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.
A Moral Dilemma A troubling experience this year moves me to seek discussion with others. The issue goes beyond the individual case and presents a moral dilemma confronting the mathematical community. I received several unsolicited letters concerning a I~rominent foreign mathematician, who has had a visiting position in the United States and is said to have applied to other Western universities for a job. My correspondents state that he actively implemented a discriminatory policy in his native country in the 1970s and 1980s, abusing his responsible position. Though he did not institute the policy, he is said to have taken an active role in it. Witnesses say he derived pleasure from humiliating his victims. What should we do about such people? Our universities are proud to be in the forefront of the fight against racial, sexual, and other forms of discrimination (though prejudice surely still exists on our campuses). Fairness towards students is agreed to be a qualification required of any candidate for a faculty position, one not to be waived just because the candidate is a talented mathematician. I believe therefore that evidence of discriminatory behavior by mathematicians in their home countries should be made available to potential Western employers. On the other hand, I fully realize that there may be abuses, and innocent people may be hurt by false accusations. I hope this letter stimulates more discussion on other mathematicians' views.
(Author's Note: My original letter named the ethnicity of the discriminated individuals, and the country and name of the discriminator. Unfortunately, the Editor of the Intelligencer has agreed to publish this letter only if these pieces of information are withheld. I felt that I would publish this watered down version because it might encourage institutions to think about the past of the individuals they are about to hire. Also those who want to get the deleted pieces of information can simply ask me, and I will gladly provide them.)
Lawrence Shepp AT&T Bell Laboratories Murray Hill, NJ 07974-0636 USA THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 (~1994 Springer-Verlag New York
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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 editorin-chief, Chandler Davis.
Admission to the Mathematics Faculty in Russia in the 1970s and 1980s A. Vershik For a number of years now, the universities of the former Soviet Union have been free of party committees (partkom). These committees were made up of individuals whose job was to see that the party line was followed, and above all to watch over the purity of the c a d r e s - the loyalty of professors and students, the blamelessness of their curricula vitae, and preferment to the necessary people. N o w the offices of party committees have been allotted to computing centers, centers for "intellectual investigations," and so on. Many of their present occupants occupied them in the past, but now they investigate problems of the interaction of science and religion, they criticize Marxism, they invite new-wave politicians and psychics, they talk about their past difficulties at work. The one thing they don't talk about is their cadre work during the period of stagnation. The higher partkom secretary of one of the finest Russian universities (Leningrad), who very carefully carried out party directives about the purity of the cadres, is n o w the director of a cultural center where he organizes evenings of Jewish culture. The prorector of another university (Moscow), once extremely active in all official campaigns and purges and in the organization of "selections" in university admissions, has now become an ardent democrat and an organizer of the most progressive projects. Of course this is w o n d e r f u l - - only, one may still ask, "Why, gentlemen, are you silent about how things were done, how you managed education, admission to universities, selection of cadres?" It would be useful for the educational community to know how and w h y the sciences lost hundreds, and possibly thousands, of indubitably talented individuals, potential leaders, hard workers profoundly dedicated to learning, whose lives have been distorted, often irreparably so. One of the most important objectives of cadre politics at the leading universities, particularly in the capi4
tal, seems to have been to limit the admission of Jews and members of certain other national minorities. Of course, this was not the only objective. It was important not to admit political pariahs (and their children!) as students, aspirants, scientific workers, and professors. Likewise, it was important to help children and relatives of the nomenklatura (party and government officials, KGB) who in reliability were classified with children of work-
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 (~ 1994 Springer-Verlag New York
e r s - - w h o , in the "proletarian" state, enjoyed a mandatory quota of admissions. Once emigration was permitted there was, so to say, an official pretext for not accepting Jews or assigning them to prestigious work, for not providing incentives, awarding degrees, etc. Thus one killed two birds with one stone: the country got rid of some of the disaffected, and at the same time one restricted them at home. But there was one more objective, perhaps the most important one, that one never talked openly about, namely holding down the number of talented people. The grayness o f official Soviet Russia during the era of mature socialism did not just happen; it was imposed from above and readily accepted below, and was of a piece with the lack of talent in the whole leadership, relieved only by isolated fluctuations. To this day we don't know the details of the secret instruction of the early 1970s which (I was told) was more or less to the following effect: restrict or delay the admission to certain post-secondary schools of individuals with ties to states whose politics are hostile to the USSR. Apparently, these could only be Jews, Germans, Koreans, Greeks, and possibly Taiwanese Chinese. Many of us know quite a few concrete stories. I