher degree (a vain hope according to N. Abel and E. Galois), the idea of Tschirnhaus (1683) was to adjoin a new equation:
See Gen. Ref. [K].
24
~/4Pg+a7q2) 1/34(27)
one adjoins Y = Q(X), where Q is a polynomial of degree strictly less than that of P, chosen expediently. In this w a y one can show that the roots of an equation of degree 5 can be expressed via the usual arithmetic operations in terms of radicals and of the solution ~b(x) of the quintic equation
X5+xX+ I
=0,
depending on the parameter x. Similarly for the equation of degree 6, the roots are expressible in the same way if we include also a function O(x,y), a solution of a 6th-degree equation depending on two parameters x and y.
For degree 7 we would have to include also a function or(x, y, z), solution of the equation X 7 + xX 3 + yX 2 + z X + 1 = O. Hence the natural question: can or(x, y, z) be expressed by superposition of algebraic functions of two variables? V.I. Arnold asked whether cr can be represented using as "irreducible branches" of the superposition mappings topologically equivalent to algebraic functions of two complex variables. Despite results of V. Lin, this question, along with the original question of D. Hilbert, remains unanswered. Let us note one more strange result in this circle of ideas: The solution of X5 + xX2 + y X + I = O cannot be expressed using algebraic functions of one variable, addition and multiplication (division is excluded!). (A. Khovanskii, see [R]; historical articles UPK] and [D].) A.N. Kolmogorov's important notions of complexity (created around 1955) were the subject of interesting comments by Yuri I. Manin at the International Congress at Kyoto (Gen. Ref. [C]). Hilbert had had another motivation for this problem: nomography, the method of solving equations by drawing a one-parameter family of curves. This problem, arising in the methods of computation of Hilbert's time, inspired the development of Kolmogorov's notion of e-entropy; among its applications is its crucial role in theories of approximation now used in computer science. Let us briefly recall the history. Contrary to the expectations of Hilbert and of contemporary mathematicians, in 1957 V.I. Arnold (then a student of Kolmogorov) showed that any continuous function f of three variables can be written
fix, y, z) = Z fi(4i(x, y), z)
i
with functions fi and ~bi continuous. A few weeks later Kolmogorov showed that any continuous function f of n variables can be written in the form 2n+1
P r o b l e m 15: "Rigorous foundation of the enumerative geometry of Schubert." Schubert calculus. Schubert's study of the number of points of intersection of varieties--for example, of the number of lines meeting four given lines in 3-space was taken up by the Italian school (Severi). The nonrigorous arguments--'principle of conservation of numbers," that is, heuristic arguments of continuity--grew into the modern theory of multiplicities of Pierre Samuel and Alexander Grothendieck (Fields Medal 1966). A topological point of view was taken by Ren6 Thorn (Fields Medal 1958): Thom-Bordman polynomials, enumerative theory of singularities. Problem 15 can not yet be regarded as solved, despite some recent progress (Demazure, Fulton, Kleiman, R. MacPherson). P r o b l e m 16: "Problems of the topology of curves and of algebraic surfaces." This is the only problem directly about topology. It falls into two distinctparts. A: Consider a nonsingular curve of degree m in p2(R). From a topological point of view such a curve is composed of ovals of which there are at most l(m - 1) (m 2) + 1, and this bound is the best possible. Curves attaining the maximum are called M-curves. The problem is to study what configurations of the ovals are possible. The problem is nontrivial beginning with m = 6, the case raised by Hilbert; this was solved by Gudkov in the 1970s. (See Fig. 1.) In 1933 Petrovskii proved inequalities concerning the topological invariants of
B = {xlf(x) >- 0}, where f determines a curve of even degree; general inequalities were proved by Petrovskii and Oleinik (1949-1951), mainly concerning surfaces in R 3, and similar results were obtained by Thorn and Milnor (1964), see [R]. In 1971 V. Arnold proved congruence relations
f
f(Xl, X2..... Xn) = i~=1fi~~j 49j(Xj)), where the 4~j are continuous, monotone, and independent of f. In the opposite direction are the results of Vitushkin (1954). When we deal with superpositions of formal series, or of analytic functions, or even of infinitely differentiable functions, we can show by an elementary enumeration technique f(~llowed by a Baire argument that, for example, almost every entire function has at an arbitrary point of C 3 a germ which is not expressible by superposition of series in two variables. So there are many more entire functions of three variables than of two. The result of Vitushkin is that for functions of smoothness c~ (with ~ larger than 1, not necessarily an integer), representability depends on n/oe. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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ooo
ooo 000 !Hamack (1878)
OO
0 iHilbert (1894)
Gudkov (1973)
Figure 1. Possible arrangements for m = 6.
connecting the number P of even ovals (contained in an even number of ovals) and the number I of odd ovals. For a curve of even degree m, P - I is congruent to (m/2) 2 modulo 8. For degree 8 the problem is still not solved. An account of the work, from O. Oleinik and I. Petrovskii (1949) to O. Viro (1979), is in [R]. The Leningrad school (Kharlamov, Gudkov, Rokhlin, Viro) turned up subtle relations between a nonsingular complex algebraic variety and its real part [V]. The current work is very technical. The results have applications in analysis (partial differential equations, completely integrable systems). B: Problem of finiteness of limit cycles. Given a vector vield V in the plane, a cycle is a periodic trajectory, and a limit cycle is a cycle on which nonperiodic trajectories accumulate. When V is analytic, a limit cycle is isolated in the set of cycles. But it is possible for cycles to accumulate at a point or more generally on a "polycycle" (concept due to H. Poincar6) formed of pieces of integral curves ending at singular points of V. Hence the natural question posed by Hilbert: Given a vector field V = (X, Y), X and Y being polynomials of degree < n, find the maximum possible number of limit cycles. To begin with, one would want to try to prove Conjecture: A polynomial vector field in the plane has only finitely many limit cycles or polycycles.
In 1923, Dulac proved this conjecture, but his proof was insufficient, as was recognized in the 1970s by Yuli II'yashenko. However, some of Dulac's ideas were revived by II'yashenko and Ecalle, in different directions. II'yashenko (1984) proved that the finiteness conjecture holds on the complement of a proper algebraic subset of the vector space of vector polynomials of degree n. Actually, the proper setting is that of singular foliations F (defined locally by analytic vector fields with isolated singular points). The return-map is defined in the neighborhood of a polycycle like C (Fig. 2). There exists an analytic curve T:[0, 1]--~M with T(0) E C such that T is transversal to F and the leaf through T(t)--for t sufficiently small--cuts T again for the first time at T(f(t)). The mapping f leaves 0 fixed, preserves orientation, and is analytic on a punctured neighborhood of 0, but need not be analytic at 0. Thus it may have a sequence of isolated fixed points (tn) which 26 THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996
converge to 0. C is then the limit of the limit cycles through the T(tn). It is to be shown that a polycycle cannot be the limit of limit cycles, which is to say that if f is not the identity, then 0 is an isolated fixed point. This would imply the finiteness by the Poincar6-Bendixson theorem. II'yashenko's proof of the finiteness conjecture and Ecalle's are given in [I] and [E]; Ecalle pursues his investigation, which uses his theory of generalized asymptotic developments (and resurgence) to study the return-map f. Let me mention finally that there is a "linearized" version of Problem B [A-I]. Problem 17: Sums of squares:
"Representation of definite
forms by sums of squares." Here "definite" means that the forms (homogeneous polynomials) in n variables with real coefficients have non-negative values for all real values of the variables.
Can they always be represented as quotients of sums of squares of forms? This has been a motivating problem in real algebraic geometry. T h e o r e m of Emil Artin (1927): Let X be an irreducible
algebraic variety over R or Q, let f be a rational function on X, and assume f is positive at the real points of X where it is defined. Then f is a sum of squares in the function field of X (functions with coefficients in Q, if X is defined over Q). T h e o r e m of Pfister (circa 1970): Let X be an irre-
ducible algebraic variety over R of dimension d, and assume f in the function field of X is positive (in the sense above). Then f can be written as the sum of at most 2a squares. Hilbert had already studied the conditions under which a homogeneous polynomial of degree m in n variables can be written as a sum of squares of homogeneous polynomials. Recently, this has been studied for analytic functions, for differentiable functions, and for Nash functions; see [B]. Real algebraic geometry is attracting renewed interest, by virtue of its relations with
Figure 2.
Polycycle (of a vector field) and return-map
logic (elimination principle of Tarski and Seidenberg, model theory) and with industrial applications (robotics), but it is intrinsically more complicated than complex analytic geometry. Problem 18: "To rebuild space with congruent polyhedra." Hilbert's problem is divided into three distinct parts. A: "In Euclidean space of n dimensions, show that there
are only finitely many different kinds of groups of displacements with a (compact) fundamental domain." In other words, one looks for discrete subgroups with compact quotient of the group E(n) of isometries of R n. The result was proved by Bieberbach in 1910. The classification of these groups is important in crystallography and it generalizes to questions about lattices in Lie groups. B: "Tiling of space by a single polyhedron which is not a fundamental domain as in A." J. Milnor's remarks in Gen. Ref. [M] still apply: This is a very lively topic today [G-S1]. Important developments have been the theory of Penrose tilings [P] and the closely related physical problem of anomalous crystal structure (see, for example, [J]), as well as Thurston's theory of self-similar fractal tilings [T]. C: "Packing of spheres. How should spheres of the same radius be arranged in space (of any dimensionality) so as to achieve the greatest density of packing?" Hilbert's text gives the impression that he did not anticipate the success and the developments this problem would have. The hexagonal packing in the plane is the densest (proof by Thue in 1882, completed by Fejes in 1940). In space, the problem is still not solved. There is very recent progress by Hales. For spheres whose centers lie on a lattice, the problem is solved in up to eight dimensions. The subject has various ramifications: application to the geometry of numbers, deep relations between coding theory and sphere-packing theory, the very rich geometry of the densest known lattices. Before entering on analysis, Hilbert poses, in general, the question of choosing the class of functions with which to work: differentiable, analytic... ? There are other, intermediate classes which in my view have not been looked at sufficiently--for example, classes of functions defined by inequalities on their derivatives (Gevrey classes) or functions which are solutions of certain types of differential equations (differentiably algebraic functions). Problems 19, 20, and 23: Partial differential equations: calculus of variations and the Dirichlet problem. The developments in calculus of variations are so diverse (finite-element method, control theory) that it is impossible to survey them here. They play a central role in the study of nonlinear phenomena (Plateau problem, equations of minimal surfaces) and, in the form of optimal control, in industrial applications of mathematics. Taking for granted the exposition by James Serrin (Gen.
Ref. [M]), I may indicate for Problem 20 (solvability of boundary-value problems) developments involving elliptic systems, both linear with nonregular coefficients and nonlinear: study of harmonic mappings; questions of regularity motivated by global geometry, as in the work of Schoen and Uhlenbeck (cf. Gen. Ref. [C], Karen Uhlenbeck's article); mechanics (elasticity), or more recently the study of liquid crystals (Gen. Ref. [I], 1988 and 1990). The study of extensions of the calculus of variations (Problem 23) has had a very wide development in recent years. Take for example the work of Bahri on the existence of periodic solutions of the three-body problem. Problem 21: Monodromy of Fuchsian equations. "Does
there always exist a linear differentia~ equation of the Fuchs class having given critical points and a given monodromy group?" One might try to prove the existence on the Riemann sphere of the following-objects with given singular points and given monodromy groups: A: Fuchsian equations (the Fuchs class): a linear first-order differential equation whose monodromy matrix has only simple poles at the given singular points; B: one requires all the solutions to have regular singular points, that is, with moderate growth in angular sectors (this is the problem treated by Deligne [D]); C" Fuchsian systems. "Fuchsian" implies singular regular but not conversely. Plemelj (1930) claimed a solution to A. However, there was an error in his proof (see Gen. Ref. [E], vol. 1), and counterexamples have been found (see exposition in [B]). Problem 22: Uniformization of analytic curves. This is a fine example of a crossroads problem: topology, complex function theory, group theory, partial differential equations . . . . . Recent developments concern the extension to higher dimensions (Griffiths). Some Reflections
One must not overlook little problems although they seem like dead ends without much future: there are many such in Hilbert's list, and they gave rise to important developments (particularly in geometry). The phenomenon keeps occurring since then: the Henon attractor; non-periodic tilings of Penrose, used and studied by physicists, by Connes ...
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If we look at the whole list of Hilbert problems and the works that have grown out of them, it is clear that they connect to an important part of this century's mathematics. They stimulated major efforts of numerous mathematicians: who in this century has not dreamed of solving a question posed by Hilbert? However, let us look at the topics locally, undistracted by the impressive figure of their author. Even aside from the absence of certain important questions (functional analysis, measure theory) and the minor part given to others (topolo g y - n o doubt Poincar6 has something to do with this), some mathematicians find the interest excited by Hilbert for certain problems to have been disproportionate. J. Dieudonn6 [LeL] judges a good portion of these problems "isolated," but this reproach can be invalidated, as we see, for example, from the recent work that "uncloisters" Fermat's conjecture. Some problems not presented at Paris (Problems 3, 5, and 15) and their sequels also took on luster from the great mathematician.
Problems, Program? As his correspondence with Minkowski shows, Hilbert must have harbored the dream of writing the program of action for the mathematicians of the 20th century--at least the first stages. This was in line with his structured, centralized vision of mathematics: G6ttingen at the center, number theory at the heart of mathematical science . . . . Didn't Ostrowski even find a curious resemblance to another great advocate of programs of action, V.I. Lenin? Rather than a program, which would inevitably be lacking in means of implementation, it was more a matter of a project, a projection toward the future, bringing out ideas dear to Hilbert: mathematicians' power to resolve crises (there are no problems without solution--contrast the dialectical point of view of Poincar6); the deep unity of mathematics, around number theory; mathematics as independent of the physical sciences. The other subject that Hilbert had thought of treating in his Paris lecture, the relationship between pure mathematics and physics, he would take up in 1930, after his retirement, when he was made honorary citizen of the great city of Koenigsberg-Kaliningrad, the city of Kant and Jacobi. In his talk, which he summarized in an interview on the local radio, the defense of pure mathematics is linked, by w a y of a reading of Kant, to his profound conviction --which he asserted as if to prove of the omnipotence of mathematics:
We must know. We will know. And on these words one hears a great burst of laughter from Hilbert. What could it mean? An intuition of what was happening? Two months later the Monatshefte fflr Mathematik received an article from the 25-year-old mathematician Kurt G 6 d e l . . . 28
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
Yesterday, Today, Tomorrow With the year 2000 coming up, it is tempting--but dangerous--to imagine an analogous effort. Indeed, other mathematicians since Hilbert have dared to envisage partial programs (A. Weil for one [LeL], or the team assembled by F. Browder [M], or finally the "sketch of a program" of A. Grothendieck [G]). Lucky chance, or sign of the end of an era? Klein's Encyclopedia of Mathematical Sciences marked the end of the century; Hilbert's lecture marked the start of a formidable development (say 1900-1970). The appearance today of the Encyclopaedia of Mathematical Sciences [E] could fill a role analogous to the former. But how could we imagine it followed by a "new Hilbert program"? In the last 20 years the progress of mathematics has been considerably transformed. Whereas mathematics in the first part of the century, up to the sixties, was motivated essentially by an internal logic, and thus came under the sponsorship of Hilbert (and his disciple Bourbaki), today "external needs"--in a global sense--are the principal motor driving mathematics as a science. Poincar6's revenge. The elaboration of electronic processing of information allows a "positive feedback" between theory and applications. This is indeed what the editors of the Encyclopaedia of Mathematical Sciences say (Vol. 1), speaking in their introduction of "the industrialization of Mathematics." Let us take a few examples: the motivating role of computer science in research in logic [I]; problems of coding, cryptography, and signal theory in number theory; "experimental" research in this area using the most powerful computers; rapprochement of theoretical physics and mathematics (Fields medals of 1990); topology and formal calculus (F. Sergeraert [I, 1990]); nonlinear systems (chaos, Korteweg-de Vries equation . . . . ); minimal surfaces of constant mean curvature (applications to interfaces of polymers). This new nature of the evolution of mathematics (which merits a deeper study comparing this era to others) is summed up by Jacques-Louis Lions in reference to optimal control theory ILl: "Herm6s, Hilbert program (Problem 23) in practice" (the Kalman filter already has an important role in controlling the Airbus). This transformation naturally carries with it sociological consequences, bringing the ways of the mathematical tribe closer to those of other subcultures: the competition for contracts, the growing weight of military contracts together with a romantic tendency in the practice of research (one dreams, "but perhaps one ought not to dream too much" [F]). In another direction, the relation between philosophy and mathematics, I would refer to [K2] but especially to
[c'].
ILl
J.-L. Lions, L'Ordinateur, nouveau D6dale, Daedalon Gold Medal Lecture, 1991. MathematicalDevelopments Arising from Hilbert Problems (F. Browder, ed.), Providence, RI: American Mathematical Society (1976). Yu. I. Manin and A. A. Panchishkin, Number Theory I: Introduction to Number Theory, in Encyclopaedia of Mathematical Sciences, Heidelberg: Springer-Verlag (1993). H. PoincarG L'avenir des math6matiques, International Congress of Mathematicians, Rome, 1908. C. Reid, Hilbert, New York: Springer-Verlag (1970).
The "we must k n o w " of Hilbert should t o d a y be acc o m p a n i e d by a "we w a n t to act." One could imagine beginning the " p r o g r a m of twenty-first-century mathematics" with a study, d o n e jointly by mathematicians and other scientists, which w o u l d list the i m p o r t a n t scientific problems deserving of mathematicians' efforts-n e w or r e n e w e d - - ( F e y n m a n integrals, turbulence, complex systems not yet studied from this point of view, theoretical biology, cognitive sciences . . . ) and w o u l d extract choices corresponding to a real social d e m a n d . Might a p r o g r a m of this sort, putting the p o w e r of mathematics to work, m a k e David Hilbert l a u g h - - b u t with delight?
References Specific to the Problems
General References
[B]
[C] [C'] [D]
[E] [F] [G] [H]
[I]
International Congress of Mathematicians, Kyoto, 1990. Proceedings, Mathematical Society of Japan/SpringerVerlag, Tokyo, 1992. P. Cartier, La pratique et les pratiques--des math6matiques, in Encyclopddie philosophique universelle, vol. 1, 1991. Ein Jahrhundert Mathematik 1890-1990, Vieweg & S., 1990. Review in Mathematical Intelligencer 13(1991), no. 4, 70-74. Encyclopaedia of Mathematical Sciences, Heidelberg: Springer-Verlag (1990) (Russian original Moscow, 1985.) J. M. Fontaine, Valeurs sp6ciales des fonctions L des motifs, Sdmin. Bourbaki (1992), Expos6 751. Astdrisque 206, (1992). A. Grothendieck, Esquisse d'un programme, Dossier de candidature au C.N.R.S., 1985. D. Hilbert, Sur les problbmes futurs des math6matiques, in Proceedings of the Second International Congress of Mathematicians, Paris: Gauthier-Villars (1902), pp. 58-114. Images des math6matiques, Annual Supplement to Courrier du CNRS, CNRS, Paris. 1985 J. F. Boutot and L. Moret-Bailly, Equations diophantiennes: la conjecture de Mordell. Images des nombres transcendants, d'apr6s P. Philippon. 1988 M. Parigot, Preuves et programmes; les math6matiques comme langage de programmation. J. M. Ghidaglia and J. C. Saut, Equations de Navier-Stokes, turbulence et dimension des attracteurs. 1990 F. Sergeraert, Infini et effectivit6: le point de vue fonctionnel. F. Bethuel, H. Brezis, J. M. Coron, and F. Helein, Probl6mes math6matiques des cristaux liquides.
J.-M. Kantor, Hilbert (probl6mes de), in Encyclopaedia Universalis (1989), vol. 11. [K2] J.-M. Kantor, L'intuition en 6quations, Report on the Kyoto Congress, Le Monde (August 1990). [LeL] A. Blanchard, Les grands courants de la pensde mathdmatique (F. LeLionnais, ed.), (1948), new edition 1962, Paris; in particular the articles by J. Dieudonn6, R. Godement, A. Weil.
[M] [Ma]
[P] JR]
Problems 1 and 2
[G]
J. Y. Bouleau and A. Louveau Girard, Cinq confdrences sur l'inddcidabilitd, Paris: Presses de l'Ecole nationale des Ponts et Chauss6es (1982). E. Nagel, J. Newman, K. G6del, and J.Y. Girard, Le thdor~me de G6del, Paris: Seuil (1989). Problem 3
[A] [B] [C] [S]
Agr6gation de math6matiques, probl6mes de math6matiques g6n6rales, Revue de mathdmatiques spdciales, 1985-1986, Paris, pp. 139-150. Vuib~rt. V. Boltianski, Hilbert's Third Problem, New York: Wiley (1978). P. Cartier, D6composition des poly6dres: le point sur le troisi6me probl~me de Hilbert, Sdmin. Bourbaki (1984-1985), No. 646. G. Sah, Hilbert's Third Problem: Scissors Congruence, London: Pitman (1979). Problem 5
[A] [G] [S]
J. Acz61, The state of the second part of Hilbert's fifth problem, Bull. Am. Math. Soc. 20 (1982). M. Gromov, Groups with polynomial growth and expanding maps, Publ I.H.E.S., no. 53. I. Sillmann, Every proper smooth action of a Lie group is equivalent to a real analytic action: a contribution to Hilbert's fifth problem, M.P.I. Mathematik, Bonn, 1993. Problem 6
[B]
P. Bohl's Fourth Thesis and Hilbert's Sixth Problem, Moscow: Nauka (1986) [in Russian].
[B-M]
A. Baker and D. W. Masser (eds.), Transcendence Theory, Advances and Applications, New York: Academic Press (1977). D. Bertrand and M. Waldschmidt (eds.), Approximations diophantiennes et nombres transcendants, Basel: Birkh~iuser (1983). A. Baker (ed.), New Advances in Transcendence Theory, Cambridge: Cambridge University Press (1988). P. Philippen (ed.), Approximations diophantiennes et nombres transcendants, de Gruyter (1992).
Problem 7
[B-W] [B] [p]
[KI
Problems 8 and 9
[B] [B-I] [G-K]
M.V. Berry, Semiclassical formula for the number variance of the Riemann zeroes, Nonlinearity 1 (1988), 399-407. E. Bombieri and H. Iwaniec, On the order of ~(12+ it), Ann. Scuola Norm. Sup. Pisa 13 (1986), 449-472. Graham and Kolesnik, Van der Corput's Method of THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
29
Exponential Sums, Heidelberg: Springer-Verlag (1991). A. Ivic, The Riemann Zeta Function, New York: Wiley (1985). E.C. Titchmarsh, The Riemann Zeta Function, Oxford: Oxford University Press (1986).
[II [T]
Problem 10
[M] [M1]
Yu. Matiyasevich, The Tenth Problem of Hilbert, Moscow: Nauka (1993) [in Russian]. Yu. Matijasevich, My collaboration with Julia Robinson, Mathematical Intelligencer 14 (1992), no. 4, 38-45.