could tell how unbelievable was my admission to the Leningrad Mekh-mat (Faculty of Mathematics and Mechanics) in 1951, at the height of Stalin's war against the cosmopolites; how crudely they used to fail capable students whom I tried to help enter the university in the 1970s by recommending them to the then dean; how I ~vas prevented from hiring talented graduate students and how these very same students eventually managed to find positions at the most prestigious Western universities; and finally, how in 1985, almost in the time of perestroika, my daughter, with a paper in her specialty accepted for publication, was not admitted to the philological faculty of Leningrad University. It is surprising that so few testimonies of the hundreds of victims and witnesses have so far appeared in print. All we have is G. Freiman's It Seems I A m a Jew with some problems and remarks by A.D. Sakharov, and materials collected by B. Kanevskii and V. Senderov (see below). It is just as surprising that so far, to the best of my knowledge, none of the hundreds of people from the cadre sections, from partkoms, from the lecturers who conducted purges at t h e examinations--no one from "the other side" has provided testimony. After all, not all of these people are naive, and not all are absolute scoundrels. Some of them were victims of circumstances. They have hardly any reason to fear revenge, even less court action. All echoes of thede tragedies are fading; justice demands confessions. But n o - - they are silent. Some have become democrats, some profess love of Jews, and some propose to emigrate and ask people whom they had earlier slighted for recommendations. Some maintain that nothing took place. And some don't deny that it all happened but insist that all was done "correctly."
There are very few documents left. The perpetrators realized that it would not do to leave traces. When I approached S.P. Merkur'ev, rector of St. Petersburg University (he died a short time ago) and asked him if it was possible to see the archives of the party committee that dealt with these matters, he offered to help me but warned that I should not overestimate the change since the putsch; almost all the organizers of these things have retained not only their former positions but also power at the University, and, for example, he was unable to remove one of the particularly odious deans. I soon saw a confirmation. When I attempted to induce two historians-- who had earlier been expelled from the University partly because they tried to object to scandalous practices of the kind I describe h e r e - - t o work in the archives, they refused, saying, "We are afraid that 'they' will get us." In 1987 1 brought an article about a case of admission to the progressive weekly Moscow News. The head of the department told me, "We can't print an arti~cle dealing with this topic. There wi'li be a flood of angry letters." But I hope that the conspiracy of silence won't last forever. I am glad that I was able to persuad~ Alexander Shen, who has worked a lot with university and secondary students, to write of the materials he has collected. Mathematical audiences (not only in the West) will find it interesting to learn some details and solve the little problems that a school graduate was supposed to solve in a few minutes. Keep in mind the young boy or girl who has made a commitment to learning, who may have good basis for this decision (participation in olympiads, math circles, and so on), and who now faces an examiner who has his instructions and his arsenal of problems. These examiners and admissions chairs were generally boorish and treated the school graduates shamefully. As is often the case, we know the names of those who carried out the instructions (the examiners) but not of those who gave them. It would make sense to list the secretaries of admissions, deans, and so off, who knew of the scandal and covered it up, right to the top of the party-KGB structure. Even these names are not such a deep secret. We have used here materials only on admissions to Mekh-mat at Moscow State University and only from the 1980s and, in part, the 1970s. There are other faculties, other universities and institutes. And there are problems of defenses of dissertations (VAK), of employment of young scholars, and many others. Is there anything surprising about the drain of Russian science, emigration, apathy, and the low prestige of official institutes and academies? All of this was vredictable from what was done. Mathematical Institute of the Russian Academy of Sciences 27 Fontanka 191011 St. Petersburg Russia THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 4,1994 5
Entrance Examinations to the Mekh-mat A. Shen
Preliminaries At one time, discrimination against Jews in entrance examinations to the leading post-secondary institutions, especially Mekh-mat at Moscow State University (MGU), was a fiercely debated subject. I think that we can now afford to look more calmly at the events and see their role in the history of Russian mathematics. This kind of discrimination was sometimes talked about as if it were the main, and virtually the only, blemish on the otherwise spotless reputation of the national party. This tone was sometimes understandable (for example, one had to talk this way in complaints about Mekh-mat submitted to the Committee of Party Control of the Central Committee). In reality, of course, this was just one of many injustices, some far worse. I entered the Mekh-mat in 1974, began m y graduate studies in 1979, and completed them in 1982. I have worked in mathematical schools from 1977 until today. I will write mostly about things I have had direct contact with. Let us hope my account will be supplemented by others. In many countries, including Russia, the proportion of Jews is appreciably greater among scholars than in the whole population. In entrance to mathematical classes and schools (with equal requirements for all applicants), the proportion of Jews among those who passed the examinations (and among those taking them) is significantly higher than in the population as a whole. Whatever the meaning of this phenomenon, it has to be kept in mind.