[R] [V] [Y]
Problem 17
[BS]
Problem 12
[K-S] K. Kato and S. Saito, Global class field theory of arithmetical schemes, Contemp. Math. 55(Part I) (1986), 5-331. IS-T] G. Shimura and Y. Taniyama, Complex multiplication of algebraic varieties and its application to number theory, Publ. Math. Soc. Japan 6 (1961). Problem 13
[D]
J. Dixmier, Histoire du treizi6me probl6me de Hilbert, Sdminaire d'histoire des mathdmatiques de l'Institut Henri Poincard, 1991-1992. [JPK] J.-P. Kahane, Le treizi6me probl6me de Hilbert: un carrefour de l'analyse, de l'alg6bre et de la g6om6trie, Cahiers Sdmin. Hist. Math S&. 1, 3 (1982). [R] J.J. Risler, Complexit6 et g6om6trie r6elle, d'apr6s A. Khovanskii, S6min. Bourbaki (1984-1985), No. 637. V.Y. Lin, Superposition of algebraic functions, Funct. [L] Anal. ego Prim. 10 (1976), 32-38. Problem 15
A. Lascoux, Anneaux de Grothendieck de la vari6t6 des drapeaux, in The Grothendieck Festschrifl, Vol. III, Basel: Birkh/iuser (1993), pp. 1-34. P. Samuel, Sur l'histoire du quinzi6me probl6me de Hilbert, Gazette des Mathdmaticiens (1975), (Oct. 1974), 22-32
[L]
IS]
[K]
Problem 18, Parts A and B
Problem 18, Part C
[C-S]
[O]
[B] [E]
[I] [K] [M] 30
V. Arnold and Yu. II'yashenko, Ordinary differential equations, in Dynamical Systems--l, Encyclopaedia of Mathematical Sciences, Heidelberg: Springer-Verlag (1988). J. Bochnak, M. Coste, and M.F. Roy, G6om6trie alg6brique r6elle, Heidelberg: Springer-Verlag (1986). J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Paris: Hermann (1992); and Six lectures on trousseries, analysable functions, and the constructive proof of Dulac's conjecture, in Bifurcations and Periodic Orbits of Vector Fields (D. Schlominck, ed.), Kluwer Acad. Publ. (1993), pp. 75-184. Yu. II'yashenko, Finiteness Theorems for Limit Cycles, American Mathematical Society Providence, Rh (1991). V. Kharlamov and I. Itenberg, Towards the maximal number of components of a non-singular surface of degree 5 in RP3, preprint (1993). R. Moussu, Le probl6me de la finitude du nombre de cycles, d'apr6s R. Bamon et Y. S. Ii'yashenko, Sdmin.
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
H. Benis-Sinaceur, De D. Hilbert a E. Artin: Les diff6rents aspects du dix-septiOme probl6me de Hilbert et les filiations conceptuelles de la th6orie des corps r0els clos, Arch. Hist. Exact Sci. 29(3) (1984), 267-286. A. Khovanskii, Fewnomials, Providence, RI: American Mathematical Society (1991).
[D-G] L. Danzer, B. Griinbaum, and G. C. Shephard, Does every type of polyhedron tile three-space? Topol Struct. 8 (1983). [D-K] M. Duneau and A. Katz, Cristaux apdriodiques et groupe de l'icosa~dre, Palaiseau, France: Publications du Centre de Physique th6orique, Ecole polytechnique, [G-S] B. Griinbaum and G. C. Shephard, Tiling with congruent tiles, Bull. Am. Math. Soc. 3 (1980), 951-973. [G-S1] B. Griinbaum and G. C. Shephard, Tilings and Patterns, New York: W.H. Freeman (1987). M. Jaric, Introduction to the Mathematics of Quasicrystals, [J] New York: Academic Press (1989). R. Penrose, Pentaplexity, Eureka 39 (1978), 16-22; [P] Pentaplexity: a class of non-periodic tilings of the plane, Mathematical Intelligencer 2 (1979), no. 1, 32-37. [T] W. Thurston, Groups, Tilings, and Finite State Automata, Providence, RI: American Mathematical Society (1989); Research Report GCG-1, Geometry Supercomputer Project, University of Minnesota, Minneapolis (1989).
Problem 16
[A-I]
Bourbaki (1985-1986), No. 655. Astdrisque, 145-146 (1987). J.J. Risler, Les nombres de Betti des ensembles alg6briques rods, une mise au point, Gazette des Mat hdmaticiens (1992), O. Viro, Progress in the topology of real algebraic manifolds, in Proceedings of the International Congress of Mathematicians, Warsaw, 1983, pp. 595-611. J.-C. Yoccoz, Non-accumulation des cycles limites, Sdmin. Bourbaki (1987-1988), No. 690.
[S]
J.H. Conway and N. J. Sloane, Sphere Packings, Lattices and Groups, New York: Springer-Verlag (1988). J. Oesterl6, Empilement de sph6res, Sdmin. Bourbaki (1989-1990), No. 727; "Les sph6res de Kepler," video (series "Mosa'ique math6matique"), Paris: Productions "Les films d'ici." F. Sigrist, Sphere packing, Mathematical Intelligencer 5 (1983), 34-38. Problems 19, 20, and 23
[V]
C. Viterbo, Orbites p6riodiques dans le probl6me des trois corps, Sdmin. Bourbaki (1992-1993), No. 774.
[B]
A. Beauville, Equations diff6rentielles a points singuliers r6guliers d'apr6s Bolybrukh, Sdmin. Bourbaki (1992-1993), no. 765. Astdrisque, 216 (1993). P. Deligne, Equations diffdrentielles ?l points singuliers rdguliers, Heidelberg: Springer-Verlag (1970), p. 136.
Problem 21
[D]
29, rue de Lacdp~de 75005 Paris, France
David Gale* This column is interested in publishing mathematical material which satisfies the following criteria, among others: 1. It should not require technical expertise in any specialized area of mathematics. 2. The topics treated should when possible be comprehen-
sible not only to professional mathematicians but also to reasonably knowledgeable and interested nonmathematicians. We welcome, encourage and frequently publish contributions from readers. Contributors who wish an acknowledgement of submission should enclose a self-addressed postcard.
Triangles and Proofs For the third time in as many years this column will be concerned with the subject of triangles, but with a difference. The earlier columns were concerned with the impact of computers on triangle geometry, whereas here we will go back to basics, taking a new look at some very classical topics. "The point of a generalization is not just to include more cases, but also to get rid of unnecessary hypotheses, and sometimes this can lead to simpler proofs." I recall being told something like this as a graduate student by one of my first mentors, Norman Steenrod. In fact there are two kinds of generalizations. One can either weaken the hypothesis or strengthen the conclusion. The examples to be treated below illustrate both of these approaches; in both cases, finding the right generalization reduces what appears to be a complicated problem to essentially a mechanical verification, a matter of just "turning the crank." For our first example we are grateful once again to Donald Newman.
points. These form the small triangle inside the starting triangle. This small interior triangle is far'from being arbitrary, however, Morley's great discovery (1899) being that it is always equilateral! When I read, or rather tried to read, Morley's proof of this startling theorem, I found it absolutely impenetrable. I told myself that maybe in future years I would return and then understand it. I never succeeded in that, and even when I read the much simpler proof based on trigonometry, or the fairly simple geometric proof due to Navansiengar, there was still too much complexity and lack of motivation. (A series of lucky breaks!) Were we to give up, forever, understanding the Morley Miracle? Or are we failing because we are asking too little? After all, Morley's theorem states that in Figure 1, the inner triangle always will be equilateral. The reason that all the proofs seem to be so difficult and unmotivated is probably because Morley's theorem is really only half the story. The full picture is in Figure 1 and this tells the whole story and indeed proves itself! (This
The Morley Miracle D. J. Newman One of the sad things about the current philosophy of mathematical education is the avoidance of plane geometry. Today's generation, and perhaps their parents as well have not heard of marvels like the 9-point circle, Descartes's theorem, Ceva'-s Theorem, or the marvel of marvels, the Morley triangle. As shown in Figure 1, one takes an arbitrary triangle and trisects its angles, obtaining three intersection
*Column editor's address: Department of Mathematics, University of California, Berkeley, CA 94720 USA.
Figure 1
THE MATHEMATICALINTELLIGENCERVOL.18, NO. 1 9 1996Springer-VerlagNew York 31
References 1. Frank Morley, Extensions of Clifford's theorem, Amer. J. Math. 51 (1929), 465-472. 2. J.M. Child, A Proof of Morley's Theorem, Math. Gaz. (1922), 171. 3. M.T. Navansiengar, Educ. Times, New Series 15 (1909), 47. 4. H.S.M. Coxeter, Introduction to Geometry, Toronto: John Wiley and Sons (1969), pp. 23-25.
A n a t o m y and Evolution of a T h e o r e m on Triangles (by the C o l u m n Editor)
Figure 2
Figure 3 happens often in induction proofs: The fuller statement is easier to prove than the restricted one.) So we turn to the "cheating" strategy as used in [4], namely, we start with the equilateral triangle and build out. The result is Figure 2. Here we have normalized matters by choosing the equilateral triangle to have side 1. Note that from symmetry it will be sufficient to prove that the indicated angle is A/3. At this point we could turn the proof over to a high school trigonometry student who has learned to "solve triangles." Observe that all the side lengths in Figure 2 are determined by angle-side-angle (ASA) of the three constructed triangles. Thus, the law of sines applied to triangle AB'C' gives AC' sin(C + ~r)/3
--
1 sin(A/3)
As every school boy used to know, the medians of a triangle meet in a point (the centroid), as do the altitudes (the orthocenter). (These days you'd be lucky to find a school boy who even knows what a median is.) A less familiar example of concurrence is the Fermat point. This is the point (of an acute triangle) which minimizes the sum of the distances from the three vertices. To find it, one constructs three equilateral triangles having as bases the three sides of the given triangle. Now connect the far vertex of each of these triangles to the opposite vertex of the given triangle. The intersection of these three lines is the desired point. Much less well known is the following theorem or pair of theorems related to a construction sometimes attributed to Napoleon. In the figure below, choose A', B', and C' to be the centers rather than the far vertices of the three triangles. Once again the lines AA', BB', and CC' turn out to be concurrent. (The so-called Napoleon's Theorem says that in this case the points A', B', and C' are themselves vertices of an equilateral triangle!) Further, the three equilateral triangles can be taken either outside or inside of the original triangle.
l
SO
AC'=
sin(C + ~r)/3 sin(A/3)
BC'=
sin(C + ~-)/3 sin(B/3)
Similarly,
Also /_AC'B = 2~r
A+~3
B+vr 3
~r 3
C+27r 3 '
so for triangle AC'B we have Figure 3. For this triangle we know two sides and the included angle, (SAS), so the remaining angles are determined, and they must be A / 3 and B/3, which one verifies by once again using the law of sines. QED 32
THE MATHEMATICAL
INTELLIGENCER
V O L . 18, N O . 1, 1996
The Fermat Point
First G e n e r a l i z a t i o n It turns out that all of these theorems are special cases of an infinite one-parameter family of theorems, which was first discovered over 100 years ago, although apparently it keeps being rediscovered. The theorem, attributed to Ludwig Kiepert, replaces the equilateral triangles of Fermat and Napoleon by any set of three similar isoceles triangles. If the base angle of these triangles is c~, then the Fermat point corresponds to the case ~ = ~r/3, Napoleon to cr = ~-/6 (plus or minus); the centroid and orthocenter are the limiting cases ~ = 0 and o~= Ir/2, respectively (check it out). The locus of the points of concurrence as o~runs from - I r / 2 to Ir/2 is a rectangular hyperbola known as Kiepert's hyperbola. For a recent exposition, see Mathematics Magazine, June 1994, pp. 188-205. Exercise (surprising but easy): Show that the Kiepert hyperbola also passes through the three vertices A, B, and C; thus it is the unique conic passing through the five points consisting of the centroid, the orthocenter, and the three vertices. (Do it. Get in on the fun. I dare you!) Second Generalization It turns out that the one-parameter family above is a special case of a triply infinite family of theorems. Referring to Figure 1 below, we have THEOREM 1. Given a triangle ABC, and points A', B', and C', such that o~ =/_BAC' = LB'AC, ]3 =/_ABC' = /_A'BC, and 3' =/_A'CB = LACB'. Then AA', BB' and CC' are concurrent. Thus, the triangles on the sides of ABC need not be isoceles. The key condition is that the base angles of the three triangles be equal in pairs, as shown in Figure 1. Note that because o~, ]3, and ~/are arbitrary we have a three-parameter family of theorems; Kiepert's result is the special case where ~ = ]3 = 7. The result is so simple and
natural in its statement that one suspects it must have been noted long ago, but the historical trail seems to be murky. A recent reference is a "Classroom Note" by D. Kirby (Am. Math. Monthly, January 1980, pp. 45-47). Third G e n e r a l i z a t i o n The rest of the story is based on observations communicated by Clifford Gardner. First some terminology: The lines a and a' in Figure I are called isogonal lines with respect to vertex A, meaning that they are mirror images of each other with respect to the angle bisector at A. Analogously, instead of reflecting in the angle bisector, one can "reflect" in the medians of the triangle. More precisely, two lines through a vertex are called isotomic if they meet the opposite side in points equidistant from its midpoint. It turns out that the analogue of Theorem 1 also holds if the lines through the vertices are isotomic rather than isogonal. In fact the result is true for any triple of concurrent lines through the vertices A, B, and C in the following form. THEOREM 2. Let p, q, and r be concurrent lines through the vertices A, B, and C, respectively, of triangle ABC. Let PA be the pencil of lines at A and let TA be the (unique) projective mapping on PA which (1) interchanges lines AB and AC, (2) leaves p fixed. Define PB, Pc, and TB, Tc similarly. For any line a in PA let a ' = T A(a). Similarly, for b in PB and c in Pc, let b ' = TB(b), c' = Tc(c). Let C' = a' n b, B' = b' n c, and A ' = c' n a. Then AA', BB', and CC' are concurrent. Note that we now have a five-parameter infinity of theorems, for the coordinates of the point P can be chosen arbitrarily. Actually, Theorem 2 is an immediate consequence of
B:/
8
B'
A'
A a'
Figure 1
--~C,
b
Figure 2 THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996 33
Figure 3 Theorem 1, using the well-known fact that a projectivity of the plane can be defined arbitrarily on a n y four independent points. Thus, in particular, Figure 1 becomes Figure 2 by leaving vertices A, B, and C fixed and taking the in-center into any point P. The usefulness of the generalization, therefore, is not to include more cases but rather to simplify the proof. Namely, we make one more projective transformation which carries C to the origin, A to the point at infinity on the y axis, B to the point at infinity on the x axis, and P to the point (1, 1) as shown in Figure 3. Then lines p, q, and r become the lines x = 1, y = 1, and x = y, respectively, and we have C'=a'nb=(1/a,b), A'=b'nc=(1/bc, B' = c' n a = (a, a/c),
PA is the set of all vertical lines, PB is the set of all horizontal lines, Pc is the set of all lines through the origin.
thus, AA' has equation x = 1/bc, BB' has equation y = a/c, and CC' has equation y = (ab)x. Then AA' n BB' = (1/bc, a/c) which lies on the line CC'.
N o w recall that any one-dimensional projective transformation is of the form y = (ax § b)/(cx § d), so TA m a p s the lines x = a to the lines x = 1/a: this is the unique projectivity which interchanges the y axis and the line at infinity, leaving the line x = 1 fixed. We will denote the vertical line whose equation is x = a simply by a, the horizontal line y = b by b, and the line y = cx through the origin by c. Then with similar definitions of TB and Tc we have
a'=TA(a)=l, a
b' = T B ( b ) = 1 b
c'
Tc(c)
1 c
It is n o w a matter of simple high school analytic geometry to calculate the coordinates of the points A', B', and C' and the equations of the lines AA', BB', and CC' and verify that they are concurrent. Namely, 34
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
1/b),
Fourth (and Last) Generalization Observe that Theorem 2 is a theorem of projective geometry; this implies that Theorem 1 is "absolute," meaning that it holds in elliptic and hyperbolic as well as Euclidean geometry (in the hyperbolic case one makes the hypothesis that the points A', B', and C' exist). Note that this is not obvious even for the simplest special case, namely, the concurrence of the medians, which was the starting point of this exposition. The usual Euclidean proof makes strong use of the parallel postulate, as one needs the line connecting the midpoints of two sides of a triangle to be parallel to the third side. It can be shown using other methods that the median theorem also holds in the non-Euclidean cases. By contrast, theorem 2 covers all three cases directly.
Symmetries of Fractals Xiang Sheng and Michael J. Spurr
Introduction
Symmetries
The Mandelbrot set, M(2), pictured in Figure 1, is a remarkable subset of the complex plane C having an extremely complicated b o u n d a r y . This b o u n d a r y appears to enclose arbitrarily small miniature "copies" of the original set, making it v e r y jagged. This behavior is typical of a fractal, and, in fact, the H a u s d o r f f dimension of the b o u n d a r y of M(2) (which we take to be a measure of jaggedness) is 2 (see [5]). One also observes that M(2) has a reflective s y m m e try about the real axis. That is, if a complex n u m b e r c = x + iy is in M(2), then its complex conjugate conj(c) = = x - iy is also in M(2). We will be interested in finding all such symmetries (reflective or otherwise) of sets which are analogues of M(2), namely, the Generalized Mandelbrot sets M ( k ) which are pictured in Figures 1 and 2 for k = 2, 3, 4, and 5. First let us define M(2). Given c E C, we take iterations of the function Pc(z) := z 2 + c to obtain P~ = z,
Re-examine the pictures of M ( k ) in Figures I and 2. One sees several lines of apparent reflective s y m m e t r y through the center of each snowflake-like Generalized Mandelbrot set. There also appear to be'rotational symmetries: if M ( k ) is rotated through an angle of 2~r/(k - 1) it appears to exactly cover the original M ( k ) . These symm e t r y motions of the complex plane (reflections and rotations) lie in two groups of interest to us, namely, G, the g r o u p of rotations and reflections of C which preserve 0,
Pc(z)
= z 2 -ff c , Pc2(Z) = ( z 2 q- c ) 2 + c ,
PB(z)
= ( ( z 2 q- c ) 2 q- c ) 2
+ c, and so on: Pn(z) := Pc(pnc -1 (z)). The sequence {Pn(z)}n=0 is called the orbit of z (under iterations of Pc). The behavior of the orbit of 0, n
oo
{Pc(0)}n=0 = {0, c, c2 + c, (c2 + c)2 + c, ((c2 + c)2 + c)2 + c. . . . }, determines whether or not c belongs to the Mandelbrot set M(2), which is defined b y M(2) := {c E C I the orbit {pn(0)} remains bounded}. One defines the generalizations M ( k ) of the Mandelbrot set M(2) similarly. For each integer k ~ 2, let Pc,k(Z) = z k + c. The orbit of 0 u n d e r iterations of Pc,k is given by {pcn,k(0)}~=0 = {O,c,c k + C, (Ck + C)k + C. . . . }. The Generalized Mandelbrot set M ( k ) is M ( k ) := {c E C I the orbit {P~k(0)} remains bounded}. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 9 1996 Springer-Verlag New York 3 5
Figure 1: M(2) is the Mandelbrot set for z 2 + c. M(3) is the Mandelbrot set for z 3 + c. FL(2) is represented in white as the cardioid main body of M(2). CL(2) consists of the point (1/4, 0) at the pinch in the cardioid. FL(3) is represented in white as the main b o d y of M(3) and resembles an open "figure 8." CL(3) consists of the two points at the pinches in the "figure 8."
G := {~b I ~b = e iO or ~b = e iO. conj for some 0 E R}, and R R , the group which includes reflections (about any line in C) and rotations (about any point in C), R R := {61 ~b = th + t, where th E G and t ~ C}.
A n y motion of the plane C lies in RR. For instance, so q~ rotates z about the point 0 t h r o u g h an angle of 0 radians. Similarly, e iO. conj(z) = ei~ is a reflection about the line L = {rei0/2 I r E R}. If qb(z) = eiOz is a rotation in G with e iO ~ 1, then ~(z) = ei~ t in R R is a rotation about the point - t / ( e i~ - 1) through an angle of 0 radians; otherwise for e iO = 1, t~(z) = z + t is a translation. If th(z) = ei~ and if ~b(t) + t = 0, then t~(z) = ei~ + t is a reflection through the line { t / 2 + i A t / 2 I A E R} if t ~ 0; if t = 0, then ~b is just the reflection & If ~ z ) = z + t or if ~ z ) = ei~ + t with ~b(t) + t v~ 0, then iterations of ~btend to infinity. Finally, w e mention G is a subgroup of R R , characterized byt = 0. qb -- e i~ takes z to ei~
36
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
To discuss the s y m m e t r y motions of M(k), we define two types of symmetries of an arbitrary set S in C. We let the G-symmetries of S, Gsym(S), be those motions in G which preserve S: Gsym(S) := {~b ~ G I th(s) E S r
s ~ S}.
Gsym(S) is a subgroup of G. More generally, the set of symmetries of a set S, Sym(S), is the set of motions in R R which preserve S: Sym(S) := {~b~ R R I ~ s ) E S r
s ~ S}
and Sym(S) is a subgroup of R R . The s y m m e t r y motions of M ( k ) a p p e a r to be precisely the same as the s y m m e t r y motions of a regular polygon having k - 1 sides and the center at the origin. These are rotations about 0 t h r o u g h any multiple of 2 r c / ( k - 1), and reflections about the lines through 0 forming an angle with the positive real axis which is an
Figure 2: M(4) is the Mandelbrot set for z 4 + c. M(5) is the Mandelbrot set for z 5 + c. FL(4) is represented in white as the main body of M(4). FL(4) resembles a 3-leaf clover or a flower with 3 petals. CL(4) consists of the 3 points at the pinches. FL(5) is represented in white as the main body of M(5). FL(5) resembles a 4-leaf clover or a flower with 4 petals. CL(5) consists of the 4 points at the pinches.
integer multiple of vr/(k - 1). For instance w h e n k = 5, then k - 1 = 4, and our regular polygon is taken to be a square as pictured in Figure 3. It appears that M(5) exhibits exactly the same s y m m e t r y properties as the square. The symmetries of a regular polygon with k - 1 sides described above form a subgroup of G called the dihedral group Dk-1, defined as { ~ = e2 ~ m / ( k - 1 ) o r
q5 =
e2 ~ m / ( k
1). conj I m
E
Symmetries
Form Dihedral
Gsym(M(k)) = Dk 1. T H E O R E M 2. For k ~ 3, the group of symmetries of the Mandelbrot set M(k) is the dihedral group Dk-l: Sym(M(k)) = Dk 1.
Z}.
In the Spring 1992 Mathematical Intelligencer Alexander, Giblin, and N e w t o n [1] s h o w e d that every motion in Dk-1 is a s y m m e t r y of M(k), just as one is intuitively led to believe b y examining the pictures in Figures 1 and 2. T h e y also conjectured that e v e r y s y m m e t r y of M(k) must belong to the dihedral g r o u p Dk-1. We will n o w show w h y this is indeed true. Mandelbrot
Mandelbrot set M(k) is the dihedral group Dk-l:
Groups
The main results on symmetries of M(k) are as follows: T H E O R E M 1. For k ~- 2, the group of G-symmetries of the
To our knowledge, Theorem 1 first a p p e a r e d in [3]. We sketch the ideas behind Theorems 1 and 2, but we refer the reader to [4] for the full proof of these theorems (and others mentioned). A key idea will involve finding points on the b o u n d a r y OM(k) of M(k) which lie closest to 0. (The partial solution by Alexander, et al. relies on points furthest from 0.) The fixed points q of Pc,k(Z) = z k + c which have derivative P~,k(q) of n o r m less than 1 are called attracting fixed points of Pc,k(Z). We wish to describe all c which have attracting fixed points. Using the fixed-point condition, we see that
qk + C = q, THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996 3 7
To indicate the d e p e n d e n c e of c on r and a w e denote it b y c = Cr,,,. We then define FL(k) b y
FL(k) : = {Cr, a [ a E [0, 2 ~ k - 1)[ and r < 1}. FL(k) is r e p r e s e n t e d in Figures 1 a n d 2 as the white flower-like m a i n b o d y of M(k). As an example, FL(2) is the cardioid m a i n b o d y of the M a n d e l b r o t set M(2). For higher values of k, FL(k) begins to resemble a flower: FL(3) looks like an o p e n "figure 8," FL(4) like a 3-leaf clover, a n d FL(5) like a 4-leaf clover. There are several reasons w h y w e are interested in FL(k). First, one can s h o w FL(k) lies inside M(k) (see [4]). Second, it is an o p e n set [because the analytic m a p p i n g f(q) = q _ qk takes the disk of radius (1/k) 1/(k-l) to FL(k)]. But m o s t importantly, w e h a v e s h o w n [4] that the b o u n d a r y OFL(k) = {c1,~ [ a E [0, 2 ~ k - 1)]} of FL(k) has precisely the s a m e set of points closest to 0 as does OM(k), the b o u n d a r y of M(k)! This follows after showing Cl,0 is in OM(k) a n d observing that
[Cl,c~[2 =
( k ) 1/(k-1) ei~
(1-keia)
2
= (k)2/(k-l) I1 -- 2k COS(~ q- ~21
~ (k)a/(k-1) (1 - k)2 = [cl,ol2,
Figure 3: The square w i t h sides parallel to the axes has symmetry motions in the dihedral group D4. M(5) has these same s y m m e t r y motions.
with equality holding precisely w h e n a = 2ran for m in Z. Thus, the set of closest b o u n d a r y points CL(k) con(1- 1/k)= sists of the points (1/k) 1/(k-1) r ei2cnn/(k-1)Cl,0 for m in Z. Refer to Figures 1 and 2 to see CL(k) as the set of "pinch points" in OFL(k). N o w if ~b = eiO or 6 = ei~ conj is a G - s y m m e t r y of M(k), then q~ m u s t also be a G - s y m m e t r y of the b o u n d ary 3M(k). Because ~bpreserves distance to 0, b takes the set of closest points CL(k) to itself. This implies that 6(Cl,0) = ei~ = ei2mn/(k-1)Cl,0 . W e conclude that ~b = e i2mn/(k-1) or q~ = ei2~-m/(k 1) . conj, a n d that a n y G-symm e t r y of M(k) m u s t lie in the dihedral g r o u p Dk-1. Let 6 = ~b + t be a m o t i o n in Sym(M(k)). We will s h o w that 0 lies in Gsym(M(k)) = Dk-1 b y s h o w i n g that t = 0. If 6 = ~b + t with 6 = ei~ conj, t h e n q~(t) + t = 0,
a n d using the condition that the derivative P',k(q) h a v e n o r m less than 1, w e h a v e
U s i n g both of these equations, w e can solve for c:
for if ~b(t) + t ~ 0, then ~ ( z ) a p p r o a c h e s infinity a n d ~b cannot preserve the b o u n d e d set M(k). N o w ~bo ei2mn/(k-1) = eiOe-i2mn/(k-1) . conj q- t also belongs to Sym(M(k)). W e conclude that
c = q - qk = q(1 -- qk-1)
ei~ + t = 0
kqk-1 = rei,~ with r < 1.
(1 for a ~ [0, 2r
- 1)] a n d r < 1.
38 THEMATHEMATICALINTELLIGENCERVOL.18,NO.1, 1996
and
eiOe-i2,mn/(k 1)~-q_ t = 0.
M(g) := {c E C I for e v e r y critical point r of g,
Therefore
el~
(t)
- e -i2mn/(k-1))
the orbit {fn(r)} of r r e m a i n s bounded}.
= O.
W h e n k m 3, (t) can o n l y be satisfied w h e n t = 0 = t. So ~b = ~b + 0 belongs to Gsym(M(k)). If ~b = q~ + t with c~ = eiO, then ~bo conj = e iO. conj + t also belongs to Sym(M(k)) a n d w e get t = 0 as above. So e v e r y s y m m e t r y of M(k) is a G - s y m m e t r y of M(k) a n d so lies in Dk 1, giving Sym(M(k)) = Dk-l! N o t e that w h e n k = 2, (1 -- e -i2mn/(k-1)) = 0 in (t) and w e cannot conclude that t=0.
N o t e that w h e n g(z) = z k, then g'(z) = k z k - 1 = 0 has only the solution 0 = r as the critical point, and the above definition of M(z k) reduces to the original definition of
M(k). W h a t are the s y m m e t r i e s of M(g) for a general polynomial g? This is a m o r e difficult a n a l o g u e of the original question p o s e d in [1] asking for the s y m m e t r i e s of M(k). The a n s w e r is p e r h a p s m o r e complicated! If N
g(z) = Z
Generalizations
any "Znj
j=l J
If one studies iterations of the p o l y n o m i a l az k + c, one also obtains a generalized M a n d e l b r o t set M(az k) := {c E C I the orbit of 0 u n d e r iterations of az k + c r e m a i n s bounded}. The s y m m e t r i e s of M(az k) f o r m a "twisted" dihedral g r o u p D(k - 1, a) as described below. T H E O R E M 3. Given a = ReiL then for k ~ 2,
Gsym(M(azk)) = D(k - 1, a), where
D(k - 1,
e [2~m-2iy]/(k-1)
a)
:=
{~b = e2wim/(k-1)
is a p o l y n o m i a l w i t h any E C \ { 0 } a n d g ( 0 ) = 0, then w e have s h o w n that G s y m ( g ) = A N I D ( n j - 1, anj)= n ~/=1 Gsym(anj znj) (see [4]). There are s o m e technicalities in defining D(0, a) but for our p u r p o s e s it will suffice to take D(0, a) to be G for a real a n d positive. As an example, let g(z) = 2z 3 + 3z 2 + 2z. Then Gsym(2z 3 + 3z 2 + 2z) = D(3 - 1, 2) N D(2 - 1, 3) fir D(1 - 1, 2) = D 3 _ 1 n D 2 - 1 n G = D 1 . W e concluded that Gsym(2z 3 + 3z 2 + 2z) = {z, 7}.
or
ch =
9 c o n j I m = 0, 1, 2, . . . , k - 2}. Also for
k>_3, Sym(M(azk)) = D(k - 1, a). This result can be d e d u c e d f r o m T h e o r e m s I a n d 2 b y m a k i n g the change of variables z = (1/a)l/(k-1)u. A key algebraic p r o p e r t y , that of c o m m u t a t i v i t y of m o t i o n s with a p o l y n o m i a l , drives the inductive argum e n t s which s h o w m a n y of the g r o u p c o n t a i n m e n t s w e h a v e mentioned. The algebraic s y m m e t r i e s of a polynomial g(z) with g(0) = 0 are those motions w h i c h comm u t e with g(z):
So conjugation and the identity are the only G - s y m m e tries of 2z 3 + 3z 2 + 2z. We show, h o w e v e r , that Sym(2z 3 + 3z 2 + 2z) is strictly larger t h a n Gsym(2z 3 + 3z 2 + 2z). Certainly, Sym(2z 3 + 3z 2 + 2z) contains the identity a n d conjugation, but it also contains the m a p p i n g ~(z) = - z - 1 because g ( ~ z ) ) = ~Kg(z)): 2 ( - z - 1) 3 + 3 ( - z - 1) 2 + 2 ( - z - 1) = - ( 2 z 3 + 3z 2 + 2z) - 1. The two m o t i o n s ~ and - z Sym(2z 3 + 3z 2 + 2z). We h a v e
1 generate the g r o u p
Sym(2z 3 + 3z 2 + 2z) = {z, ~, - z - 1, - z -
1}.
Gsym(g) := {c~ ~ G I g(ch(z)) = ~b(g(z)) for all z E C}, Sym(g) := {~bE RR I g(~(z)) = ~K~(z)) for all z E C}. Both Gsym(g) and Sym(g) are groups. W e observe that every element of Dk-1 is in G s y m ( z k) because (e2"a'im/(k-1)Z)k = e2"a'irn/(k-1)Z k a n d (e 2 ~ m / ( k 1)~)k = e 2 ~ m / ( k - 1 ) Z k. It is s t r a i g h t f o r w a r d to c o m p u t e b o t h G s y m ( z k) and Sym(z k) a n 4 see that they equal Dk-1 for k m 2. Theorems 1 and 2 give that, for k m 3, G s y m ( z k) = Sym(z k) = Dk-1 = Gsym(M(k)) = Sym(M(k)). Based on this fact, one might conjecture similar behavior for m o r e general polynomials. H o w e v e r , s o m e care m u s t be taken. First, if g(z) is a n y p o l y n o m i a l with g(0) = 0, w e let fc(Z) = g(z) + c. The M a n d e l b r o t set M(g) associated to g(z) is defined as
N o w let us c o m p a r e our algebraic s y m m e t r i e s for 2z 3 + 3z 2 + 2z c o m p u t e d in the p r e c e d i n g p a r a g r a p h with the geometric s y m m e t r i e s for the M a n d e l b r o t set M(2z 3 + 3z 2 + 2z). By examining M(2z 3 + 3z 2 + 2z) in Figure 4(a), w e see that Sym(M(2z 3 + 3z 2 + 2z)) should consist of: reflection a b o u t the x-axis (~), reflection a b o u t the y-axis ( - ~ ) , rotation about 0 b y vr ( - z ) , and the identity (z). W e will s h o w that these are, in fact, elements of Sym(M(2z 3 + 3z 2 + 2z)) in T h e o r e m 4. But m a t t e r s are complicated. The geometric s y m m e tries Sym(M(2z 3 + 3z 2 + 2z)) cannot m a t c h either of our algebraic s y m m e t r y g r o u p s Gsym(2z 3 + 3z 2 + 2z) or Sym(2z 3 + 3z 2 + 2z). For instance, - z belongs to Sym(M(2z 3 + 3z 2 + 2z)) but not to Gsym(2z 3 + 3z 2 + 2z), so Gsym(2z3 + 3z 2 + 2z) is strictly contained in THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996 39
F i g u r e 4: (a) M ( 2 z 3 + 3 Z 2 + 2z), the M a n d e l b r o t s e t for g(z) = 2 z 3 + 3Z 2 + 2z. &(z) = - z - 1 is a s y m m e t r y o f g, h e n c e ~b(z) = - z is a s y m m e t r y o f M ( 2 z 3 + 3Z 2 + 2z). 0 is the c e n t e r of M ( 2 z 3 + 3 z 2 + 2z). (b) T h e f i l l e d - i n Julia set for 2Z 3 + 3Z 2 + 2z. Jo is the b o u n d a r y o f the f i l l e d - i n Julia set. Jo h a s s y m m e t r i e s : ~ z ) = - z - 1, c o n j u g a t i o n (and h e n c e r e f l e c t i o n t h r o u g h the l i n e x = - 1/2).
Sym(M(2z 3 + 3z 2 3- 2z)). But also one can s h o w that the algebraic s y m m e t r y &(z) = - z - 1 in Sym(2z 3 + 3z 2 § 2z) does not lie in the geometric s y m m e t r y g r o u p Sym(M(2z 3 + 3z 2 + 2z)). This follows because 0 belongs to M(2z 3 § 3z 2 § 2z) but ~ 0 ) = - 1 d o e s not belong to M(2z 3 + 3z 2 + 2z). So Sym(2z 3 + 3z 2 + 2z) Sym(M(2z 3 + 3z 2 3- 2z)). W e are left to h u n t for another algebraic candidate to give Sym(M(g)). We propose
S(g) := {q~ E G I 3 6 = ~b + t E Sym(g)}, w h i c h is obtained b y taking the "G c o m p o n e n t " ~b of each algebraic s y m m e t r y ~b = ~b + t. Note that for g(z) = 2z 3 3- 3z 2 3- 2z, we h a v e Sym(2z 3 + 3z 2 + 2z) = {z, E, - z - 1, - E - 1}. Thus, S(2z 3 3- 3z 2 3- 2z) = {G-component(z), G-component(E), G - c o m p o n e n t ( - z - 1), and Gc o m p o n e n t ( - E - 1 ) } . So S(2z 3 3- 3z 2 + 2z) = {z, E, - z , -E}. These are precisely the geometric s y m m e t r i e s w e s a w for Sym(M(2z 3 + 3z 2 + 2z)) in Figure 4(a). W h a t we can p r o v e is:
point of g(z) is equivalent to ~(r) being a critical point of g(z). N o w fC~(c)(~z)) = g ( ~ z ) ) + 4~(c) = tldg(z)) + 4~(c) = ch(g(z)) + t + oh(c) = ch(g(z) + c) + t = ~b(fc(z)), and b y induction f~(c)(~(z))= ~fc(z)), k Vk E N. Therefore, {fck(r)} remains b o u n d e d for all critical points r of g (qdfk(r))} remains b o u n d e d for all critical points r of g <=~ {f~(c)(~r))} remains b o u n d e d for all critical points r of g <=~{f~(c)(r)} remains b o u n d e d for all critical points r of g. F r o m this, w e see that c E M(g) ~ oh(c) E M(g), giving ~b r Sym(M(g)). The fact m e n t i o n e d earlier that S(2z 3 + 3z 2 + 2 z ) = {z, E, - z , -E} is contained in Sym(M(2z 3 + 3z 2 + 2z)) is a special case. History is repeating. Just as Alexander, et al. k n e w that Dk-1 = S(z k) is contained in Sym(M(zk)) and asked w h e t h e r S(z k) = Sym(M(zk)), w e n o w k n o w that S(g) is contained in Sym(M(g)) and ask if S(g) = Sym(M(g)).
Symmetries
of Mandelbar
Sets
If one studies iterations of the function Pc,k(Z) := ~k + C, one can ask if the orbit {PY,k(0)} = {0, C, ~k + C, (Ck + ~)k + C. . . . }
4. If g ( z ) = ~']=1 ajzJ with g ( 0 ) = 0 and degree(g) -> 2, then S(g) is contained in Sym(M((g)), where S(g) := {~b E G I 3 ~ = ~b + t E Sym(g)}.
THEOREM
To see this, let ~b = e iO or e iO. conj a n d let ~b = ~b + t be in Sym(g). Then g ( ~ z ) ) = ~Kg(z)). Differentiating this w i t h respect to z if ~b = e i~ one obtains that g'(6(z)) = g'(z) [if 4~ = e i~ -conj, differentiating with respect to E gives g'(6(z)) = conj(g'(z))]. Therefore, r being a critical 40
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
remains b o u n d e d . If so, then w e say c lies in the MandelBAR set M(k) := {c ~ C I the orbit {PY,k(0)} r e m a i n s bounded};
M(k) is r e p r e s e n t e d in Figure 5 for k = 2, 3. By examining the c o m p u t e r g r a p h s of M(k), Alexander, Giblin, and N e w t o n [1] w e r e led to conjecture that Sym(M(k)) is the
dihedral group Dk+l. They proved parts of this, and we can complete the confirmation. m
m
THEOREM 5. Sym(M(k)) = Gsym(M(k)) = Dk+lfor k ~ 2. The proof is comparable to those of Theorems 1 and 2 (see [4]). The idea is to find an open set BFL(k) in M(k) which is the analogue of FL(k) in M(k). We do this by taking those c for which Pc,k has a fixed-point q with k~ k-1 of norm less than 1. As before, we have shown that the closest~r)oints on OBFL(k) are precisely the closest points of OM(k). These closest points turn out to be those of the form
ei2cnn/(k+l) (k)l/(k 1) ( 1 - k ) = ei2~Tm/(k+l)cl,0, m E Z.
Julia Sets
The article by Alexander, et al. [1] also raised and partially answered some questions about symmetries of Julia sets. For a polynomial g with g(0) = 0, let fc(z) = g(z) + c and take Ac(oo) to be the set of points escaping to ~ under iterations of ft. Then the Julia set Jc of fc is just the boundary OAc(~) of Ac(~). The filled-in Julia set is the complement of Ac(o~). See Figure 4(b) for a picture of the Julia set J0 of 2z 3 + 3z2 + 2z. Beardon [2] has completely characterized the rotational symmetries of Julia sets of polynomials fc, but some questions on the reflective symmetries remain open. We briefly describe Beardon's results and then some ways of supplementing them. For a polynomial f(z) = ~ = 0 akzk,o f degree n ~ 2 with leading coefficient an # 0, the centroid Kof the Julia set of f is
Again every element q~of Gsym(M(k)) is in Gsym(OM(k)) and preserves distance to the origin. So 4~(c1,0)= ei2mn/(k+l)Cl,0 = ei~ implies that ei2mn/(k+l) = eie; then ~b ~:= --an-1 must be of the form ~b = e2Vrim/(k+l)o r q~ = e 2"trim/(k+l) .__conj nan and so belongs to Dk+ 1. We conclude Dk+ 1 Gsym(M(k)). To show that Sym(M(k))= Dk+l, one proceeds as in The rotational symmetries (of form ei~ + t) of the Julia Theorem 2. set of f form one of these groups: the ffivial group, a ~-
m
Figure 5: M(2) is the MandelBAR set associated with 32_ + c. BFL(2) is the white set inside M(2). M(3) is the MandelBAR set associated with ~3 + c. BFL(3) is the white set inside M(3). THE
MATHEMATICAL
INTELLIGENCER
VOL.
18, NO.
1, 1 9 9 6
41
F i g u r e 6: (a) T h e f i l l e d - i n J u l i a s e t f o r z 9 + z 5 + 0.6 + 0.2/. (b) T h e f i l l e d - i n J u l i a s e t f o r z 9 + z 5 - 0.2 + 0.6i.
cyclic g r o u p of rotations a b o u t ~', or the g r o u p of all rotations a b o u t ~. We say f is in n o r m a l f o r m if an = 1 and an-1 = 0. One can conjugate f(z) b y some linear polynomial az + b to obtain [(1/a)z - (b/a)] o f o (az + b) in norm a l form. S u p p o s e t h e n that f has been written in norm a l form. If the g r o u p of rotational s y m m e t r i e s is infinite, then f(z) = z n. Otherwise, the order of the cyclic g r o u p of rotational s y m m e t r i e s a b o u t the centroid is the largest integer m such that f(z) can be written as zrQ(z m) for s o m e p o l y n o m i a l Q a n d s o m e natural n u m b e r r. W e illustrate B e a r d o n ' s result on f(z) = 2z 3 + 3 Z 2 q2z. The centroid of f is just ~ := - a n - 1 / ( n a n ) = - 3 / 3 9 2) 7 - 1 / 2 . Conjugating f(z) = 2z 3 + 3z 2 + 2z b y [ ( 1 / V 2 ) z - (1/2)]_to obtain the n o r m a l form, we h a v e [V'2 z + (1/X/~)] o f o [ ( 1 / V 2 ) z - (1/2)] = z 3 qZ/2. NOW m = 2 is the largest integer such that z 3 + 2:/2 = zrQ(zm). We conclude that the rotational s y m m e tries of the Julia set J0 associated to f(z) = 2z 3 + 3z 2 + 2z f o r m a cyclic g r o u p of rotations of order 2 a b o u t the centroid ~ = - 1 / 2 . C o n s u l t i n g Figure 4(b), w e see that these indeed describe all the rotational s y m m e t r i e s of J0, b u t that there a p p e a r to be s o m e reflective s y m m e t r i e s as well. We will describe a w a y to obtain t h e m shortly. W e return to o u r algebraic symmetries. It turns out that u n d e r the correct circumstances these m a p p i n g s can be s y m m e t r i e s of a Julia set. W e will use: T H E O R E M 6. For a polynomial g with g(O) = O, let fc = g + c. Then if ~ = cb + t is in Sym(g), one has 6(Jc) = J4(c). For the proof, observe that f~c) (~z)) = ~(fk(z)) for all k E N. Then f~c) (~z)) approaches infinity ca ~fck(Z)) approaches infinity ca fkc(Z) approaches infinity. This implies that z E Jc ca ~ z ) E Jcxc), which is the desired conclusion. Returning to our e x a m p l e 2z 3 + 3z 2 + 2z, if ~b = ~ + t is a s y m m e t r y of M(2z 3 + 3z 2 + 2z), then ~b(0) = 0, a n d b y T h e o r e m 6, ~J0) = J4(0) = J0 and 6 is a s y m m e t r y of J0. N o w w e h a v e a l r e a d y o b s e r v e d that z, $, - z - 1, and - $ - I are s y m m e t r i e s of 2z 3 + 3z 2 + 2z. T h e o r e m 6 tells 42
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
us that z, ~, - z - 1, a n d - ~ - 1 are also s y m m e t r i e s of Jo. N o w z (id) and - z - 1 (rotation t h r o u g h angle ~a b o u t z = - 1/2) are accounted for b y Beardon's results, whereas ~(reflection a b o u t the x-axis) and - ~ - 1(reflection a b o u t x = - 1 / 2 ) are (along w i t h z and - z - 1) accounted for b y Sym(2z 3 + 3z 2 + 2z)! As a final illustration of T h e o r e m 6, we examine Julia sets for certain fc(z) = z 9 + z 5 + c. N o w G s y m ( z 9 + z 5) = D9-1 n D5-1 = D8 n D4 = D4 is contained in Sym(z 9 + zS). N o t e that the 90 ~ rotation ~b = i = e ir belongs to D4 a n d that the point c = 0.6 + 0.2i satisfies ~b(c)= ~b(0.6 + 0.2i) = - 0 . 2 + 0.6i, that is, - 0 . 2 + 0.6i is a 90 ~ rotation of 0.6 + 0.2i. T h e o r e m 6 says that the Julia set for z 9 + z 5 - 0.2 + 0.6i = z 9 + z 5 + ~b(0.6 + 0.2i) is a 90 ~ rotation of the Julia set for z 9 + z 5 + 0.6 + 0.2i. This is illustrated in Figure 6.
References
1. Alexander, C., Giblin, I., Newton, D., Symmetry groups of fractals, Mathematical Intelligencer 14 (1992), No. 2, 32-38. 2. Beardon, A.F., Symmetries of Julia Sets, Bull. London Math. Soc. 22 (1990), 576-582. 3. Sheng, X., Symmetries of Generalized Mandelbrot Sets and their Julia Sets, Masters thesis, East Carolina University, November, 1990. 4. Sheng, X., Spurr, M.J., Symmetries of Mandelbrot and Mandelbar Sets, to appear. 5. Shishikura, M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, SUNY Stony Brook Institute for Mathematical Sciences, preprint, July 1991. Blue Cross of Western Pennsylvania 5th Avenue Place Suite 1092 Pittsburgh, PA 15222 USA Mathematics Department East Carolina University Greenville, NC 27858 USA mjs@mathl .math.ecu.edu
Symmetries of Julia Sets A.F.
In a recent article [1], the symmetries of Julia sets were discussed. In this article we shall describe a simple algorithm which enables one to compute the Euclidean s y m m e t r y group of the Julia set of a given polynomial and also to construct a polynomial with a Julia set having a prescribed s y m m e t r y group. Throughout this article, P(z) = an zn + 999+
alz + a0,
where n -> 2 and an ~ 0, and Je is the Julia set of P. For the moment, we restrict ourselves to the direct symmetries z ~-~ az + b, where lal = 1, of Je. It is easy to see that the group Ep of such symmetries is a rotation group. Indeed, EP cannot contain translations (as Je is bounded); and as the group generated by two rotations with different centres contains translations, the elements of Ep are rotations with a c o m m o n fixed point. Except for the case of extreme symmetry, namely, P(z) = z n (or a polynomial conjugate to this), Ep is, thus, a finite cyclic rotation group Cq (of order q); the problem is to find the c o m m o n fixed point ~ and the order q of Ep in terms of the coefficients aj. Surprisingly, perhaps, ~ is easy to find. For any w, let ~(w) be the centre of gravity of the n solutions Zl. . . . . Zn of P(z) = w. Identifying coefficients of z n-1 in the identity an zn + " ' " + a l z + a o -
w=P(z)
=an(Z
Beardon
tries of Jp (if any) take the simple form z ~ e iOz. We now write P in the form P(z) = z p Q ( z q ) ,
where Q is a polynomial and where the integers p and q are maximal. One can now show that ~p is the rotation group Cq of order q about the origin. It is easy to see that ~p D Cq (to prove that Ep C Cq is harder, and again we refer to [2] for the details). Let" g(z) = coz,
co = exp(2cr//q),
so that Ig(z)l = Izl. It follows that Pg(z) = P(oaz) = ~oPP(z) = gPP(z);
so for each integer k there is an integer m such that pkg(z) = gmpk(z), and hence that
Ipkg(z)l = ipk(z)l.
-w --Zl)''"
(Z --Zn) ,
we see that an-1 = - n a n , ( W )
so that ~(w) is independent of w. Henceforth, we write for ~'(w). It follows automatically that, for every positive integer k, ~ is the centre of gravity of the n k solutions of Pk(z) = w, and as these solutions converge to Jp, it is at least intuitively clear that ~ must be both the centre of gravity of Jp and the centre of all rotations in ~p. This is indeed true (see [2] for details). By a linear change of variable, we m a y assume that an-1 = O, and this has the advantage that the symmeTHE MATHEMATICAL INTELLIGENCERVOL. 18, NO. 1 9 1996 Springer-Verlag New York
43
W e n o w see that, as k ~ % P k ( z ) ~ o~ if a n d only if pkg(z) --, ~: g is a s y m m e t r y of the basin of attraction A~ of ~. As Jp = OA~, g is in s and so s D Cq. Let us illustrate this algorithm b y considering the polynomial
P(z) = z 4 q- 4/z 3 - 6z 2 + (1 - 4i)z + 1. For this p o l y n o m i n a l , ~"= - i , and so, writing ~b(z) = z + i [so that q~(O = 0], w e find that q ~ P ( ~ - l ( z ) = z ( z 3 -4- 1).
W e deduce that s is the rotation g r o u p of o r d e r 3 about the centre - i . As each quadratic p o l y n o m i a l P is conjugate to z 2 + c, the algorithm s h o w s that, for quadratic polynomials, s = C2. As far as cubic p o l y n o m i a l s are concerned, the three p o l y n o m i a l s z 3 + 1, z(z 2 + 1), and z 3 + z + 1 h a v e Ep equal to C3, C2, a n d C1 (the trivial group), respectively. Finally, for a general p o l y n o m i a l P, the full Euclidean s y m m e t r y g r o u p Fp of Jp m a y include indirect s y m m e tries z ~ a~ + b, w h e r e la] = 1. H o w e v e r , Ep is either Fp or a s u b g r o u p of index 2 in Fp, so that Ep is either Fp or the corresponding d i h e d r a l g r o u p Dq (of order 2q). We leave the reader to explore this case and to verify that in the example above, s = D3.
References 1. C. Alexander, I. Giblin, and D. Newton, Symmetry groups of fractals, Mathematical Intelligencer 14 (1992), no. 2, 32-38. 2. A.F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc. 22 (1990), 576-582.
Department of Mathematics and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 1SB United Kingdom
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THE MATHEMATICALINTELLIGENCERVOL. 18, NO. l, 1996
Symmetries of Fractals Revisited Eike Lau and Dierk Schleicher
Introduction
Multicorn: M~ := {c ~ C: the sequence z0 := 0; Zn+l := f*a,c(Zn) is bounded}.
In the Spring 1992 issue of The Mathematical Intelligencer, Alexander, Giblin, and Newton [1] describe the symmetry groups of certain Julia sets and their corresponding parameter spaces (as they call them, the generalized Mandelbrot and Mandelbar sets). Besides beautiful pictures, they give proofs that these "fractals" are invariant under certain symmetry groups and conjecture that there are no other symmetries. Proofs of many of these statements have been known before in a more general form (Beardon [3], Sheng and Spurr [18]). In this note, we want to describe some simple geometric observations proving the cases considered by Alexander et al., along with a few additional relations.
In the simplest case d = 2, this includes the standard Mandelbrot set M2, whose structure is quite well understood. This work was pioneered by Douady and Hubbard [8, 9]; see also Br'anner [5] and Sehleicher [16]. The sets M~ have also been called "Mandelbar sets" (compare Crowe, Hasson, Rippon, and Strain-Clark [7]); we call them "Multicorns" in analogy with Milnor's suggestion "Tricorn" for M~ [13]. They have been explored only recently: Nakane [14] has proved that the Tricorn is connected; many structural similarities between the Mandelbrot sets and the Multicorns can be found in Nakane and Schleicher [15] and in the references
Complex Dynamics All of the "fractals" that we investigate here come from iterating polynomials of the type fa, c:z~-~ z d + c or f~,c:z ~ ~a + c defined on the complex plane; here c is a complex parameter, d -> 2 an integer, and the bar denotes complex conjugation. For every degree d, there is one holomorphic and one antiholomorphic family of polynomials, labelled by the parameter c. Choose one such polynomial and watch what all the complex numbers z do under iteration. Clearly, points that are far enough away from the origin escape after many iterations towards infinity--for example, if Izl > max{Icl,21/(d-1)}. Points with bounded orbit under iteration make up the filled-in Julia set, which we call Ka,c for the holomorphic and K~,c for the antiholomorphic case. The topological boundaries of these (bounded) sets are the Julia sets. There are many equivalent definitions available for Julia sets of polynomials. One is that every neighbourhood of any point in the Julia set will cover every point in the complex plane, with at most one exception, after sufficiently many iterations. In order to classify parameter values c, we define the following subsets of parameter space (see Figs. 3 and 5): Mandelbrot set: Md := {c E C: the sequence z0 := 0; Zn+l := fd,c(Zn) is bounded} THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 9 1996 Springer-Verlag New York
45
therein. However, one of the principal conjectures about the Mandelbrot set (that it is locally connected) is false for the Multicorns: they are not even pathwise connected (see Hubbard, Nakane, and Schleicher [10]). In the definitions above, the decision about boundedness of the iterated sequence is simplified by the fact that a single z n with IZnl > 21/(a-~) already implies that the sequence is u n b o u n d e d . Since Zl = c, this implies in particular that all Me and M~ are contained in a disk around zero with radius 21/(a-~). The complex n u m b e r 0 is the only critical point (in C) of our maps: the only point at which these m a p s are not locally injective, i.e., where their derivatives vanish. This Figure 1. Julia sets of polynomials. Left: Holomorphic case, number is singled out as the starting value of the itera- d = 6, c = 1.106926e i~/5 E M a (Ka, c connected); it has d-fold rotational symmetry and, c a - 1 being real, d axes of reflection because a filled-in Julia set Kd,c or K~,c is connected if tion. Right: Antiholomorphic case, d = 3, c = - 0 . 4 6 7 2 8 + and only if the critical point 0 has bounded orbit under fd,c 0.3201i ~ MCa (/Ca,c completely disconnected); it has d-fold roor fd,c, respectively; otherwise, it consists of uncountably tational s y m m e t r y but, since c a+~ is not real, no reflectional many disconnected (yet non-isolated) points. Therefore, the symmetry. sets Me and M~ are the loci of connected Julia sets. This fact illustrates an important principle in complex dy- Ka,c, there are reflection symmetries if and only if c d-1 is real; namics: m a n y important properties of the dynamics of in the antiholomorphic case K~,c, reflection symmetries exist a holomorphic map are closely related to the orbits of if and only if c a§ is real. In both cases, if axes of reflection the critical points. We will later meet another instance exist, there are d of them at equal angles; one of them is the line through 0 and c. of this principle. By a symmetry of a subset of the complex numbers we mean a rigid motion of this set onto itself, orientation- In the next section, we will see that the parameters c for preserving or not. More specifically, symmetries are which Kd,c has reflection symmetries form the reflection translations, rotations, line reflections, and compositions axes of Md, and similarly for K*d,cand M~. It is a familthereof. All our sets are easily seen to be compact; they iar observation in complex dynamics that interesting therefore have unique (smallest) circumscribed circles features of Julia sets reappear in the corresponding pawhich have to remain invariant under the symmetries. rameter spaces. H o w can one prove such s y m m e t r y statements? In A n y such symmetry group thus is a compact subgroup of the symmetry group of the circumscribed circle. Unless our case, this turns out to be quite simple. First note that it contains the full symmetry of the circle, it is finite: ei- the filled-in Julia sets are completely invariant: if some z ther a cyclic group C~ or, if there exist reflection symme- iterates a w a y towards infinity, then clearly flz) and all tries, a dihedral group Dn. There can be no translation; all f-l(z) also do, and only then. In other words, fd,c(Ka,c) = -1 " " * * * *-1 * rotations have a common center, which also lies on all Kd,c = fa,c (Ka,c) and similarly f3,c(Ke;) = Ke,c = f
1. Excluding this trivial case, we s h o w the only one center of rotational symmetry; for c = 0 this reasoning is void and we have, in fact, the full rotational following: symmetry. This settles the claims about rotations. N o w let us assume there exists a reflection s y m m e t r y For nonzero c, the rotational symmetry of every filled-in Julia set Kd,c or K*d,caround the origin is exactly d-fold, so their with respect to some axis through the origin, making an symmetry groups are either Ca or De. In the holomorphic case angle ~ with the positive real axis. Then taking d-th pow46
T H E M A T H E M A T I C A L I N T E L L I G E N C E R V O L . 18, N O . 1, 1996
ers produces a set which has a reflection symmetry with respect to the line at angle d~ (in the antiholomorphic case, we have to conjugate and obtain the angle -d~). Adding c produces a reflection axis through the point c. But taking d-th powers and adding c is just the action of fe,c and leaves Ke,c invariant. The constructed reflection axis through c is thus a reflection axis of Ke,c and contains 0. The same statement holds in the antiholomorphic case. We see that every axis of reflection is mapped by the dynamics onto the line joining the origin to c. In particular, if ~ was the angle between the real axis and this line, we see that +d~ and r must differ by an integer multiple of vr, or ~ = Irk/(d u 1), k an integer (upper sign for the holomorphic, lower one for the antiholomorphic case). This reasoning not only shows the necessity of the condition to have a reflection symmetry but also that in these cases the symmetry does in fact exist: reflected points z (with respect to any axis of reflection) land on reflected images (with respect to the line through 0 and c), so they both escape to infinity or both stay bounded. Since ~ is the argument of the complex parameter c (up to addition of a multiple of ~-), we see that this condition is equivalent to ce;1 being real. This completes the characterization of the symmetries of our Julia sets. In general, for a polynomial f(z) = ~ akz k it is easy to see that its Julia set has q-fold rotational symmetry around the origin provided we can write f(z) = zn•bkZq k with some coefficients bk and integer n; see Beardon [4]. The converse is also true; see Beardon [3] or [2], Section 9.5. The idea of this proof, at least for connected Julia sets, is to construct a conformal isomorphism from the exterior of the unit disk onto the exterior of the filledin Julia set, fixing ~ (which turns out to be quite easy). The coefficients of its power series w + bo + blW 1 q_ b2 w-2 q- . . . exhibit the symmetries of the image region: it has a q-fold rotation symmetry around a point a if and only if bo = a and all other coefficients vanish, except possibly bq-1, b2q-1, b3q 1. . . . Details are in Lau [11]. For an antiholomorphic polynomial f(~), it turns out that its Julia set has the same rotational symmetries as the one for f(z). For a general holomorphic or antiholomorphic polynomial to have a Julia set with reflection symmetry, the arguments of its coefficients have to satisfy certain conditions which can be derived by similar methods [11]. Symmetries
here. First, we need to look a little more closely at what happens if we iterate such a polynomial. One particularly nice way to guarantee boundedness of the sequence (Zn) in the definition of the sets Me and M~ is to show that it converges to a fixed point. If a fixed point zf of a holomorphic function f has the property that If'(zf)l < 1, then points in a neighbourhood of zfwill get closer and closer to zf under iteration because f(zf + E.) = Zf q- ~ ' ( Z f ) q- O(IEI2); such fixed points are called attracting. Things are simplified by the following theorem which Pierre Fatou and Gaston Julia proved independently early this century: If a rational map f of degree d 2 has an attracting fixed point, a critical point off converges to it under iteration. The proof is not hard; it can be found in Milnor [12], w or Beardon [2], Section 9.3. Since 0 is the only critical point of our maps fe,c in C, this result has several consequences: for one, there can be at most one attracting fixed point for each c. Moreover, if one exists, zero has bounded orbit, and so c is contained in Me. For the maps fa,c, the same, thing is true, because the second iterate f~,c 9 fd,c is holomorphic and has the same Julia set. Its critical points are zero together with those points which are mapped to zero by f~,c. Now consider the following classical cycloid construction: draw a circle with radius Re = d -1/(d-1) around the origin and imagine a small disk with radius re = Re/d and center at Re on the positive real axis (see Fig. 2). This disk touches those two circles around the origin whose radii are Re - re and Re + re. Now attach a pen to the leftmost point of the small disk (where it touches the innermost circle) and let this disk move in
of Parameter Sets
Now let us investigate the symmetries of the Mandelbrot sets Ma and the Multicorns M*a. The former have d - 1-fold rotational syr~metries around the origin and d - I axes of reflection while the symmetries of the latter turn out to be d + 1-fold with d + 1 axes of reflection. In both cases, the axes of reflection make equal angles, one of them being the real axis. The proof that all these symmetries exist is easy [1, 18]; we give it below for reflections and in the next section for rotations. Sheng and Spurr [18] also show the converse; we give a much shorter proof
Figure 2. Constructing the boundaries of the regions having attracting fixed points for Ma (upper half) and MCa (lower half), shown here for d = 4. The small disk is moving between the two circles drawn, rolling along the inner or outer one, respectively. THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1 1995 47
the gap between the innermost and outermost circles. If y o u make our disk roll on the surface of the inner circle, the pen will d r a w a curve with d - 1 peaks to the inside, between which there are identical smooth curves, each touching the outer circle once in the middle. If, however, the disk rolls on the surface of the outer circle (whose circumference is d + 1 times bigger than the one of the disk), w e get d + 1 peaks to the outside, between which there are again identical s m o o t h curves, each touching the inner circle once in the middle. It turns out that we have just d r a w n the boundaries of the sets of parameters c such that the (anti-)holomorphic map of degree d has an attracting fixed point. The first curve (peaks to the inside) is for the holomorphic and the second one for the antiholomorphic case. Before proving this w e show h o w it will settle the rotational symmetries of the Mandelbrot sets and Multicorns: according to the theorem of Fatou and Julia mentioned above, the domains s u r r o u n d e d b y our curves belong to the sets Md and M~, respectively. They have d ~ 1-fold rotational symmetries a r o u n d the origin as required. Their b o u n d a r y points closest to the origin are the points ~ = Ra - rd on the positive real axis and their s y m m e t r y partners. These points are in fact b o u n d a r y points of M d and M~: assuming the d -T- 1-fold symmetries, it suffices to consider the positive real points 8a. If, say, c = 8d + E (E > 0), then fa;(z) ~ z + c for all real z ~ 0 (for a proof, just find the m i n i m u m of fe,c(Z) - z; our special choice of Rd was designed to make this computation work). This means z grows at least by c in each iteration step and hence escapes to ~; in particular, 0 has u n b o u n d e d orbit. And since for real c and z, f~,c(Z) = fd,c(Z) is real, this is also true for the antiholomorphic maps. We see that Me has exactly d - 1 b o u n d a r y points closest to the origin and M~ has exactly d + 1 of them. This excludes further rotations. This also answers the question about reflection symmetries. The real axis is a reflection axis (complex conjugated c-values have conjugated orbits for conjugated starting values, and we always start iterating at 0), so b y composing this with the d u 1-fold rotations w e can construct all the s y m m e t r y operations of a regular d T1-gon, namely the dihedral groups De ; ~. Additional reflections cannot exist without additional rotations. In order to validate o u r reasoning, we have to prove the proposition that the cycloid construction above surr o u n d s a d o m a i n of parameters c for which there exists an attracting fixed point of the maps fe,c or f ~,c. We first give a proof for the holomorphic case. Let v = re i~ be a complex n u m b e r in the closed unit disk (i.e., Ivl = r ~ l ) , zf=d 1/(cl-1)v, and c =c(v) = zf - zy = d-1/(a-1)(v - v e / d ) . Note that for r = 1, zf is just the position of the center of the small rolling disk w h e n this center is at an angle of ~ with the real axis. The second term zy in c simulates the small disk, having a d times smaller radius and a d times higher angular velocity. For points on the disk nearest to the origin, 48
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
the two velocity contributions just cancel out which means that the small disk rolls exactly on the inner circle. Every time the pen gets there it draws a cusp. By construction, Zf is a fixed point of fd,c(Z) = z d + c. Moreover, the derivative of fd,c there is v d-1. Hence this fixed point is attracting for r < 1. It is easy to verify that the map c(v) is a holomorphic bijection between the disk and the described parameter domain, and that every parameter with an attracting fixed point is obtained in this way. For the antiholomorphic case the reasoning is, mutatis mutandis, the same. Let c = zf - -~ so that the small disk n o w turns the other w a y round, rolling along the outer circle and drawing cusps to the outside. For the standard Mandelbrot set M2, however, there is no reason for the center of rotation to be the origin, so a different proof is required. D o u a d y and Hubbard [8J have constructed a conformal isomorphism from the exterior of the unit disk onto the exterior of M2; it begins c(w) = w 1/2 + (1/8)w -1 - (1/4)w -2 + . . . (with all coefficients dyadic rationals). N o w the result from the end of the previous section shows that there is no rotational symmetry.
R e m o v i n g the S y m m e t r i e s Knowing the d - 1-fold rotational s y m m e t r y of the Mandelbrot sets, the obvious idea is to take c to the p o w e r d - 1. Doing this, we define A = dc d-1 for c E Md, the additional factor d preventing the main components from getting smaller: n o w the radii Rd d -1/(d-1) m a p to 1 for e v e r y d. The resulting sets, which we call Ud, are all symmetric with respect to the real axis but no longer have rotational symmetries. They are displayed in Fig. 3 for several values of d. Their central components (the "bodies") always have one cusp but slightly different shapes for different d. The components attached to t h e m (the "heads", the "arms", and so on) get bigger and bigger and have d - 2 cusps each. M a n y more structural similarities a m o n g these sets can be discovered; c o m p a r e Devaney, Goldberg, and H u b b a r d [6]. There is a dynamical relation between the c-plane and the A-plane. Define mappings gd,x : z ~ M1 + z/d) a. Then the maps fa,c have the same dynamics as gd,• p r o v i d e d A = d c a-1 (although in another coordinate system, differing b y a linear change of coordinates: ga,x = ~b 9 fa,c 9 ~b-1 with ~b(z) = dz/c - d; the critical point in the ~plane is n o w - d ) . Therefore, for e v e r y A and its corresponding d - 1 values of c, the critical points either all escape or all remain bounded. This is the reason for the rotational symmetries of the M a n d e l b r o t sets. The similarities between the d e s y m m e t r i z e d Mandelbrot sets for different d are stronger yet: look at the maps gd,x again and recall that they approximate, as d ~ ~, the exponential function g~,A(z)= Aez. In the same w a y as for finite d, we define: =
U~ = {A ~ C: the sequence z0 := 0; Zn+ l = g~,,~(Zn) is bounded}.
Figure 3. The (generalized) Mandelbrot sets Ma for d = 2, 3, 8 and the d e s y m m e t r i z e d sets Ida for d = 3, 8, 100 (M2 and U2 coincide up to a scaling constant).
The set U~ thus defined is shown in Fig. 4. The impression that it is a kind of "limit set" of the Ua for d --* can be given a precise meaning; this is still an object of research (see, for example, Devaney, Goldberg, and Hubbard [6], or Schleicher and Shishikura [17]). Having explained the situation for the desym-
metrized Mandelbrot sets, we should a d d that the same is true for the Multicorns: here, the functions corresponding to f~,c(Z) = ~d + c are g~,• = M1 + ~ / d ) a for A = d-ga/c, which establishes the d + 1-fold rotational symmetries of the Multicorns M}. To stress the analogy, we could also write A = dcd/c in the holomorphic case.
Figure 4. Left: The parameter set U= for the exponential map z ,--', Ae z. The solid orange regions to the fight are artifacts due to extremely fast growth of numbers; the black lines should extend all the w a y to infinity. Right: Detail. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 1995
49
Figure 5. The Multicorns 3d~a for d = 2, 3, 8 and the Unicorns/aVa for d = 2, 3, 8, 100. The boundaries of the central regions for which there are attracting fixed points are artificially highlighted.
We like to call the d e s y m m e t r i z e d Multicorns U n i c o r n s LFe;some of them are s h o w n in Fig. 5. Their central parts are again a little different for all the d, but they all have one cusp to the outside. The sets U~ again have a limit
set U* for d --~ ~, corresponding to the m a p z ~ Ae z. It is s h o w n in Fig. 6. Here is one last curiosity which w e want to mention. Take any d -~ 2 and d r a w the boundaries of the central
Figure 6. Left: The parameter set U*~ for the antiholomorphic exponential map z ~ Aez, again with artificially highlighted central part. The solid orange regions are artifacts like those in U~ above. Right: Detail. (See cover.) 50
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
[4] A. Beardon: Symmetries of Julia Sets. Mathematical Intelligencer, 18(1) 43-44. [5] B. Branner: The Mandelbrot Set. In: R. Devaney (ed.): Chaos
[6]
[7] [8] [9] [10] [11] [12] Figure 7. The boundary of the central components of the sets Ua and ~ (rotated), containing those values ;~ for which there exists an attracting fixed point (shown here for d = 4). These curves are images of each other under reflection in some circle.
[13] [14] [15] [16]
regions in the A-plane of the d e s y m m e t r i z e d Mandelbrot set Ud and Unicorn U~; turn the latter a r o u n d so that both cusps are on the right (see Fig. 7). Then for each ray starting at the origin, the distances of the points w h e r e the ray intersects the two lines have constant p r o d u c t (1 - 1/d2) d i, no matter which ray was taken. In other words, one curve can be obtained from the other b y inversion in a circle of appropriate radius. This property, which also holds in the case d ~ % does not use iteration theory, just disks rolling on circles, and is a nice exercise in elementary geometry. H a v i n g seen the similarities between the p a r a m e t e r sets after dividing out the symmetries, it is tempting to do the same for the Julia sets, too, b y replacing z with z e (respectively ~d). This does in fact r e m o v e the symm e t r y a r o u n d the o r i g i n - - h o w e v e r , the picture does not really change because a d d i n g c restores the original Julia set again: recall that the Julia sets are invariant u n d e r the iteration process. All that glitters is not gold.
[17] [18]
and Fractals: The Mathematics Behind the Computer Graphics. Proceedings in Applied Mathematics 39, American Mathematical Society (1988). R. Devaney, L. Goldberg, J. Hubbard: A Dynamical Approximation to the Exponential Map by Polynomials. Preprint, Mathematical Sciences Research Institute, Berkeley (1986). B. D. Crowe, R. Hasson, R. J. Rippon, P. E. D. Strain-Clark: On the Structure of the Mandelbar Set. Nonlinearity 2 (1989), 541-553. A. Douady, J. Hubbard: Itdration des polyn6mes quadratiques complexes. Comptes Rendus Acad. Sci. Paris 294 (1982), 123-126. A. Douady, J. Hubbard: Etude dynamique des polynSmes complexes. Publ. Math. Orsay (1984/85). J. Hubbard, S. Nakane, D. Schleicl~er: Multicorns are not Pathwise Connected. Manuscript, in preparation. E. Lau: Symmetrien in der Komplexen Dynamik. Jugend forscht (1993). J. Milnor: Dynamics in O~e Complex Variable: Introductory Lectures. Preprint, IMS Stony Brook, #5 (1990). J. Milnor: Remarks on Iterated Cubic Maps. Experimental Mathematics, Vol. 1 (1992), No. 1, 5-24. S. Nakane: Connectedness of the Tricorn. Ergod. Th. Dyn. Sys. 13 (1993). S. Nakane, D. Schleicher: On the Dynamics of Antipolynomials. Manuscript, in preparation. D. Schleicher: The Structure of the Mandelbrot Set. Manuscript, in preparation. D. Schleicher, M. Shishikura: Topology and Combinatorics of Postsingularly Finite Exponential Maps. Preprint. X. Sheng, M. Spurr, Symmetries of Fractals. Mathematical Intelligencer, 18(1) 35-42.
Holsteiner Chaussee 313b D-22457 Hamburg Germany Mathematisches Institut, Technische Universita't M~inchen ArcisstraJ3e 21 D-80333 M~inchen Germany e-maih [email protected]
Acknowledgment
The authors wish to thank Axel Hundemer for his help in preparing the photographs. References
[1] C. Alexander, I. Giblin, D. Newton: Symmetry Groups of Fractals. The Mathematical Intelligencer 14 (1992), 32-38. [2] A. Beardon: Iteration of Rational Functions. Springer (1991). [3] A. Beardon: Symmetries of Julia Sets. Bull. London Math. Soc. 22 (1990), 576-582. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 1995
51
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 cafd where the famous conjecture was made, the desk where the
famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.
A Walk Around Leibniz Elisabeth Miihlhausen
"Leibniz biscuits, Leibniz biscuits," the man with his mobile confectionery trolley used to shout, as he walked up and down the platform in Hannover Central Station. Nowadays you see Leibniz biscuits slowly revolving on an illuminated advertising sign in the centre of the station concourse (Fig. 1). Perhaps you should buy yourself a packet of these crunchy biscuits to keep you going on the following walk around the haunts of Gottfried Wilhelm Leibniz (1646-1716). Polymath that he was, Leibniz became involved in looking at the suitability of long-life biscuits such as rusks for provisioning soldiers in the field, but he was not the inventor of such biscuits. In 1891 the Hannover firm "Cakes-Fabrik H. Bahlsen" named its butter biscuits after the most famous inhabitant of Hannover, and ever since, it has printed his name on the packet and baked his name into every biscuit. Leibniz came to the court of the Dukes (later Elector-
*Column Editor's address: Mathematics Institute, University of Warwick, Coventry, CV47AL England. 52
Princes) of Hannover in 1676 and spent the remaining 40 years of his life there. During this time he put his versatile talents at the service of three very different masters. Leaving the station, you cannot miss the equestrian statue of the second. Elector-Prince Ernst August of Hannover (1629-1698) engaged Leibniz the engineer and physicist to improve mining in the nearby Harz Mountains, burdened Leibniz the historian with extensive research on the history of the House of Guelph, and also called on his services as legal and political adviser. Leibniz, however, saw himself primarily as a mathematician. During his years in Hannover he published his discovery of differential calculus, developed integral calculus, and established the theory of determinants. He also pursued his interest in applied mathematics. Three years before coming to Hannover he had already impressed the Royal Society of London by demonstrating his calculating machine. Soon afterwards he was elected a Member of the Society. In Hannover he continued this work by constructing several calculators. Under his first master, Duke Johann Friedrich of
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 9 1996 Springer-Verlag New York
The Leibniz walk through Hannover.
Braunschweig-Li~neburg, he was the director of the Court Library of Hannover which was housed in the Leine Palace. If you leave the station along the aptly named Bahnhofstrat~e, you come to this palace on the bank of the river Leine at the corner of Leinstrat~e. It is now the seat of the Parliament of Lower Saxony. At one time it housed the ducal court, and so Leibniz lived and worked here. When in 1698 the library was relocated because of lack of space in a rented aristocratic house (dating from 1652), Leibniz moved with it and lived there until his death.
Figure 1. Rotating Leibniz biscuit.
This "Leibniz House" was destroyed in the Second World War but meticulously reconstructed on another site ten years ago. It stands between the Parliament and the Historical Museum, in the small Holzmarkt Square with its ornate Goose Girl fountain (the G~inselieselbrunnen). It has the most beautiful Renaissance gable in Hannover and so you cannot miss it. Since 1983 it has been the guest house of the Universities in Hannover and the venue for scientific conferences. Try your luck and ring the bell to the left of the massive front door (Fig. 2). Maybe you can visit the Leibniz exhibition on the ground floor, officially only open on Sundays from 10 a.m. to 1 p.m. and 1:30 p.m. to 6 p.m. On a huge wall map you can follow Leibniz's travels as he covered nearly 20,000 km on his visits to Paris, Rome, Florence, Berlin, Vienna, London, and elsewhere. On almost every occasion he took with him his travelling chair--you can admire the original! Another map shows the location of about 1,100 correspondents of Leibniz the avid letter-writer. 15,000 of his letters have survived. He wrote mainly in French, Latin, and German, but also in Italian, Dutch, and English. If you are interested in even more detail, you have to go to the nearby international centre for Leibniz research in the Regional Library of Lower Saxony (8, Waterloostrat~e), in whose manuscript department you will find Leibniz's papers with their own printed catalogue. In their Leibniz Research Library are to be found the collected works and publications on Leibniz from all over the world. A comprehensive Leibniz bibliography is kept up to date. THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996 53
Figure 2. Leibniz's front door.
In the entrance hall, somewhat relegated to a corner, stands a bust of Leibniz by Gustav Seitz (Fig. 3) showing him without the long, curly wig and lace cravat that he wore at this period. These do, however, appear on a 1980 German postage stamp (see The Mathematical Intelligencer, 13(3), 1991, p. 84). This was based on the considerably more attractive 1703 oil painting by Andreas Scheits, the Hannover court painter.
Figure 3. Leibniz by Gustav Seitz.
54 THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996
Resisting the temptations of the wealth of material on society in Leibniz's day held by the Historical Museum, we will stroll across the River Leine and on the other side of Leibniz Embankment (Leibnizufer) come to Neust/idter Church where Leibniz is buried (Fig. 4). For the last two years of his life, Leibniz was ill and lonely but still worked to the point of exhaustion. His third master, Elector-Prince Georg Ludwig (1660-1727), who came to the English throne in 1714 as George I, had left Leibniz behind in Hannover when he moved with his court to Great Britain. Georg Ludwig had no need of Leibniz as a political adviser and actually forbade him to travel; he insisted he finish the historical work started under Ernst August. He cared little for Leibniz's talents or personal needs, and the court was not represented at Leibniz's funeral in November 1716. Inside the church the only memorial is a plain slab bearing the words "Ossa Leibnitii". Let's leave this sad place and continue our walk along the Leibnizufer. What a contrast to see the three blowzy sculptures of Beryl Cook-like ladies (the Nanas) by the French artist Niki de St. Phalle! Crossing Goethestrat~e to the other bank of the Leine, walk down Brfihlstrat~e till you come to a spot where an impressive avenue of linden trees (the Herrenhauser Allee) leads to the Royal Gardens of Herrenhausen (the Herrenhauser Garten). To the right of this avenue is the Technical University, surrounded by the Welfen Gardens. Inside the main building of the University there is nothing to remind
Figure 4. Neust~idter Church.
Figure 5. Bahlsen lecture hall.
you of Leibniz--at least I haven't been able to find anything. However, a lecture hall in the mathematics department (Fig. 5) was funded by Bahlsens Keksfabrik, maybe in retrospective homage to Leibniz, the founder of their fortunes. On my visit I was told to use a pair of binoculars to scan the outside of the building for a sculpture of Leibniz carved in sandstone. My attempt was not successful, I must confess, and I used the binoculars to watch the birds in the English landscape gardens (Georgengarten) opposite the Technical University. Here in a picturesque glade by the lake lies an open colonnaded Grecian temple dedicated to Leibniz (Fig. 6). This memorial was built in 1787, a long time after Leibniz's death, to house a bust designed by the
Figure 6. Leibniz Temple.
Figure 7. Leibniz by Christopher Hewetson.
Irishman Christopher Hewetson in 1790 (Fig. 7). Originally it was situated on the city wall near Waterloo Place but was moved, together with the bust, to the Georgengarten in 1935. Today you can still see "Genio Leibnitii" in gold letters above the columns, but the bust is no longer there. It was the frequent target of spray paint attacks, so it was decided to remove it to a safer place, the entrance hall of a building formerly owned by the Hannover firm Continental. This building (7, Vahrenwalder Strage), close to the Werderstrat~e subway station, now houses Hannover Technology Centre. Continuing, we como to one of the largest and most beautiful baroque gardens in Germany, called simply the "Big Garden" (Grot~er Garten). And it really is big. That it is also beautiful, you will discover for yourself. Begun in 1666, the garden was developed over three decades by Martin Charbonnier. Sophie (1630-1714), the wife of Elector-Prince Ernst August, devoted a lot of energy to the extension and maintenance of the garden. "Le jardin, c'est ma vie," she once said. But she had other interests too; she particularly enjoyed discussions with Leibniz. He was full of advice and ideas for the garden and invented a new system for powering fountains. Contact with this well-educated woman with her cultural and philosophical interests was one reason that Leibniz the enthusiastic traveller always came back to Hannover and made that city the centre of his life. Sophie's daughter Sophie Charlotte, wife of King Friedrich I of Prussia, also took a special interest in Leibniz's philosophy and called herself his pupil. THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996 55
Figure 8. Leibniz walking for Bahlsen (courtesy Bahlsen Museum).
Leibniz often visited her in the Charlottenburg Palace in Berlin to discuss philosophical questions at her salons. When she died in 1705, Leibniz was deeply affected and wrote a poem dedicated to her which is counted as the most significant piece of baroque poetry in the German language. Walking like Leibniz in the Bahlsen advertisement of 1912 (Fig. 8) between the hedges which divide the garden into many smaller ones, each with its own character, you must take the opportunity to compare the two statues of Sophie. As you entered the garden from the Leibniz Temple, you will have passed the yellowpainted Gallery which was built at the instigation of Sophie. For the numerous members of the court who met there in summer, it was not far from there to the open-air theatre where they enjoyed ballet and plays. Between the gallery and the theatre, hidden behind hedges, is a group of statues of the royal personages responsible for the construction of the Big Garden. Besides father-in-law Georg, husband Ernst August, and son Georg Ludwig, there is Sophie herself, carved in sandstone. Her hair dressed in a truly amazing coiffure of towering rolls and curls, she wears a marvellously ornate gown, trimmed with bows and flowers. In one hand she holds up the train of her dress, in the other she carries a closed fan. You won't find a statue here of Johann Friedrich, the first to engage Leibniz, who also cared a lot for the garden. When his brother Ernst August had these statues erected at the end of the 17th century he left him out, for the simple reason that he did not like him. Sophie is the only person who has a second memor56 THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996
ial in the garden, that of 1878. This is made of marble and situated near the centre of the entire garden, at the spot where she died while taking a stroll. Here she sits on a chair, her hair in a simpler style this time, as is the dress. Now she holds a book in her hand, her forefinger marking the place she has reached. For you cannot read without taking a break now and then, especially if you are reading Leibniz. So she sits and watches the fountains play. If you too feel like having a break, you could have coffee in the garden restaurant and try the German cheesecake. Suitably refreshed, you'll be able to walk with renewed energy across the Herrenh~iuserstrat~e to the Mountain Garden (the Berggarten). This was constructed on a moraine during the same period as the Big Garden. It started as a kitchen garden to grow fruit and vegetables for the ducal table. Later it became a botanical garden famous for the collection of palms and orchids in its 36 glasshouses. There are wonderful old trees in the garden; my favourite is the ginkgo tree from China, planted in 1834. Here you can pretend you are one of the courtiers who tried to disprove a statement by Leibniz. He maintained that every individual is unique and even doubted whether there exist two identical leaves on a tree. The courtiers explored these philosophical ideas not only theoretically but also practically by comparing leaves in the Herrenhausen Gardens. But Leibniz was right, so please don't pick all the leaves in your own attempt to prove him wrong.
Acknowledgment I am indebted to Christine Hillam of the University of Liverpool for advice on the translation of this article.
Bibliography Amt ffir Fremdenverkehrs- und Kongret~wesen der Landeshauptstadt Hannover (ed.), Hannover: Der rote Faden zu den Sehenswardigkeiten in der Innenstadt, Hannover: Grfitter 1991. R. Finster/G.v.d. Heuvel, Gottfried Wilhelm Leibniz, Reinbek bei Hamburg: Rowohlt 1990. J. Linnewedel, Der k~nigliche Garten zu Herrenhausen, Hannover: Schl6tersche 1991. J. Schilgen/H.-C. Hoffmann, Herrenhduser Gdrten, Grasberg: Sachbuchverlag Karin Mader, 1987. H. Schroeder (ed.), Sophie & Co. Bedeutende Frauen Hannovers. Biographische Portraits, Hannover: Fackeltr~iger-Verlag1991. G. Schnath, Das LeineschloJ~, Hannover: Hahnsche Buchhandlung, 1962. E. Stein/A. Heinekamp (eds.), Gottfried Wilhelm Leibniz, Hannover: Gottfried-Wilhelm-Leibniz-Gesellschaft 1990. K.-H. Weimann, Gottfried Wilhelm Leibniz 1646-1716, Ausstellung im Leibniz-Haus Hannover, Hannover: Ohle 1983. Donaustrasse 102 D-12043 Berlin Germany
Jeremy J. Gray* Beppo Levi and the Arithmetic of Elliptic Curves t Norbert Schappacher and Ren4 Schoof
Introduction
Most students of mathematics encounter the name of the Italian mathematician Beppo Levi in integration theory when they learn "Beppo Levi's Lemma" on integrals of monotone sequences of functions. The attribution of this result is historically correct, but it by no means exhausts Beppo Levi's mathematical accomplishments. Between 1897 and 1909, Beppo Levi (1875-1961) actively participated in all major new mathematical developments of the time. He was a man of great perseverance and energy, with an independent mind and a wide mathematical and philosophical culture. His list of publications includes more than 150 mathematical papers. Apart from his lemma, Beppo Levi is known for his work (at the very beginning of this century) on the resolution of singularities of algebraic surfaces. N. Bourbaki's Elements d'histoire des mathdmatiques mention Beppo Levi as one of the rare mathematicians to have recognized the Axiom of Choice as a principle used implicitly in set theory, before Zermelo formulated it. As we shall see below, the role of Beppo Levi set theory seems sometimes overrated. On the other hand, his work on the arithmetic of elliptic curves has not received the attention it deserves. He occupied himself with this subject from 1906 to 1908. His investigations, although duly reported by him at the 1908 International Congress of Mathematicians in Rome, appear to be all but for-
gotten. This is striking because in this work Beppo Levi anticipated explicitly, by more than 60 yedrs, a famous conjecture made again by Andrew P. Ogg in 1970, and proved by Barry Mazur in 1976. Shortly before his retirement, Beppo Levi faced a tremendous challenge which he more than lived up to: he was forced to emigrate, and devoted the last 20 years of his long life to building up mathematics in Rosario, Argentina.
*Column editor's address: Facultyof Mathematics,The Open University, MiltonKeynes,MK7 6AA, England. tThis article is based in part on a colloquiumlecturedeliveredby the first author at the Edmund Landau Center for Research in Mathematical Analysis (Department of Mathematics, The Hebrew University, Jerusalem) on 29 April 1993. The Landau Center is supported by the Federal Republicof Germany. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1 9 1996 Springer-Verlag New York
57
In this article we briefly describe Beppo Levi's life and mathematical work, with special emphasis on his forgotten contributions to the arithmetic of elliptic curves. For more detailed biographical information the reader is referred to an extremely well researched article [Coen 1994]; a convenient list of Beppo Levi's publications is in [Terracini 1963, pp. 601-606]. 1 Beppo Levi's collected papers are about to be published by the Unione Matematica Italiana.
Family and Student Years Beppo Levi was born on May 14, 1875 in Torino, Italy, the fourth of 10 children. His parents were Diamantina Pugliese and Giulio Giacomo Levi, a lawyer and author of books in law and political economics. Perhaps the greatest mathematical talent in the family was Beppo's brother Eugenio Elia Levi, who was his junior by 8 years. By the time Eugenio became a "normalista" at the elite Scuola Normale Superiore in Pisa, Beppo was already an active mathematician. He took great interest in the mathematical education of his younger brother, and Eugenio had a brilliant career [Levi 1959-1960]. In 1909 the 26-year-old Eugenio was appointed professor at the university of Genova. He worked in complex analysis and the theory of Lie algebras. The "Levi condition" on the boundary of a pseudoconvex domain and the "Levi decomposition" of a Lie algebra are named after him. In World War I, Eugenio Levi volunteered for the Italian army. He died a captain, 33 years old, when the Italian army was overrun by the Austrians at Caporetto (October 21, 1917). He was the second brother Beppo lost in the war: Decio, an engineer and the last child of Beppo Levi's parents, had been killed on September 15, 1917 at Gorizia. Beppo enrolled in the university of his home town Torino in 1892, when he was 17. His most influential teachers were Corrado Segre, Eugenio d'Ovidio, Giuseppe Peano, and Vito Volterra. Although he maintained an interest in all the mathematics he learned, he became most closely affiliated with Segre, and thus grew up in the famous Italian school of algebraic geometry. In July 1896 he obtained his degree, the laurea, writing his tesi di laurea on the variety of secants of algebraic curves, with a view to studying singularities of space curves.
Singularities of Surfaces While completing his tesi di laurea, Beppo Levi was also helping his teacher Corrado Segre proofread Segre's important article "Sulla scomposizione dei punti singolari delle superficie algebriche." There Segre defines the in-
1Seealso Homenajea Beppo Levi, Revista Uni6n Matematica Argentina 17 (1955),as well as the special volume Mathematica~Notx 18 (1962). 58
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
finitely near multiple points of a singular point on an algebraic surface. The question arose whether a certain procedure to eliminate singularities would eventually termin.ate. More precisely: Under which conditions does the sequence of multiplicities of the infinitely near points obtained by successive quadratic transformations x = x'z', y = y'z', z = z', reach 1 after finitely many steps? Segre thought this was the case unless the starting point lies on a multiple component of the surface, and he wanted to deduce this from a result by the Scandinavian geometer Gustaf Kobb. But that result was not correct in full generality, and Segre's corollary was justly criticized by P. del Pezzo. It was the young Beppo Levi who supplied--with a proof that even satisfied Zariski in 1935-a complete solution of the problem, which forms the content of his first publication [Levi 1897a]. So much for the substance of this particular issue between Segre and del Pezzo. But it was only a tiny episode in a ferocious controversy between the two mathematicians. It is probably fortunate that most of this exchange of published notes (four by del Pezzo, the last one being entitled in ceremonial latin "Contra Segrem," and two by Segre) appeared in fairly obscure journals. They were not included in Segre's collected works. We understand that the exchange reflects a personal animosity predating the mathematical issues [Gario 1988]; cf. [Gario 1992]. Beppo Levi continued to work for a while on this subject. He attacked the problem of the resolution of singularities of algebraic surfaces and claimed success. Beppo Levi's method is that of Segre: to alternate between quadratic transformations (as above) and monoidal transformations of the ambient space, followed by generic projections. This procedure typically creates new, "accidental" singularities of the surface. The main problem then is to control this procreation of singular points, showing that the process eventually terminates. Beppo Levi's solution was published in the paper [Levi 1897b] which appeared in the Atti dell'Accademia delle Scienze di Torino, Segre's house paper. It was acknowledged to be correct and complete by E. Picard in the early 1900s, by Severi in 1914, by Chisini in 1921--but not by Zariski in 1935. In his famous book on algebraic surfaces [Zariski 1935], Zariski points out gaps in all the proofs for the resolution of singularities of algebraic surfaces that were available at the time, including Levi's. In a dramatic climax, Zariski closes this section of his book with a note added in proof to the effect that Walker's function-theoretic proof (which had just been finished) "stands the most critical examination and settles the validity of the theorem beyond any doubt." See also the remarkable introduction to the article [Zariski 1939]. To to sure this was not the end of the history . . . . We will not discuss the completeness of Beppo Levi's proof; but we may quote H. Hironaka's [1962] footnote
about the papers [Levi 1897a, 1897b] from his lecture at the International Mathematical Congress in Stockholm on his famous theorem on the resolution of singularities of arbitrary algebraic varieties (in characteristic 0)-a result for which he obtained the Fields Medal later. 9 the most basic idea that underlies our inductive proof of resolution in all dimensions has its origins in B. Levi's works, or, more precisely, in the theorem of Beppo Levi. . . . Here, Hironaka alludes to the main result of B. Levi's [1897a], which we discussed above. Let us conclude this section with a slightly more general remark: It is commonplace today to think of the Italian algebraic geometers as a national school which contributed enormously to the development of algebraic geometry, in spite of their tendency to neglect formal precision--a tendency which is seen as the reason for the many "futile controversies" which mark this school. 2 It seems to us that this view is rather biased9 An historically more adequate account would have to measure the fundamental change of paradigm introduced in the thirties by Zariski and Weil. One has to try and imagine what it must have been like to think about desingularization without the algebraic concept of normalization. As for the controversies, they do not seem to be a result of formal incompetence so much as of personal temperament and competition; examples of such feuds can be found long after the end of the Italian school of algebraic geometry9 Finally, even the very name of ,9 "Italian School of Algebraic Geometry" can be misleading, in that it does not bring out the strong European connection of this group of mathematicians.3
Axiom of Choice and Lebesgue's Theory of Integration In spite of his beautiful work on algebraic surfaces, Beppo Levi gave up his assistantship to the chair of Luigi Berzolari at the University of Torino in 1899 (the year that the latter moved to Pavia), and accepted po2For instance D. Mumford (Parikh 1990, xxvf): "The Italian school of algebraic geometry was created in the late 19th century by a half dozen geniuses who were hugely gifted and who thought deeply and nearly always correctly about their field . . . . But they found the geometric ideas much more seductive than the formal details of the proofs . . . . So . . . they began to go astray. It was Zariski and ... Weil who set about to tame their intuition, to find the principles and techniques that could truly express the geometry while embodying the rigor without which mathematics eventually must degenerate to f a n t a s y . " ~ r Dieudonn6 [1974, 102f]: "Malheureaasement, la tendance, tr6s r6pandue dans cette 6cole, a manquer de pr6cision dans les d6finitions et les d6monstrations, ne tarda pas h entrainer de nombreuses controverses futiles,...." 3For instance, an obituary notice for Corrado Segre in Ann. di Mat. Pura et Appl. (4) 1 (1923), p. 319f. describes the "second phase" of Italian geometry (initiated by Segre and a few Italian colleagues) as a wonderful synthesis of ideas by "Cremona, Steiner, v. Staudt, P16cker, Clebsch, Cayley, Brill, Noether, and Klein."
Figure 1. The youthful Beppo Levi.
sitions at secondary schools in the northern Italian towns of Vercelli, Piacenza, and Torino, and also in faraway places like Bari and Sassari. He probably accepted these somewhat mediocre but better-paid positions in order to contribute to the finances of the family in Torino, after his father's untimely death in 1898. In 1901 Beppo Levi was candidate for a professorship (at Torino); the position was given to Gino Fano, with Beppo Levi ranking third on that occasion9 From this period dates Beppo Levi's contact with early variants of the Axiom of Choice, which earned him a mention in a footnote of Bourbaki's Eldments d'histoire des mathdmatiques [Bourbaki 1974, p. 53]. Thanks to Moss [1979], and in particular to the extremely thorough historical study by Moore [1982, especially w "Italian Objections to Arbitrary Choices"], we may be historically a little more precise than Bourbaki (and also safely dismiss the apocryphal story told by Abraham Fraenkel in [Fraenkel and Bar-Hillel 1958, p. 48]). Apparently following the local Torino tradition, which had been started very early by G. Peano, of criticizing uncontrolled applications of arbitrary choices in set theory, Beppo Levi published a criticism of Felix Bernstein's thesis, pointing out a certain partition principle that Bernstein used [Levi 1902]9In this sense Beppo Levi belongs to the prehistory of Zermelo's famous arTHE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996 59
ticle [Zermelo 1904]. Moore makes it clear, however, that Beppo Levi cannot be said to have already possessed Zermelo's Axiom of Choice. Indeed, Beppo Levi was never ready to admit this axiom in general; see [Moore 1982, w for a discussion of Beppo Levi's later attempts to regulate the use of this and other problematic principles of set theory. Alternatively, the reader might wish to consult the easily accessible letter to Hilbert [Levi 1923] to get acquainted with Beppo Levi's peculiar idea of deductive domains. A little later, Beppo Levi tried to come to terms with the new theory of integration and measure of Henri Lebesgue. In a letter to Emile Borel postmarked June 1, 1906, Lebesgue writes [Lebesgue 1991, p. 148f]:
2. Cib post0, dimostrerb ta proposizione seguente: Se una successione non decrescente di f u n z i o n i f~ (x) positive ed integrabili nell'aggreyalo '~ ha in '~ un limile f(x), e se esisle ed finito il lira f
f~(x) d x ,
la funzione f (x) ~ inteyrabile in ~ e si ha
f J(x) ctx=li=m f , f.(x) dx. Si ponga ~=~limd f x f " ( x ) d x =
(2)
A.
Figure 2. Facsimile from page 776 of [Levi 1906a1.
My dear Borel. . . . . My theorems, invoked by Fatou, are now criticized by Beppo Levi in the Rendiconti dei Lincei. Beppo Levi has not been able to fill in a few simple intermediate arguments and got stuck at a serious mistake of formulation which Montel earlier pointed out to me and which is easy to fix. Of course, I began by writing a note where I treated him like rotten fish. But then, after a letter from Segre, and because putting down those interested in my work is not the way to build a worldwide reputation, I was less harsh . . . . Lebesgue's reply to Beppo Levi's criticism, published in Rendiconti dei Lincei [Lebesgue 1906], makes it quite clear who is the master and who is the apprentice in this new field. This somewhat marginal role of Beppo Levi's first papers on integration may explain w h y his name is often lacking in French accounts of integration and measure theory. Even DieudonnG in chapter XI (written by himself) of the "historical" digest [DieudonnG et al. 1978], fails to mention Beppo Levi's works altogether. In English- and German-speaking countries however, a course on Lebesgue's theory will usually be the one occasion where the students hear Beppo Levi's name mentioned. His famous Lemma was published in the obscure Rendiconti del Reale Istituto Lombardo di Scienze e Lettere [Levi 1906a]. The article provides the proof of a slight generalization of one of Lebesgue's results. The statement of Beppo Levi's Lemma which we give in the inset is a resum6 of sections 2 and 3 of [Levi 1906a]. The lemma was quoted (and thereby publicized) by G. Fubini in his important paper in Rendiconti Acc. dei Lincei [Fubini 1907], which contains the proof of what mathematicians still know as Fubini's theorem, in the case of a rectangle domain. Levi's lemma is similar to Fatou's lemma, which coincidentally also dates back to 1906 (Acta Math. 30, 335-400). One can thus build up the theory without reference to Beppo Levi's result. It is nevertheless difficult to understand why Dieudonn6 omits Levi's name from his "historical" account of Lebesgue's theory. Fubini's theorem for a rectangle, to quote Hawkins [1975, p. 161], "marked a real triumph for Lebesgue's ideas." As Fubini said, the Lebesgue integral "is now necessary in this type of study." In fact, Fubini's theo60 THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996
rem had been anticipated by Beppo Levi, albeit without a detailed proof, in a footnote of a very substantial paper on the Dirichlet Principle [Levi 1906b, p. 322]. Here Beppo Levi observed that Pringsheim's careful investigation of double integrals in Riemann's theory of integration carries over to Lebesgue's theory, yielding, in fact, a simpler and more general statement. This paper has inspired a number of developments in functional analysis and variational calculus. Thus Riesz [1934] derived from its section 7 the idea for an alternative proof (not using separability) of the existence of orthogonal projections onto a closed subspace of a Hilbert space. The proof uses what other authors isolate as "Beppo Levi's inequality"--see for instance [Neumark 1959, w Furthermore, a certain class of functions studied in [Levi 1906b] led Nikodym to define the class of what he called (BL)-functions [Nikodym 1933]. This idea was carried further in the study of so-called "spaces of Beppo Levi type" [Deny and Lions 1953]. It is also possible that a remark in [Levi 1906b] inspired some of Lebesgue's later contributions to Dirichlet's Principle. A passage in another letter of Lebesgue to Borel (12 February 1910) seems to suggest this. But the history of Dirichlet's Principle at that time is very intricate, so we do not go into details here.
Elliptic Curves In December 1906, 10 years after his laurea, Beppo Levi was appointed professor for geometria proiettiva e descrittiva at the University of Cagliari on the island of
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Sardinia. Because of the Italian system of concorsi, this somewhat isolated place has been the starting point of quite a number of academic careers. For instance, in the 1960s, the later Fields medallist Enrico Bombieri, Princeton, also was first appointed professor at Cagliari. At the end of 1906, Beppo Levi was a candidate for the Lobachevsky prize of the Academy of Kazan, on the basis of two papers on projective geometry and trigonometry. In spite of the positive scientific evaluation of the works he only received an "honorable mention" because, it was said, the prize was reserved for contributions to non-euclidean geometry. In fact, the prize was not awarded at all that year [Kazan 1906]. The year 1906 may have been the richest year for Levi's mathematical production. It was probably early that year that he began to work on the arithmetic of cubic curves. We have seen how Beppo Levi often became acquainted with a new theory by a critical reading of seminal papers. The subject of the Arithmetic of Algebraic 62
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Curves had been defined and christened by Henri Poincar6 in his momentous research program [Poincar6 1901]. This program is best understood as the attempt to reform the tradition of diophantine analysis, whose practitioners were perfectly happy every time a certain class of diophantine equations could be solved explicitly (or shown to be unsolvable) by some trick adapted to just these equations. N o w Poincar6 proposed to apply some of the notions developed by algebraic geometry during the 19th century. More precisely, Poincar6's idea was to study smooth, projective algebraic curves over the rational numbers up to birational equivalence. The first birational invariant that comes to mind is of course the genus, and a typical first problem studied is then the nature of the set of rational points of a curve of given genus g. Poincar6 starts with the case of rational curves, g = 0. Their rational points (if there are any) are easily parametrized, and a remarkably complete study of this case (although Poincar6 does not
interrupt his flow of ideas by references) had in fact been published by Hilbert and Hurwitz in 1890. The case of curves of genus 1 with a rational point, i.e., of elliptic curves, occupies the bulk of Poincar6's article. Without loss of generality one may assume the elliptic curve is given as a curve of degree 3 in p2, by a nonsingular homogeneous cubic equation. It is the theory of the rational points of these curves, as sketched by PoincarG that Beppo Levi is picking up, criticizing, and developing further in 1906. Any line in p2 meets a cubic curve E in three points (counting multiplicities). If the line and the curve are both defined over Q, and two of the three points of intersection are rational, Le., have rational homogeneous coordinates in p2, then so is the third. This defines a law of composition E(Q) x E(Q) ~ E(Q), called the chord and tangent method (see boxed text). Except in rather trivial cases [when the set E(Q) is very small], this does not afford a group structure on the set of rational points.
But it may always be turned into an abelian group, essentially by choosing an origin. Neither Poincar6 nor Beppo Levi takes this step toward the group structure, and both work with the chord and tangent process itself. 4 Flex points are then special, in that starting from such a point, the method does not lead to any new point. This basic method of the arithmetic of elliptic curves had been used by Fermat when working with certain diophantine problems, and in particular in some of his proofs by infinite descent. Its geometric meaning seems to have been first observed by Newton--see [Schappacher 1990] for a more detailed history of the method.
4Although an abstract notion of group was defined already in H. Weber's article "Die allgemeinen Grundlagen der Galois'schen Gleichungs-Theorie," Mathematische Annalen 43 (1893), 521-524, and differently in J.A. de S6guir's Eldments de la thdorie des groupes abstraits, Paris, 1904, the mathematicians of the time were certainly not trained the way we are today to look for this structure. THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996 63
During 1906-1908 Levi published four remarkable papers on the subject in Atti della Reale Accademia delle Scienze di Torino [Levi 1906-1908]. The first paper is rather general. Beppo Levi avoids the very difficult question whether a plane cubic curve possesses a rational point or not, by assuming, once and for all, that the curves under consideration do. He classifies these elliptic curves up to isomorphism, not only over C, but over Q. Generalizing the chord and tangent process, Beppo Levi also considers deducing new rational points on E from given ones by intersecting E with curves over Q of degree higher than 1. He knows, as did Sylvester and others before him, that this apparently more general notion of rational deduction of points does not yield any more general dependencies than the chord and tangent method. In the first paper, Levi gives, in particular, a birationally invariant definition of the rank with respect to the chord and tangent process of the set of rational points on an elliptic curve over Q, under an assumption which amounts to saying that E(Q) is a finitely generated abelian group. He justly criticizes Poincar6 for having overlooked that, given two birationally equivalent curves, one may and the other may not have a rational point of inflection--this actually makes Poincar6's notion of rank not birationally invariant! Beppo Levi's notion of rank does not coincide with what we call today the Z-rank of the finitely generated abelian group E(Q): Beppo Levi adds to the free rank the minimum number of points needed to generate the torsion subgroup. (To be precise, the minimality condition that he writes down for his (finite) basis of the set of rational points is not strong enough to make the rank uniquely defined; he only asks that a basis be minimal in the sense that none of its points be expressible in terms of the others.) In a footnote he stresses very explicitly that the assumption of finite generation for E(Q) was not proved (it was established only in 1922, by L.J. Mordell): [The finite rank assumption] may be doubtful: either there might exist a cubic curve with a basis consisting of infinitely many rational points: in this case one would say that the rank is infinite; or no basis exists at all in the sense that, for any given set of rational points, one can obtain these points rationally from other points which themselves cannot be obtained rationally from the given set: this would occur if every rational point were on the tangent of another rational point. Thus Beppo Levi is more explicit than Henri Poincar6, who did not let the possibility of E(Q) not being finitely generated enter into his discussion. Like Beppo Levi's footnote, Mordell's proof of 1922 has two parts: it is shown that the rank cannot be infinite, and then, by means of the theory of heights, it is shown that the second possibility indicated by Levi does not occur. The last part of Beppo Levi's first note is devoted to elliptic curves all of whose points of order 2 are rational. For these curves Beppo Levi seems to embark on a general 2-descent, but he does not quite conclude it. (A 64
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more historically minded study of Levi's notes in the context of the development of the method of descent is given in [Goldstein 1993].)
Ogg's Conjecture Starting from a given rational point, other rational points on a given elliptic curve may be constructed applying successively the chord and tangent method, but only to the given point or to points constructed in previous steps. As Levi puts it [Levi 1906-1908, w usually one would in this way obtain infinitely many rational points, but in certain exceptional cases the procedure "fails" in the sense that one ends up in some kind of a loop. In modern language, this means that the point of departure has finite order in the group E(Q). Beppo Levi sets out to classify these "failures" of the chord and tangent method; that is, to determine what the structure of the subgroup of points of finite order on an elliptic curve over Q can be. The last three papers in the series are devoted to this problem. Beppo Levi's method is straightforward. He takes a general nonsingular cubic curve and writes down explicitly what the "failure" of the chord and tangent method for a given rational point on the curve means for the coefficients of the equation. The chord and tangent method can of course "fail" in various ways, and each way gives rise to a certain finite configuration of points and lines. Levi distinguishes four types: configurazioni arborescenti, poligonali, poligonali misti, and, finally, configurazioni con punti accidentali. Let us translate this into modern terminology: Levi fixes a finite abelian group A and computes under what conditions on the coefficients of the curve the group of rational points admits A as a subgroup. He is aware of the complex analytic theory of elliptic functions and he exploits this as well as the restrictions on the structure of A coming from the fact that the elliptic curve is already defined over R (even over Q): if A occurs, it is either a cyclic group, or a cyclic group times Z/2Z. Beppo Levi's configurazioni arborescenti, poligonali, and poligonali misti correspond, respectively, to subgroups of the form A = Z / n Z where n is a power of 2, n is odd, or n is even but not a power of 2. The configurazioni con punti accidentali correspond to the groups A = Z / n Z x Z/2Z, with n even. Beppo Levi's method yields explicit parametrizations of elliptic curves with given torsion points; the problem of their existence then comes to solving certain diophantine equations. Sometimes this is very easy. Thus Levi shows that the groups
Z/nZ Z/nZXZ/2Z
f o r n = l , 2. . . . . 10, 12,~, forn
all occur infinitely often.
2, 4, 6, 8,
J
What is more remarkable, Beppo Levi can also show that certain configurazioni do n o t o c c u r : the group A = Z / n Z does not occur for n = 14, 16, 20, and A = Z / n Z • Z / 2 Z does not occur for n = 10 and 12. In these cases he must study some thorny diophantine equations, defining plane curves of genus 1 or 2. He concludes by infinite descent, very much in the spirit of Fermat. In today's jargon of the arithmetic of elliptic curves, the infinite descent involves a 2-descent [Levi 1906-1908, w In some other cases Beppo Levi does not overcome the technical difficulties; for n = 11 and n = 24 he gives the equations but cannot rule out the existence of unexpected solutions. The equations Levi finds, say, for the groups Z / n Z , are equations for the modular curves Xl(n) that parametrize elliptic curves together with a point of order n. The modular curves called today X0(N) and X(N) and their explicit equations--which we would regard as fairly similar to Xl(n)--had actually been studied in the 19th century; see for instance [Kiepert 1888-1890]. There is no indication that Beppo Levi was aware of this connection. He seems to have looked at his equations only the way he obtained them: parametrizing families of elliptic curves with given torsion points. Neither upper half-plane, modular groups, nor modular functions are evident in his work. The "easy" cases, where examples of elliptic curves with a torsion point of order n abound, are precisely those cases where the genus of Xl(n) is zero. The equation that Beppo Levi finds for n = 11 is (what we recognize today as) a Z-minimal equation for the curve X1(11) of genus 1:
y 2 X - Y 2 Z - X2Z + YZ 2 = 0 . Its five obvious rational points are all "cusps" of X1(11). Of course, Beppo Levi does not see these cusps as boundary points of the fundamental domain of F~(11) acting on the upper half-plane. For him they are simply the solutions to the equation which correspond to degenerate cubic curves. At the 1908 International Mathematical Congress in Rome, Beppo Levi reported on his work on elliptic curves [Levi 1909]. There he also explained what he thought would happen for the other values of n: he believed that the above list exhausts all possibilities. This is how he states it: for the configurazioni arborescente he has proved that Z / n Z where n is a power of 2 cannot occur for n = 16 and therefore n " . . . cannot contain the factor 2 to a power-exceeding 3." For the configurazioni poligonali he writes regarding the group Z / n Z with n odd: "It is very probable that for n > 9 there do not exist any more rational points . . . . " A s far as the configurazioni poligonali misti are concerned, he remarks that he has shown that Z / n Z cannot occur when n = 20 and that therefore n = 5(2k) cannot occur for any k - 2. He has shown that Z/2nZ does not occur for n = 7 and "it
is probable that they do not exist either for larger odd values of n." It is not difficult to see that these conjectures already imply that the above list for the groups of the type Z / n Z • Z / 2 Z should be complete: "Such configurations exist for n = 2, 4, 8, 6, but not for n = 12 and 10; one can argue that they do not exist for larger values of n." Apart from the group Z/24Z which he does not mention, this means precisely that Beppo Levi believed the list(*) to be complete. More than 40 years later T. Nagell made the same conjecture [Nagell 1952a]. s In our days the conjecture became widely known as Ogg's conjecture, after Andrew Ogg, who formulated it 60 years after Beppo Levi. The problem studied by Beppo Levi and later by Billing, Mahler, Nagell, and Ogg has been very important in the development of arithmetic algebraic geometry. In the years following 1970, rapid progress was made by invoking the arithmetic theory of modular curves. The cases n =-11, i5, and 24 had ~lready been taken care of before the fifties [Billing and Mahler]; [Nagell 1952b]. (It is touching to read Beppo Levi's review of [Billing and Mahler 1940] in Mathematical Reviews where he recognizes the equation of X1(11) that he had already published in 1908 but was unable to solve completely at the time.) Ogg shows that n = 17 does not occur, in his important paper [Ogg 1971] where the connection between this problem and the theory of modular curves is spelled out. Ligozat [1975] and Kubert [1976] take care of several small values of n. In 1973 Mazur and Tate [1973] show that there do not exist rational points of order n = 13 on elliptic curves over Q. This exceptional case is eliminated by means of a 19-descent (!), performed in the language of flat cohomology, on a curve of genus 2. Finally in 1976, Barry Mazur proves the conjecture. It is a consequence of his careful study of the modular curves X0(n), which are closely related to the curves Xl(n). His paper [Mazur 1977] is a milestone. The techniques developed therein are basic in the proof of the main conjecture in Iwasawa theory by Mazur and Wiles [1984], Ribet's reduction [Ribet 1990] reducing Fermat's Last Theorem to the Conjecture of Taniyama, Shimura, and Weil and Wiles' recent proof of this conjecture. The latest development concerning torsion on elliptic curves is Lo~c Merel's proof (Spring 1994) of the general boundedness conjecture. This is the statement that the K-rational torsion of any elliptic curve defined over a field K of degree d over Q is bounded in terms of d alone. Beppo Levi's conjecture is an explicit version of the special case d = 1 of this new theorem. Merel's proof builds upon Mazur's work and subsequent refinements by S. Kamienny, combining them with other recent results in the arithmetic of elliptic curves. SWe t h a n k P r o f e s s o r A. S c h i n z e l for b r i n g i n g t h i s p a p e r to o u r a t t e n tion. THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1, 1996
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Figure 5. Barry Mazur. ParmamBolognamRosario Having treated the first 10 years of Beppo Levi's professional life rather extensively, we will be much shorter with the remaining 50 (!) years. This half century from 1908 to the end of the 1950s falls naturally into three periods: almost 20 years in Parma, 10 years at Bologna, and a good 20 years in Argentina. In 1909 Beppo Levi married Albina Bachi. She was from the town of Torre Pelice in Piemonte, the alpine north west of Italy, as was Beppo Levi. He had started visiting his future in-laws in 1906, the year of his nomination at Cagliari. Three children, Giulio, Laura, and Emilia, came from the marriage. At the end of 1910 the family left Cagliari: Beppo Levi was appointed at the university of Parma, on Italy's mainland. He stayed there until 1928. Among his uninterrupted production (increasingly also on questions of mathematics teaching), there is one remarkable number-theoretic contribution from this Parma period: a paper on the geometry of numbers in Rendiconti del Circolo Matematico di Palermo [Levi 1911] where he claimed to give a proof of a conjecture of Minkowski's concerning critical lattices in R n. However, [Keller 1930] mentions a letter of Beppo Levi in which he acknowledged a gap in his proof. A complete proof of this result was given only in the 1940s [Haj6s 1942]. It was in Parma that the Levis lived through World War I and the ensuing political transformation of Italy and of Europe. A reflection of these events--immediately painful for Beppo Levi through the death of two 66
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of his younger brothers---can be found in his remarkable speech (11 January 1919) at Parma University for the opening of the academic year 1918-1919, on "Nations and Humanity" [Levi 1919]. Another nonresearch publication of his from roughly the same period is Abacus from One to Twenty [Levi 1922], a booklet conceived and illustrated by Beppo Levi himself. It is designed to introduce children to the first numbers and elementary arithmetic operations. In the explanatory notes at the end of the booklet the author sketches his general ideas about the concept of number as being the fundamental example of mathematical representation, distinct from the process of counting, as well as about teaching elementary arithmetic. In the 1920s, the first decade of Mussolini's rule, Beppo Levi was explicitly antifascist. He signed the Croce manifesto in 1925. Around that time his situation at Parma University became increasingly difficult because more and more disciplines had to be suppressed for budgetary reasons. In the end, the sciences were reduced to chemistry, and Levi was the only professor left at the mathematics department. It therefore came as a great relief when he obtained his transfer to Bologna-a town with a traditionally famous university--at the end of 1928, after all obstacles to his nomination there had finally been overcome. While in Bologna he held various posts in the Italian Mathematical Society (U.M.I.), and took care of the Bolletino dell'Unione Matematica Italiana for many years. It may have been through correspondence related to a paper submitted to this journal that Beppo Levi first entered into contact with a mathematician from Argentina. In spite of his personal opposition to fascism, Beppo Levi took the oath to fascism in 1931, like most other Italian mathematicians. This oath was generally considered a mere formality; even the church held that it was a legitimate claim by the government for obedience. The mathematician Levi-Civita added a private reservation, and the government showed that it was quite prepared to accept the substance without the form. Of roughly 1200 professors, only 11 refused to sign. The 71year-old Vito Volterra was one of them. Volterra, by the way, stayed in Italy, where he died in 1940--so the SS car that came to his house in 1943 to deport him had to leave empty . . . . It was only after the rapprochement between Hitler and Mussolini, in 1938, that Italian fascism adopted some of the racial policies of the Nazis which at that time were building toward their monstrous climax in Germany and German-controlled Europe. Thus LeviCivita, Beppo Levi, and a total of 90 Italian Jewish scholars lost their jobs in 1938, and most of them had to start looking for a country of refuge. Beppo Levi was 63 years old when he lost his professorship in Bologna. At age 64 he started as the director of the newly created mathematical institute at the Universidad del Litoral in Rosario, Argentina.
The founding of this institute at Rosario, upstream from Buenos Aires, 6 took place at a time of cultural expansion of several provincial Argentinian cities, mainly Rosario, C6rdoba, and Tucum~n. A relative prosperity helped in the development of more substantial groups of professionals, mainly lawyers, medical doctors, and engineers, who promoted local cultural activity in these cities and invited leading intellectuals and artists from Buenos Aires to lecture or visit there. These professionals were financially better off, and their clients were richer yet. Societies, orchestras, art galleries, and publishing houses began to emerge in this period in Rosario. The official opening ceremony of the mathematical institute in Rosario was held in 1940. Lectures were delivered by Cort6s PlY, Rey Pastor, and Beppo Levi. These two men had been the key to Beppo Levi's arrival in Argentina. PI~ was an engineer who taught physics and had an active interest in the history of science. PI~ was a friend and admirer of Rey Pastor, the Spanish mathematician who founded the Argentine mathematical school.
6We are grateful to Eduardo L. Ortiz, London, for the information on Argentina contained in the following paragraphs. For the opening of the Rosario institute, see Publicaciones del Instituto de Matemdticas de la Universidad Nacional del Litoral 35 (1940); cf. E. Ortiz (ed.), The works of Julio Rey Pastor, London 1988.
Figure 6. Beppo Levi: the Argentine years.
Figure 7. Beppo Levi (front row, fourth from left) at a meeting of the Argentine Mathematical Union, 1948.
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Beppo Levi was extremely active in Rosario, and the n e w responsibilities w h i c h he happily e m b r a c e d right f r o m the v e r y beginning of the job gave m o r e satisfaction to his life in emigration than one w o u l d expect. 7 Apart from organizing and managing the Institute (assisted by Luis A. Santal6) he f o u n d e d and edited a journal and a book series of his institute. The journal app e a r e d for the first time already in 1939, and as of 1941 was called Mathematica~ Nota~ (Boletin dej Instituto de Matematica). Roughly one-third of Beppo Levi's publications are in Spanish. These are his papers from the Argentinian period, m a n y of which a p p e a r e d in Mathematica~ N o t x . Beppo Levi continued teaching at Rosario until the age of 84. In 1956 (shortly before he turned 81) he was a w a r d e d the Italian Premio Feltrinelli. Unfortunately the official text of the prize committee [Segre 1956] shows a somew h a t uncertain appreciation of some of Beppo Levi's works. At the end of the evaluation the committee of this prize for Italian citizens congratulates itself that Beppo Levi has highly h o n o r e d the n a m e of Italy by his w o r k in A r g e n t i n a . . . Beppo Levi died on 28 August 1961, 86 years old, in Rosario, where his institute is n o w n a m e d after him. He was probably the shortest mathematician in o u r century, with the longest professional activity. Acknowledgments We w o u l d like to thank all colleagues and friends w h o gave us hints or sent us documents while w e were preparing this article. We are particularly indebted to Professor Salvatore Coen, Bologna, for freely sharing his extensive k n o w l e d g e of Beppo Levi's life and w o r k with us, and for making available some of the less accessible publications of Beppo Levi; and to Dr. Laura Levi, Buenos Aires, for a most interesting correspondence which conveyed a vivid impression of the personality of her father. Thanks to R a y m o n d Seroul, Strasbourg for drawing the figures on his computer. References Billing, G. and Mahler, K. (1940): On exceptional points on cubic curves, J. London Math. Soc. 15, 32-43. Bourbaki, N. (1974): Eldments d'histoire des mathdmatiques, Collection histoire de la pens6e, Paris: Hermann. Coen, S. (1991): Geometry and complex variables in the work of Beppo Levi; in Geometry and Complex Variables, Proc. Int. Meeting on the Occasion of the IX Centennial of the University of Bologna (S. Coen, ed.), New York: Marcel Dekker 111-139. Coen, S. (1994): Beppo Levi: la vita; in Seminari di Geometria 1991-1993, Universitt~ degli Studi di Bologna, Dipart. Math., 193-232. 7This is reflected already in some of the first letters back to relatives in Italy. The same letters also show that his wife found her life rather more difficult in Rosario. We owe this information to Dr. Laura Levi. 68
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Deny, J. and Lions, J.L. (1953): Les espaces du type de Beppo Levi, Ann. Inst. Fourier 5, 305-370. Dieudonn6, J. (1974): Cours de gdomdtrie alge'brique, 1: Aperr historique sur le developpement de la gdomdtrie alge'brique, Paris: PUF. Dieudonn6, J. et al. (1978): Abrdgd d'histoire des mathdmatiques 1700-1900, tone II, Paris: Hermann. Fraenkel, A. and Bar-Hillel, Y. (1958): Foundations of Set Theory, Amsterdam: North-Holland. Fubini, G. (1907): Sugli integrali multipli, Rend. Acc. Lincei 16, 608-614. Gario, P. (1988): Histoire de la r6solution des singularit6s des surfaces alg6briques (une discussion entre C. Segre et P. del Pezzo), Cahiers Sdmin. Hist. Math. Paris 9, 123-137. Gario, P. (1992): Singolarita e geometria sopra una Superficie nella Corrispondenza di C. Segre a G. Castelnuovo, Arch. Hist. Ex. Sci. 43, 145-188. Goldstein, C. (1993): Preuves par descente infinie en analyse diophantienne: programmes, contextes, variations, Cahiers Sdmin. Hist. Math. Inst. Henri Poincard 2(3), 25-49. Haj6s, G. (1942): Uber einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem Wfirfelgitter, Math. Zeitschr. 47 427-467. Hawkins, T. (1975): Lebesgue's Theory of Integration, Its Origins and Developments, 2nd ed., New York: Chelsea. Hironaka, H. (1962): On resolution of singularities (characteristic zero), in Proceedings of the International Congress of Mathematicians 1962, Djursholm, Sweden: Institut MittagLeffier, pp. 507-521. Hurwitz, A. (1917), Uber tern~ire diophantische Gleichungen dritten Grades, Vierteljahresschrifl der Naturforschenden Gesellschafi in Zfirich, 62, 207-229 (Math. Werke II, 446-468). Kazan (1906): Izvestiya fiziko-matematicheskogo obshtshestvo pri imperatorskom kazanskom universitet (Bulletin de la Socidtd physico-mathdmatique de Kasan) (2) 15, 105-118. Keller, O.-H. (1930). Uber die lfickenlose Erffillung des Raumes mit Wfirfeln. J:. Reine Angew. Math. 163, 231-248. Kiepert, L. (1888-1890): Uber die Transformation der elliptischen Funktionen bei zusammengesetztem Transformationsgrade, Math. Ann. 32 (1888), 1-135; Uber gewisse Vereinfachungen der Transformationsgleichungen in der Theorie der elliptischen Funktionen, Math. Ann. 37 (1890), 368-398. Kubert, D. (1976): Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33, 193-237. Lebesgue, H. (1906): Sur les fonctions d6riv6es, Rend. Acc. Lincei (5) 152, 3-8. Lebesgue, H. (1991): Correspondance avec Emile Borel, Cahiers Sdmin. Hist. Math. Paris 12. Levi, B. (1897a): Sulla riduzione dei punti singolari delle superficie algebriche dello spazio ordinario per trasformazioni quadratiche, Ann. Mat. Pura Appl. 26, 218-253. Levi, B. (1897b): Risoluzione delle singolarita puntuali delle superficie algebriche, Atti Reale Acc. Sci. Torino 33, 66-86. Levi, B. (1902): Intorno alla teoria degli aggregati, Rend. Reale Istit. Lombardo Sci. Lett. 35, 863-868. Levi, B. (1906a): Sopra l'integrazione delle serie, Rend. Reale Istit. Lombardo Sci. Lett. 39, 775-780. Levi, B. (1906b): Sul Principio di Dirichlet, Rend. Circ. Mat. Palermo 22, 293-360. Levi, B. (1906-1908): Saggio per una teoria aritmetica delle forme cubiche ternarie, Atti Reale Acc. Sci. Torino 42 (1906), 739-764, 43 (1908), 99-120, 413-434, 672-681. Levi, B. (1909): Sull'equazione indeterminata del 3~ ordine, in Atti del IV Congresso Internazionale dei matematici, Acc. dei Linc., Roma 1908 2, 175-177. Levi, B. (1911): Un teorema d el Minkowski sui sistemi di forme lineari a variabili intere, Rend. Circ. Mat. Palermo 31,318-340.
Levi, B. (1919): Nazioni e Umanit?l, Parma: E. Pelati. Levi, B. (1922): Abaco da uno a venti, Parma, B. Levi editore (60 pages, of which 44 illustrated). Levi, B. (1923): Sui procedimenti transfiniti (Auszug aus einem Briefe an Herrn Hilbert). Math. Ann. 90, 164-173. Levi, E.E. (1959-1960): Opere, 2 vol., ed. Cremonese: Unione Matematica Italiana (a cura di M. Picone). Ligozat, G. (1975): (Gourbes modulaires de genre 1. Thbse, Bull. Soc. Math. France, Mdmoire 43. Mazur, B. (1977): Modular curves and the Eisenstein ideal, Publ. Math. IHES 47, 33-186. Mazur, B. and Tate, J. (1973): Points of order 13 on elliptic curves, Invent. Math. 22, 41-49. Mazur, B. and Wiles, A. (1984): Class fields of abelian extensions of Q, Invent. Math. 76, 179-330. Moore, G. H. (1982): Zermelo's Axiom of Choice, Its Origins, Development, and Influence, Berlin: Springer-Verlag. Moss, B. (1979): Beppo Levi and the Axiom of Choice, Hist. Math. 6, 54-56. Nagell, T. (1952a): Problems in the theory of exceptional points on plane cubics of genus one; Den llte Skandinaviske Matematikerkongress, Trondheim 1949, Oslo: J. Grundt Tanums Forlag, pp. 71-76. Nagell, T. (1952b): Recherches sur l'arithm6tique des cubiques planes du premier genre dans un domaine de rationalit6 quelconque, Nova Acta Re. Soc. Sci. Upsaliensis, 15, 1-66. Neumark, M.A. (1959): Normierte Algebren, in Hochschulb/icher ffir Mathematiker 45, Berlin (VEB Verl. Wiss.). Nikodym, O. (1933): Sur une classe de fonctions consid6r6e dans l'6tude du probl6me de Dirichlet, Fund. Math. 21, 129-150. Ogg, A. (1971): Rational points of finite order on elliptic curves, Invent. Math. 22, 105-111. Parikh, C. (1991): The Unreal Life of Oscar Zariski, Boston: Academic Press.
Poincar6, H. (1901), Sur les propri6t6s arithm6tiques des courbes alg6briques, J. Math. 7,161-233 (CEuvres V, 483-550). Riesz, F. (1934): Zur Theorie des Hilbertschen Raumes, Acta Sci. Math. Szeged 7, 34-38. Ribet, K.A. (1990): On modular representations of Gal(Q/Q) arising from modular forms, Invent. Math. 100, 116-139. Schappacher, N. (1990): D6veloppement de la loi de groupe sur une cubique, in Sdminaires de Thdorie des Nombres, Paris 1988-1989, Boston: Birkh/iuser. Segre, B. (1956): Relazione per il conferimento del Premio "Antonio Feltrinelli" per la matematica per il 1956, riservato a cittadini italiani, di L. 1.500.000. Commisione: Francesco Severi (Presidente), Giu'lio Krall, Modesto Panetti, Mauro Picone, Enrico Pistolesi, Giovanni Sansone, Beniamino Segre (Relatore), Atti Acc. Lincei, Vol. 29, 663--664. Terracini, A. (1963): Commemorazione del Corrispondente Beppo Levi, Rend. Acc. Lincei 34, 590-606. Zariski, O. (1935): Algebraic Surfaces, Berlin: Springer-Verlag. Zariski, O. (1939): The reduction of the singularities of an algebraic surface, Ann. Math. 40, 639-689. Zermelo, E. (1904): Beweis, dat~ jede Menge wohlgeordnet werden kann, Math. Ann. 59, 514-516. U.F.R. de mathdmatique et informatique Universitd Louis-Pasteur 7 rue Rend Descartes F-67084 Strasbourg Cedex, France e-maih [email protected] Dipartimento di Matematica 2 a Universitdt di Roma "Tor Vergata" 1-00133 Roma, Italy e-maih [email protected] THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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Jet Wimp*
The Interpretation of Quantum Mechanics by Roland Omnes
phenomena. It was therefore assumed, by analogy, that an underlying medium, called the "ether," was necessary to support the propagation of electromagnetic Princeton NJ: Princeton University Press, 1994, 550 pp. waves. It was then possible to estimate the properties US $39.50 (softcover); US $95.00 (hardcover) of this medium. Estimates revealed very bizarre properties (more solid than steel, as unresisting as a vacuum). Reviewed by Robert Gilmore All attempts to measure its properties were trumped by The 20th century has witnessed two major revolutions mysterious effects which "exactly countered" the effect in physics that have radically changed our understand- being measured. The most famous experiment to proing of the world in which we live. These are the Theories duce a null result was the Michelson and Morley failure (1887) to measure the speed of the earth through of Relativity and of Quantum Mechanics. The Special Theory of Relativity has forced us to re- this medium. "Etherists" advanced a number of ingealize that space and time are not independent but rather nious explanations for the null result of the experiment. observer-dependent projections of a four-dimensional These ideas were all swept away by Einstein's radical continuum, space-time. This has generated a number of reformulation of the problem, in which he took as axparadoxes. These are predictions of the theory which iomatic that c (the speed of light) is the same in all inrun counter to our human intuition developed from ex- ertial reference frames. The consequences of this reforperience at speeds much less than the speed of light. We mulation were equally as bizarre as the properties of the have had to learn to live with these "paradoxes," be- "ether": "Ether" was a superfluous concept; one had to give up the idea of simultaneity; time was not absolute; cause all which have been tested have been verified. Quantum Mechanics (QM) has forced us to revise our one had to relinquish the idea of space and time as sepinterpretation of microscopic phenomena. The problem arate entities and replace them by a space-time continis that there was no consensus on the proper interpre- uum. Many consequences of the new theory were "paratation of QM when it was developed (-1925), nor is doxical" but, in the long run, insights obtained by abandoning the analogy between electromagnetic and there any consensus now. The source of the interpretation problem can be stated sound waves far outweighed the difficulties. We face the same problem with QM. Our interpretasimply. We develop theories from data and they must provide predictions consistent with experiment. tion depends on analogy. Our analogy involves shrinkHowever, we often interpret new theories by analogy ing macroscopic concepts down to the microscopic level. with older ones. Analogy can be very useful, but it can There are two principal concepts on which the analogy sometimes be a bad guide. Physicists have made bad is built: particles and fields. Particle motion, described analogies in the past; we are at present guilty of draw- by classical mechanics, is properly the quantum meing poor analogies in interpreting QM; future genera- chanical behavior of N (>>1) "particles" with m > 0. The electromagnetic field is properly the quantum metions will undoubtedly be guilty of this sin. A recent faulty analogy can serve as a cautionary tale: chanical behavior of N (>>1) quanta (photons) with the interpretation of Maxwell's equations (1864). These m = 0. We forget this--that N > > 1 in either case equations predicted that both light and radio waves when we want to interpret what a single quantum (N = were electromagnetic wave phenomena differing only 1, either m > 0 or m = 0) is and how it behaves. Is it a in frequency. Wave phenomena were already familiar "particle"? Is it a "wave"? How could it be both? Our from the propagation of longitudinal and transverse problem is that this analogy is not correct, like "ether" waves in fluids and solids. In the case of sound waves before it, and to the extent it is not correct, these quesan underlying medium was necessary to support these tions don't even make sense. Beginning shortly after QM evolved to the point at which predictions of experimental results could be *Column Editor's address: D e p a r t m e n t of M a t h e m a t i c s , D r e x e l U n i v e r s i t y , P h i l a d e l p h i a , P A 19104 USA. made and were found to be correct, the midwives of the 70 THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 1 9 1996 Springer-Verlag New York
theory embarked on a discussion of how to interpret it, how to "understand" what is happening at an atomic level. This discussion is nowhere near resolution today. Interpretations of QM were proposed by (among others) Dirac, Feynman, Bohr and Heisenberg, and Einstein and Schr6dinger. In addition, there is a lunatic fringe not covered by these broad areas. Some of the fringe ideas turn out to be not so lunatic. The view of Dirac is not widely known and even less widely adopted. So far as I know, it has two adherents-Dirac and myself. Dirac points out that there is in QM a dimensionless ratio, the fine structure constant a = e2/hc = 1/137.03604. He suggests that in any final theory of matter, all dimensionless ratios must be derived quantities, not inputs. Thus, at some point in the future we must be able to derive one of the three quantities e (fundamental charge), c (speed of light), or h (Planck's con-stant/2~r). By not entirely convincing arguments, Dirac rates e and c more fundamental than h, so that h must be derivable from them. But Planck's constant is the lower bound for the uncertainty relations; Dirac concludes that it is fruitless to argue about the ~nterpretation of uncertainty relations in particular, and QM in general, until such time as we can derive th~s physical constant from a more basic theory. Feynman took a more pragmatic view. He warned young physicists that the question is a trap. The different interpretations give the same predictions and therefore cannot be distinguished by physical experiment. The vast majority of working physicists have adopted Feynman's viewpoint and Bohr's Copenhagen Interpretation (see below) as a default. Important arguments about the interpretation of QM have revolved around the view of Bohr and Heisenberg, on the one hand, and Einstein and Schr6dinger, on the other. Bohr and Heisenberg established an interpretation based on observables. This view evolved from Bohr's early attempts, and failures, to describe atoms more complicated than hydrogen in terms of planetary models (e.g., two electrons in orbit around a helium nucleus). He imposed the condition on his younger colleagues, most notably Heisenberg, that classical concepts such as position, momentum, orbit, and angular momentum, were not reliable constructs on which to build an atomic theory. A microscopic theory should be based only on observables. Heisenberg obliged, and constructed a theory in which observables were identified with matrices. Heisenberg's atomic theory was called Matrix Mechanics. Schr6dinger came to his formulation from a different route. Einstein observed (1905) that light of wavelength behaved, in some respects, like a collection of particles with momentum p = h/,~ (h = Planck's constant). A surprising time later (1923), de Broglie proposed that a particle of momentum p behaved, in some respects, like a wave of wavelength A = h/p. Schr6dinger decided that if deBroglie were to be taken seriously, then asso-
ciated with each particle there would be a wave and that wave must satisfy some equation. What equation? To guess this, he reached back to Maxwell's equations, deduced an approximate equation satisfied by each of the three components of the electric and the magnetic fields, and made identifications between 1/A and p/h in a particular representation (Hamilton-Jacobi) of classical mechanics. The result of this ad hoc process is now called the Schr6dinger equation. The approximate correctness of this equation for describing low-energy atomic phenomena, and its ease of use (compared with Heisenberg's matrices), guaranteed its central role in the modern description of matter. Schr6dinger also showed (1925) that his wave mechanics was equivalent ~to Heisenberg's matrix mechanics. Bohr and Heisenberg wished to say as little as possible about a system. They would discuss only observables. If "it" wasn't observable, "it" didn't exist. Even if "it" was observable but wasn't observed, "it" didn't exist. On the contrary, Einstein and Schr6dinger felt that particles were endowed with wavelike properties as well as particlelike properties, which only enriched the discussion of their spectrum of possible behaviors. As the discussion about the interpretation of QM continued, Bohr refined his interpretation. Whenever multiple interpretations of an experiment were possible giving different outcomes, Bohr's interpretation was always correct. It was as if he had a direct telephone connection to the atomic level. He acted as if he never lost sight of Hilbert's observation that a system governed by PDEs is not even defined until the boundary conditions are specified--and as if the observer formed part of the boundary conditions on a quantum system. Bohr's interpretation grew into what we now call the Copenhagen Interpretation. Einstein and Bohr continued their dialogue for the rest of their lives. The problem haunted Bohr, and at his death in 1962, equations were found on his blackboard which reflected his continuing obsession with it. The Copenhagen Interpretation was elegantly summarized in D. Bohm's book (1952). Every quantum system Q is described by a state vector 1@ in some Hilbert space. This state vector contains all the information that can ever be known about Q. The state vector evolves dynamically according to Schr6dinger's equation. In order to extract information about Q, an interaction must take place between Q and a measuring apparatus A. The measuring apparatus is a classical object and gives classical results. It is outside the scope of analysis. The entire system A + Q interacts to produce a measurement of Q. Replacing A by a different apparatus A' changes the entire system A + Q to a different system A' + Q, and the results of the measurement need not be similar. Only quantities which have been measured can be considered real. The only values of an observable which can be observed in a measurement are eigenvalues a, of the observable, which is a hermitian operator on the Hilbert THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996 71
space. The probability of observing eigenvalue an when the experiment is carried out is I(~nl~)l 2, where I~n) is a unit eigenvector with eigenvalue an. After the measurement, if it gave the value an, the state vector of Q is I4~n) instead of I@. Von Neumann replaced Bohr's classical measuring apparatus by an apparatus obeying the laws of QM. This resolves some questions while raising others. For example, where is the dividing line between A and Q when both are quantum mechanical? There are three questions which should be addressed if the Copenhagen Interpretation is to be accepted as a satisfactory interpretation of QM. 1. What is the origin and meaning of the uncertainty relations, or more generally of the noncommutativity of observables? 2. Where is the boundary in A + Q? 3. How does the apparatus force the "collapse of the wave function" I@ ~ I~n~ during a measurement? Omnes's monograph is the latest in a long line of efforts to address these and other philosophical issues. It is not for the lay reader or the faint of heart. It is directed to physicists with a working knowledge of this field who would like to know how to interpret what they practice. It is also directed to philosophers who realize that profound questions posed by Greek and Renaissance philosophers may finally be answerable that answers may be provided by a deeper understanding of the world as understood by physicists. Question 1 above is Dirac's question. It is not even raised in Omnes's book. Neither is Question 3: Omnes simply states that 3 above is what happens and need not be explained by theory. In essence, this book studies only half of the interaction between A and Q. I found this to be a major disappointment. Even at this level, fundamental questions arise. One was raised by Schr6dinger. Suppose a radioactive nucleus has probability 0.5 of decaying in an hour. At the end of an hour the state of the nucleus is a linear superposition of undecayed and decayed states. One such nucleus and a cat are placed inside a box. The decay of the nucleus will initiate a series of reactions that will ultimately kill the cat. How does Copenhagen interpret the cat after one hour? Von Neumann would say that the cat is in a linear superposition of live and dead states: only our conscious observation of the cat after an hour would force its wave function to collapse to live or dead. Heisenberg would say that only when the box is open does it become true that the cat is live or dead. Bohr argued that the cat is a classical object, and as such must be live or dead at any time, and that we determine which when we open the box. All three contributed to the "Copenhagen Interpretation." To see how Omnes's approach differs from or improves on the Copenhagen Interpretation, it is first useful to state the Copenhagen Interpretation in axiomatic 72 THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996
formulation. Postulates are presented in many of the texts available today. In general, all presentations are equivalent, but no two are identical. A useful approach is modeled on Newton's formulation of Classical Mechanics. It has three postulates dealing with kinematics, dynamics, and measurements. 1. Kinematics. The playing field on which QM takes place is a Hilbert space. A quantum system Q is represented by a (state) vector I~) in this Hilbert space, or more generally by a density operator p on this space. Observables are represented by hermitian operators acting on this space. The Hilbert space for two particles, or more generally two systems, is the tensor product of the Hilbert spaces for the individual systems. If the particles are identical bosons (fermions) the Hilbert space carries a symmetric (antisymmetric) representation of the permutation group. 2. Dynamics. The state I@ or p presenting a quantum system evolves under the Schr6dinger equation (or Pauli-Schr6dinger equation, or Dirac equation . . . . ). If two systems are noninteracting, the Hamiltonian for the combined system is the direct sum of the Hamiltonians for the two subsystems. If the particles are identical, the Hamiltonian commutes with the permutation group. 3. Measurement. The quantum system Q and the apparatus A measuring the value of an observable interact with each other during a measurement. Q acts on A to produce a unit eigenvalue an with probability Pn = I(q~nl~/)l2, where I~n) is a unit eigenvector of the hermitian operator representing the observable with eigenvalue an. A back reaction of A on Q guarantees that Q is in the eigenstate Ibn) of the hermitian operator immediately after the measurement. Omnes formulates his version of QM in five Rules, which differ somewhat from the postulates of the Copenhagen Interpretation. He accepts the Kinematic Postulate in his Rules I and 3, and the Dynamics Postulate as his Rules 2 and 3. He changes the first half of the Measurement Postulate (the effect of Q on A) in his Rule 4, but accepts the second half of the Measurement Postulate, dealing with the back reaction of A on Q and the "collapse of the wave function," as his Rule 5. Omnes's Rule 4 differs in a very substantial way from the Measurement Postulate. It is not stated at all in terms of a quantum system and a measuring apparatus. It is stated instead in terms of only the quantum system and a calculus of logics. A quantum system may obey more than one logic. For example, a system may obey logics L1 and L2. These two logics are compatible if there exists a third logic which contains both L1 and L2 as sublogics (possibly not proper sublogics). If no such logic exists, L1 and L2 are called complementary. Roughly speaking, a measuring apparatus specifies a logic. If two logics are compatible, the observables commute and their values can simultaneously be known; if comple-
mentary, the observables do not commute and their values cannot simultaneously be known. There is more to it than this. The probability that a quantum system undergoes event I is pr(1) = Tr PlpP1, where p is the system's density operator and P1 is the Hilbert space projection operator which defines the event 1. The probability that event 1 occurs at time tl and event 2 occurs at time t2 is pr(2, 1) = Tr P2PlPP1P2, where Pi is the projector associated with event i at time ti. This sequence of events is called a history. If one of these events is further resolved into two alternatives [such as the particle is in arm 1 of an interferometer at time tl (P~) or in arm 2 of the interferometer at time tl (P~')], then the projectors obey P1 = P~ + P~' and P~ Pi' = 0. The requirement pr(2, 1) = pr(2, 1') + pr(2, 1") places the following constraint on the projectors:
tem obeys the laws of classical mechanics is I - 9 where 9 - 10-Pq, and pq is a very large number. This formalism is then applied to the measurement problem. The measurement apparatus is assumed to be described by the laws of QM. Omnes assumes that there is a small number of "classical" degrees of freedom, and the remaining degrees of freedom are not accessible. The trace involved in determining the outcome or probabilities for an experiment is over these nonclassical degrees of freedom. Since they are never really seen, these degrees of freedom should be regarded as "hidden variables" in Omnes's presentation of measurement theory, although this term is usually reserved for other problems in the interpretation of QM. For example, for the Schr6dinger cat paradox, the state of the cat is described by the 2 • 2 density matrix
Tr P2P{PP~'P2 + Tr P2P~'PP{P2 = O. Logics L1 and L2 are compatible only if all such consistency conditions (for P~ and PI' obeying L1, P2 obeying L2) are satisfied. This reduces the compatibility or complementarity of two logics to a computation of consistency conditions, where the result of a consistency computation ranges from 0 (compatible) to 1. It then makes sense to consider nearly compatible logics for which the consistency conditions hold within 9 where 9 is a small number. This provides a major advance over the standard Copenhagen Interpretation, where the observables either commute or do not, and are simultaneously measurable or are not. This also provides a mechanism to answer the question: Under what conditions can we reliably talk about a quantum particle taking a trajectory from spacetime point a to point b? Bohr would say it makes no sense; Einstein would demur. Omnes responds with a nice computation for a particle traveling from the origin at time t = 0 and observed within a small sphere centered at 2d on the z axis at time 2t. He asks, "How reliable is it to speak about a trajectory from the origin to the region of observation?" This is done by computing the consistency condition for the particle within a somewhat larger sphere centered on the z axis at d at time t, subject to the boundary conditions: created at a, observed near b. This consistency condition computation is roughly equivalent to a Feynman sum over restricted paths between these regions of space and time. The result is that in many instances it makes a great deal of sense to talk about trajectories for quantum particles. Omnes asks if we can in fact regard classical mechanics as a limit of QM. His answer is in no way equivalent to the standard Copenhagen Interpretation approach to the classical limit given by the Ehrenfest equations, where Newton's equations of motion are satisfied by expectation values of quantum observables. The results of Omnes's computation are presented in a way which at first seems unsatisfactory. He argues that, for the system considered, the probability that the sys-
TrPlpPI=
[ pu Pld] PdZ
Pdd
"
Here pll is the probability that the cat is still alive at time t: Pu = 2-(t/T), T = 1 hour, Pdd is the probability that the cat is dead: Ped = 1 - PU. The off-diagonal matrix elements Pld = P~I represent quantum correlations. A density operator describing a classical result must be diagonal, with no quantum correlations. This is the nub of the cat paradox. Omnes estimates Pld ~ l O - t / D , where D is a very small number. He thus concludes that quantum correlations drop off very rapidly. His computation resolves the conflict among the views of Bohr, Heisenberg, and von Neumann within the Copenhagen Interpretation itself; once again, Bohr's intuition was on target. Regrettably, this approach still leaves unresolved the question: Given that the cat is either alive or dead, why can we only compute probabilities? Is it for the same reasons that we compute probabilities in statistical mechanics: insufficient knowledge of initial conditions? Or is there something yet deeper involved? Following the codification of the Copenhagen Interpretation, as for example in Bohm's book (1952), three major developments in the interpretation of QM have been made by Bohm, Bell, and Leggett. Bohm developed a version of the quantum theory in which it was possible to speak about the position of a particle, and the trajectory which it followed. Such a theory had previously been developed by de Broglie (1927), but was abandoned by him after criticism by the principals involved in the Copenhagen Interpretation. This development by Bohm was in remarkable contrast to a theorem by von Neumann stating that "hidden variables" could not occur in any correct version of quantum theory. Hidden variables were values assigned to dynamical variables (position, momentum, angular momentum, spin) before or between observations--things Bohr warned Heisenberg not to build a theory on; things that Einstein and Schr6dinger deeply felt should exist. Einstein felt this so deeply that he proposed an experTHE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
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iment (the Einstein-Podolsky-Rosen experiment, 1935) to show that one could speak of such variables. The basic idea was to consider two correlated particles, to measure the position of one and the momentum of the other, then argue that this information meant that the position and momentum of both could be known. After extended discussion with Bohr, he conceded that the correctness of QM would force some kind of long-range (faster-thanlight) exchange of information between the two particles. Disappointed, he found it difficult to believe in any kind of "spooky action at a distance." Bohm's representation spurred Bell to wonder just how it was that von Neumann's theorem could be circumvented. This theorem had put a damper on searches for better interpretations for 20 years. It must be regarded as the first of the "no-go" theorems in QM. The problem with no-go theorems is that there may be somebody in the wings who is smarter than you--even if you are von N e u m a n n - - a n d finds some way to make "go" what you think cannot. Bell identified the point at which Bohm's representation of quantum theory evaded the constrictions of von Neumann's theorem. There is an assumption of locality implicit in the proof of the theorem. Bell formulated (1966) a number of inequalities that must be satisfied by any system for which locality holds. He showed that under certain conditions the results of quantum mechanical calculations could violate these inequalities. For once there was a clear-cut test to distinguish different philosophical interpretations of QM. The tests have since been done by Clauser (1972) and others. They show that Bell's inequalities are violated. They show more: the quantum mechanical predictions are correct well within experimental error. They also show that correlations among quantum systems do not obey locality. So Einstein's fear of "spooky action at a distance," influenced by his close connection with the causality and finite propagation speeds of Special Relativity, is in fact not shared by Nature. Somehow signals travel at one speed, correlations (symptoms of boundary conditions) travel at another. The third advance, proposed by Leggett, was that certain devices (SQUIDS) could be very large and still remain in a quantum state. Bohr's original dictum that a measuring device was classical because it was "large" could no longer be supported. Omnes's discussion of these points leaves something to be desired. For example, the manner in which Bohm's model evaded von Neumann's theorem is not clearly stated. He does not make clear precisely what hidden variables are: they remain mysterious throughout the discussion. His discussion of Bell's theorem leaves the impression that Bell proved hidden variables could not exist, whereas the crucial assumption is locality. Bell's theorem shows that locality implies inequalities; QM predicts these inequalities could be violated; experiment shows QM is correct. Therefore, quantum correlations do not satisfy locality. This feature is already present in 74
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
Bohm's model. So once again Bohr was correct and Einstein not, but everybody is surprised at this action at a distance, which is now a clearly recognized part of the quantum interpretation. This is not the first time that action at a distance has created a problem for interpretation. Newton's force of gravitational attraction acts at a distance. This was a great embarrassment to Newton, who was able to coexist with this force but never comfortably. It remained for Einstein to resolve Newton's dilemma more than 200 years later with his own theory of gravitation. Will we have to wait another 200 years for some future savant to resolve Einstein's dilemma? Omnes suggests that the existence of macroscopic quantum systems can aid in our study of the decoherence phenomenon. Decoherence, to which Omnes devotes an entire chapter, plays a central role in the measurement problem. It would have been nice if Omnes suggested how SQUIDS could be used to measure decoherence times. The measurement of such phenomena would add credibility to his interpretation. This book contains a number of other errors, less serious but annoying nonetheless. The first time a hamiltonian is presented, the equation is incorrect. The description of the spin-echo experiment is not correct. Expressions for the Wigner distribution are presented incorrectly more often than correctly. It has been the fashion, these days, to lay out the formulation of QM in an axiomatic format. Earlier presentations simply described the computational processes to be carried out. The push toward axiomatic formulation derives from our desire to make the assumptions of our subject precise so that, if disagreements occur, the possible culprits can more easily be identified. These axiomatic formulations of QM are modeled on earlier axiomatic formulations which underlie other branches of physics. So it would be useful, at this point, to investigate just how successful such formulations have been. The archetypical axiomatic formulation goes back to Classical Mechanics and Newton's three laws of motion: (1) a particle moves with constant velocity unless acted upon by forces; (2) F = dp/dt; (3) Fij + Fji = 0. The functions of these three laws are as follows. The first defines a coordinate system (inertial coordinate system) in which the actual laws of dynamics (F = dp/dt) are true. The third is a conservation (of momentum) law. Now it turns out that Newton's first law, as stated by Newton, isn't true: consider an observer on a merry-goround. A proper statement of this law is: In an inertial coordinate system a particle moves with constant velocity unless acted upon by forces. Only much later was it understood that it defined inertial coordinate systems. Then comes the pragmatic question: how do you know the forces on an object are zero? The standard answer "when the object is sufficiently far from other stuff" won't do because of the long-range nature of the gravitational force. Instead, it is more reasonable to
adopt Mach's principle (Boundary conditions for ODEs? Hilbert would be shocked!) to resolve the definition of inertial frames; that is, an inertial frame is defined by the distribution of matter in the universe. This not only defines an inertial frame, it does what Michelson and Morley failed to do, and what Einstein stated axiomatically cannot be done. It defines an absolute rest frame. We have recently been able to do this in practice as well as principle from our observation of the 3 ~ microwave background blackbody radiation. From the observed dipole distribution of this radiation we know that we are moving about 370 k m / s in the direction of Leo with respect to the cosmic background radiation (late 20th century ether). This radiation is statistically distributed, as would the graviton background radiation be, could we measure it. This reduces us to the situation that Newton's inertial rest frame can only be defined as a statistical average. Nor is it clear that the rest frame defined by the microwave background (m = 0, JP = 1-) and the gravitational background (m = 0, JP = 2 +) have any relation to each other (m = mass of photon or graviton, J = helicity, P = parity). The archetype for all axiomatic formulations is of course Euclid's formulation of the rules of surveying. The origins of this scientific endeavor are lost in antiquity. The honor of codification goes to Euclid, who came perhaps a millenium after the field became economically important in the postdiluvian Nile river valley. Even though Euclid's axiomatic formulation has been available for -2500 years, the consistency of this set of axioms has not always been clear. For example, it is only relatively recently that the role of the fifth postulate has (apparently) been settled. As a branch of mathematics, Euclidean geometry should not be subject to the messy kind of probability or approximation statements which the physical disciplines share. As an axiomatization of some practical endeavor, however, it is. For example, Euclid proves that the sum of the interior angles of a regular figure with four equal sides ("square") is 2r radians, where a surveyor would actually measure 2~1 + ~) radians, with E -IO-PE, pE small. This suggests that the axiomatic formulation of a field of physics as intuitive as Classical Mechanics amounts to a moving target, with perhaps no satisfactory resolution, even though it has been going on for 300 + years. In short, the axiomatization of any area of practical endeavor is filled with potholes and surprises. Approximations and probabilities apparently cannot be avoided. The statement t h a t Classical Mechanics holds with probability 1 - e, 9 = lO-Pq is far less shocking when we realize that pq > > PE. That is, Classical Mechanics is a much better approximation of QM (for a macroscopic system) than thermodynamics is of statistical mechanics, and this is a much better approximation than Euclidean geometry is of surveying. However, it is clear that the axioms provided by
Omnes are incomplete. How can they give the Classical Mechanics in the limit when they do not take into consideration either the first or third laws of Newton? How can they be complete when they ignore the collapse of the wave function? The current volume provides a useful contribution to the literature of interpreting QM. However, it will probably only convince those who need no convincing. The author presents many useful arguments, most of them correctly; but many more are not presented at all. It seems that there is yet another uncertainty relation lying just below the surface which no one has yet stated. This is a relation between the precision with which we do calculations and make predictions (Predictability) and our understanding of the underlying physics (Understanding). QM is the most precise theory ever developed; A(Predictability) is very small. As the dialogue of which this book is a part shows, &(Understanding) is very large. Perhaps A(Predictabitity) z~(Understanding) -> 1 in some suitable system of units, and this is what ultimately limits our ability to interpret QM.
Department of Physics & Atmospheric Science Drexel University Philadelphia, PA 19104 USA
Spirals: From Theodorus to Chaos by Philip J. Davis Wellesley, MA: A.K. Peters, Ltd., 1993. • + 237 p. ISBN 1-56881-010-5. US $29.95
Reviewed by Michele Emmer It would only be possible to imagine life or beauty as being "strictly mathematical" if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them. Theodore Andrea Cook [1]
Spirals Everywhere I was particularly happy to receive Davis's book on spirals to review. Recently, I have been working with spirals and helices both for exhibitions on art and mathematics and for movies on visual mathematics. Moreover, in an introduction to a chapter on spirals of a book dedicated to Venice, I quoted a phrase ascribed to Madame de Sta61 [2]: L'esprit humain fait toujours des
progr~s mais, ce progr~s est spirale. The same phrase is quoted on the first page of Davis's book. I do not know whether he, like me, found the phrase in Theodore Cook's famous book on spirals, THEMATHEMATICALINTELLIGENCERVOL.18,NO.1,1996 75
written in 1914 [1, p. 417]. There is a more important reason for my current interest in the work of Davis. For the past 2 years I have held a position at the University Ca" Foscari (Ca' means casa, or palace) in Venice. The math department is located in a wonderful 18th-century palace called Ca' Dolfin near the second bend of the Canal Grande that divides Venice in two. The Canal Grande consists of two large bends of a spiral, one in one direction, from the railway station to the Rialto bridge, the second, from the Rialto bridge to Piazza San Marco. (See Figure 1.) So this review on spirals is written from the inside of a spiral, one of the most famous! Davis's book is divided into several parts: Part One consists of the text with amplifications of the three Hedrick Lectures delivered on the occasion of the 75th anniversary of the Mathematical Association of America (MAA); Part Two contains some technical developments by Walter Gautschi and Arieh Iserles; Part Three consists of some historical supplements. Finally, there are the notes. The organizing principle of the book? I have planned this book so as to exhibit a variety of things: some history, some philosophy, some anecdotes, a fair amount of mathematics, naturally; some things old, some things absolutely new, some things proved, and many things that invite exploration at a variety of levels of sophistication; many things that invite discussion, conjecture and proof. (p. 4) An essential part of the book is the notes, wherein the author has taken the liberty of free-associating on some of the ideas contained in the text. The notes are the more personal part of the volume; the author hopes he has been able to moderate his natural effusiveness so that it would not be possible to say of him, as it was of George Eliot's Casaubon, that "he dreams footnotes" (p. 5).
The starting point of this fascinating and amazing book is the question: "What is a Spiral?" It would seem to be a simple question, capable of a simple answer. We find examples of spirals everywhere. Davis, looking on the compact disc that contains the Mathematical Reviews of the past 50 years, turned up about 250 entries under spirals and an equal number under a variety of related designations. A few examples of what he has found: pythagorean spirals; lambda spirals; spiral galaxies; spiral waves; spirals and convexity theory; spirals and quasiconformal mappings; spirals generated by Diophantine Gauss sums; spiraMike analytic functions; spirals in stochastic processes; spirals in Hilbert space. There are the spirals of genetics, in the internal structure of a tornado, and so on. (See, for example, a recent conference and exhibition on spirals in science, art, and culture [3].) How is it possible to find one's way in this infinity of spirals? Only by starting from a clear definition! Davis begins with dictionaries; they do not help much. They give definitions for which counterexamples are easily provided. In the Encyclopaedia Britannica, 11th edition, a spiral is defined as a curve that winds around a fixed point. In the American Heritage Dictionary, it is the locus of a point moving around a fixed center at a monotonically increasing or decreasing distance. In the James and James Mathematical Dictionary there is no entry for spiral. We can add other questions to the first one: What is a discrete spiral? What is a fractal spiral? So the question is how to proceed from a common, perhaps visual, conception to a mathematical definition, or at least, to a common but perhaps unspoken set of agreements. These questions can be posed for entities other than spirals. What is a curve? What is a number? What is a set? What is probability? What is chaos?
Figure 1. Perspective map of Venice, in M. Giulio Ballino,
Disegni delle pill illustri citt~, Venice, 1569. (2 Vernou) 76
THE MATHEMATICAL INTELL1GENCER VOL. 18, NO~ 1, 1996
We could answer questions of this kind by making normative definitions, that is, definitions that preserve desirable properties. However, there must be a sense in which a definition, once promulgated, proves fruitful and stable. "This process may take centuries, millennia; and since a definition is necessarily a limitation--finitization--the process is always unsatisfactory and never ending" (p. 23). Which properties must characterize a spiral? Would you say that the equation of a spiral cannot, in rectangular coordinates, be algebraic? Would you say that the product of two spirals is a third spiral? And so on. The number of possible questions grows. There is the problem of distinguishing spirals from what some writers have called volutes, whorls, meanders, wanderings, doodles, noodles, tangles, or explosions (Fig. 2). Why all the wishywashiness about a definition? Isaac Barrow wrote (p. 25), "[mathematicians] take up for contemplation those features of which they have in their minds clear and distinct ideas; they give these appropriate, adequate and unchanging names ..."
Figure 2. Is this a spiral?
Is This a Spiral?
Why is it that, whereas, for example, the notion of a group has been quite stable for almost two centuries, mathematicians do not have clear and distinct ideas about spirals? The story is complex because, as Charles Sanders Peirce wrote (p. 26), technical words should be introduced that are "so unattractive that loose thinkers are not tempted to use them." The term spiral is an ancient one. "Everybody loves a spiral, wants to have a spiral of their own." Of course, mathematics has ways of accommodating all tastes. Mathematics slaps on its subjects such adjectives as almost, sub, semi, super, quasi, alpha-quasi, pseudo, true, faithful, fuzzy, ultra, meta, strong, weak degenerate, standard, generalized, pre-, -like, -oid, incomplete, and hundreds more. We can produce a weak, generalized, fuzzy, alpha-quasi pre-spiral, if we want. Another possibility is to look at the forms of spirals and try to understand how we should distinguish between spirals and nonspirals, perhaps performing experiments. We could use a modern approach to decide what a spiral is. Teach a computer what a spiral is. Build yourself a "spiral recognizer"; that is write a program that allows the computer to discriminate between spirals and nonspirals. If one of the major problems of the computerization of mathematics could be to get a computer to recognize automatically what is mathematics and what is not, Davis in a note (p. 211) states, "One thing is certain: mathematics is far too important a subject to be left to the mathematicians either for definition, extension or promulgation." The starting spiral of Davis's investigation is the spiral of Theodorus. Theodorus was born in Cyrene, approximate dates 465 to 399 BC. He was the teacher of Plato and Theaetetus. In Plato's dialogue Theaetetus,
Theaetetus tells Socrates, "Theodorus wag writing out for us something about roots, such as roots of 3 or 5, showing they are incommensurable by the unit: he selected other examples up to seventeen; there he stopped. N o w as there are innumerable roots, the notion occurred to us to include them all under one notion or class." A discussion of what is knowledge and what is abstraction follows, the problem of the particular and the general and the relationships between the two (see note 3, p. 194). So w h y does Theodorus stop at V ~ ? An answer was provided 70 years ago by J.H. Anderhub. He imagined that Theodorus constructed V2, V3, by a sequence of contiguous right-angled triangles. In each triangle, each outer leg is of length 1. Anderhub observed that the resulting snail-like figure is such that V ~ arises from the last triangle for which the total figure is nonself-overlapping. This spiral is, for Davis, the Ur-spiral, the granddaddy of mathematical spirals (see Fig. 3). If we now consider the spiral of Theodorus in the complex plane and define its vertices Zn in iterative fashion, we have Zn
.
Zn+l = zn + i iZnl ,
(1)
or replace it by a linear homogeneous difference equation with nonconstant coefficients, Zn+l = 1 +
Zn.
(2)
We can study on the basis of (2) what Davis calls the discrete spiral of Theodorus. H o w we can draw a curve through the sequence of points z0, zl, z2. . . . ? There are an infinity of ways to do it, for example, by connecting up the points automatically with straight-line segments. The resulting curve will then have corners. Smoothness THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
77
ematics may become impresarios of hardware and software instead of playwrights, composers or actors, with the more creative roles in the total mathematical production preempted by a small cadre of mathematical elite who work in conjunction with a large group of people skilled in visuals.
Figure 3. The Spiral of Theodorus, exhibiting the "solution" of Anderhub.
might be required; it might be required that the interpolant satisfy the difference equation (2) for noninteger values of n. The interpolant could be required to be analytic, and so on. Mathematicians should always generalize; so one writes down Zn+l = aZn + b zn
JZnl
(3)
for a and b arbitrary complex numbers. The dynamics of (3) are non-trivial and are considered in lecture 111, "Theodorus Goes Wild." I hope it is clear by now that the book of Davis is a genuine source of stimulating ideas on many topics. As the author himself writes (p. 170): "This small volume on spirals is an assemblage of theorematic material both old and new, theorematic conjecture, computer experience, computer graphics, historical monuments of mathematics that span two and one-half millennia, historical and philosophical animadversions." As I have already pointed out, it is in the notes that Davis puts together his reflections on mathematics, culture, and science. It is impossible to give an idea of the variety of stimulating observations contained in them. I suggest you read them while reading the text. Don't avoid them!
In another note (p. 225) he speaks of the possible recognition of a class of computer-generated visual theorems that might be incorporated into our mathematical experience, and poses the question whether this contradicts the idea that mathematics is a process of verbal and symbolic communication at both the input and output ends. He asks himself, "Do you think that you can see something in the figure that cannot be described in words? Can this visual theorem contribute to one's emotional life? And if visual theorems, why not auditory theorems? One might listen to a spiral, via its discrete Fourier transform" (see also Refs. 4 and 5). This book provides many questions, starting with the concrete question "What is a spiral?"--so apparently trivial. In some ways, the book bridges the gap between the two cultures, the humane and the scientific. The computer is the protagonist of this tale--a character both loved and detested, and in any case absolutely necessary to our age. The book ends on an optimistic note: "The computer and computer graphics have clearly widened old and opened up new creative sources of the mathematical spirit. A retrospective show of spirals in the year 2500 would amaze us. One hope that both the visual and the symbolic will endure in their individual aspects as well as in fruitful partnership."
References 1. Theodore Andrea Cook, The Curves of Life, London: Constable & Co. (1914); reprinted by Dover Publ. Inc., New York (1979), p. 428. 2. Michele Emmer, La Venezia perfetta, Venice: Centro Internazionale della Grafica (1993), p. 59. 3. Hans Hartmann and Hans Mislin (eds.), Die Spirale im menschlichen Leben und in der Natur: eine interdisziplin~'re Schau, Basel: Birkh/iuser (1985). 4. Michele Emmer (ed.), Visual mathematics, special issue, Leonardo 25(3/4) (1992); reviewed by Harold L. Dorwart, Mathematical Intelligencer 16(4) (1994), 70-72. 5. Michele Emmer (ed.), The Visual Mind, Cambridge, MA: The MIT Press (1993). Dipartimento di Matematica Universitdl Ca' Foscari Dorsoduro 3825/E, Ca' Dolfin 30123 Venice, Italy
Visual Mathematics?
A Photo A l b u m for Weierstrass edited by Reinhard Biilling
One of Davis's most trenchant observations concerns what could be called the imperialism of the visual.
Vieweg Verlag, Wiesbaden, 1994. ISBN 3-528-06602-4, DM 98
The tendency of our students to think only in terms of colored overlays in polywindowed displays, the decline of poetry and recitation, the condemnation of old-fashioned oratory as flatulent and suspicious, are all independent indications of a transition . . . . Our future teachers of math78
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
Reviewed by R.B. Burckel
This book is a reproduction of two photo albums presented by students, friends, and colleagues to Karl
Weierstrass on 31 October 1885, his 70th birthday. They lay slumbering in a Berlin museum since 1904, and the first one was serendipitously discovered in about 1980. The whole fascinating story of their provenance, plus many details about participants, was told to readers of this journal by Reinhard B611ing in an article entitled "A birthday present," Mathematical Intelligencer vol. 11 (1989), no. 4, 20-25. That article is the best possible review of this book, all the more since Dr. B611ing is a noted Weierstrass scholar. In that article he declared his intention to publish an edited version of the recently found albums. Now, some five years later, he has brought this to fruition with the help of Vieweg Verlag, and we are all in his debt. On the album's cover is a photo of the beautiful copper engraving that adorned the original. There is a short but intimate biography of Weierstrass, including the story of how the album came into being. These are given in both German and English. The original plan, proposed by Mittag-Leffier, was for a sculpted bust, together with a medallion if funds permitted. Solicitation was not confined to Germany, as Weierstrass's former students and admirers were well distributed throughout Europe and Russia. Somewhat more than 5000 marks were ultimately contributed (all from individua l s - n o big foundations in those days), enough money to finance the marble bust, a gold medallion, and the photo album, the idea for which came up rather late in the planning. Due to the well-known rivalries among the Berlin mathematicians and their adherents, some friction was anticipated over this project. There was
some, but it was not significant. Besides this saga, a bilingual account of the birthday celebration itself is included, as are many letters (untranslated) among t h e prominent organizers. Then the editor has undertaken the time-consuming task of ferreting out biographical data to accompany (most of) the photos. Here these are all approximately 4 x 6 cm., although the originals in the second, shorter album are considerably larger. There are altogether 334 of them. The frontispiece is a photo of Weierstrass, the latest known of him (taken sometime between 1865 and 1870), but unfortunately no photo of the marble bust is offered. However, the mathematical tourist can see the bust itself in the Weierstrass lecture hall of the Humboldt University in Berlin. In 1912 the Acta Mathematica (founded in 1882 by the great organizer and internationalist Mittag-Leffier) published a special 179-page issue which comprised the tables of contents of its first 35 volumes, plus photos and professional biographies of most of its contributing authors to that point (204.altbgether). Many of these appear in younger photographs in the Weierstrass album, and comparing photos from the two treasuries is fun. There is another excellent short biography of Weierstrass in English, which puts his mathematical accomplishments in perspective, by K.-R. Biermann; it appears on pp. 219-224, vol. XIV of the Dictionary of Scientific Biography (Charles Scribner's Sons, N e w York, 1971).
Department of Mathematics Kansas State University Manhattan, KS 66506-2602 USA
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THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 1, 1996
79
Robin Wilson* Mathematical Physics III Keith Hannabuss and Robin Wilson A quarter of a century after Planck's original quantum hypothesis, physicists still had no coherent understanding of quantum theory, but in 1925 two independent discoveries revolutionised the subject. Towards the end of that year, Erwin Schr6dinger (1887-1961) learned of de Broglie's idea that particles could also behave like waves, and found the appropriate partial differential equation to describe those waves. He showed that the energies of the light quanta emitted by a hydrogen atom were differences between eigenvalues of his partial differential operator. This resolution of the problem had, however, been preceded six months earlier by the more algebraic theory of Werner H e i s e n b e r g (1901-1976). Heisenberg realised that quantum-theoretical quantities like position, momentum, and energy could be represented by infinite arrays, which Max Born later recognised as matrices. The non-commutativity of matrix multiplication led Heisenberg to his famous
Erwin Schriidinger: wave functions for hydrogen
Uncertainty Principle. In the previous year, W o l f g a n g Pauli (1900-1958) explained the structure of the periodic table of elements by the introduction of an Exclusion Principle, which forbade the simultaneous occupation of the same state by two electrons. He showed how Heisenberg's new ideas could also explain the radiation properties of the hydrogen atom. In 1928 Paul Dirac (1902-1984) derived an equation for the electron which, unlike those of Heisenberg and Schr6dinger, was consistent with Einstein's Theory of Relativity. This equation explained electron spin, and led him to predict the existence of antiparticles, such as the positron which was detected four years later. Schr6dinger discovered that, despite the apparent differences, his theory was equivalent to that of Heisenberg. John y o n N e u m a n n (1903-1957) then showed that both theories are special cases of a more abstract approach based on the theory of linear operators in Hilbert spaces.
Werner Heisenberg: body-centered cubic lattice
Paul Dirac: cloud chamber track of a positron-electron pair
John yon Neumann
Erwin Schr6dinger * C o l u m n editor's address: Faculty of Mathematics, The O p e n University, Milton Keynes, MK7 6AA, England.
8 0 THE MATHEMATICALINTELLIGENCERVOL.18, NO. 1, 1996
Wolfgang Pauli