Elimination of Undesirable School Graduates After certain events in 1967 (the well-known letter of 99 mathematicians in defense of Esenin-Volpin) and especially in 1968 (mathematicians protesting the intervention in Czechoslovakia), the situation at the Mekh-mat worsened significantly. I.G. Petrovskii ("the last nonparty rector of MGU'), who had done many good things, died in 1973. His successor, R.V. Khokhlov ("the last decent rector of MGU'), perished in 1977. By 1973, the "special program" of elimination of undesirable graduates, especially Jews, was in full swing. The category of "undesirables" included the (small) group of those who didn't belong to the Komsomol. From that time on and until 1989-1990, when this practice was halted, the situation stayed much the same. The number of victims did change: in later years, the potential victims, aware 6
of the barriers, didn't try to apply. Also, in the mid-80s there was a time when Mekh-mat s t u d e n t s - - unlike students at other institutions--were drafted. This reduced the number of applicants to Mekh-mat. Yet another form of discrimination began in 1974. It was open but no less unjust. It involved a two-stage competition for Muscovites and non-Muscovites (the same number of places were reserved for each group although non-Muscovites were more numerous). The ostensible reason was the shortage of rooms. An applicant who did not ask for a place in a hostel (but had no close relations in Moscow) was, however, also classified as a nonMuscovite. The harm from this discrimination was offset by the lower level of the competition for non-Muscovites. During the period of anti-Jewish discrimination the following people were among the responsible officers of the admissions committee (in various capacities): Lupanov (current dean of the Mekh-mat), Sadovnichii (current rector of MGU), Maksimov, Proshkin, Sergeev, Chasovskikh, Tatarinov, Shidlovskii, Fedorchuk, I. Melnikov, Aleshin, Vavilov, and Chubarikov.
How Things Were Done: The Procedure Direct discrimination was a natural concomitant of the shabby conduct of the examinations. The written part of the examination in mathematics consisted of a few simple problems that required only computational accuracy, and one or two very involved and artificial problems (the last problem was usually of this kind). Only "pure plusses" were counted. A flaw in the solution (sometimes invented and sometimes due to the checker's failure to understand the work) meant loss of most of the credit for the problem. As a result, most of the applicants got threes and twos (out of five); the examination was almost totally uninformative. Now we come to the oral part of the examination in mathematics. Even if there were no discernible discrimination, it is virtually impossible for all examiners to make the same demands on applicants. The questions on the tickets are very general and imprecise, and the requirements of the examiners are necessarily not comparable; all the more so because, as a rule, the examiners had no school contact with the students. The examinations included writing a composition and passing an oral test in physics. The physics exam was given by members of the MGU Physics Faculty. It was not a particularly brilliant faculty, and the task of giving examinations was assigned to its less brilliant members.
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 @1994 Springer-Verlag New York
S o m e Examples. In 1980 no credit was given for the solution of a problem (an equation in x) because the answer was in the form "x = 1;2" and the written answer was "x = 1 or x = 2" (the school graduate was Kricheskii, the senior examiner in mathematics was Mishchenko; source: B.I. Kanevskii, V.A. Senderov, Intellectual Genocide, Moscow, Samizdat, 1980). In 1988, during an oral examination, a student who defined a circle as "a set of points equidistant-- that is, at a given distance-- from a given point" was told that his answer was incorrect because he hadn't stipulated that the distance was not zero (the textbook had no such stipulation). The graduate's n a m e was Arkhipov and the names of the examiners were Kovalev and Ambroladze. The 1974 examination in physics included the question: What is the direction of the pressure at the vertical side of a glass of water. The answer "perpendicular to the side" was declared to be incorrect (pressure is not a vector and is not directed a n y w h e r e - - graduate Muchnik).
Sometimes the questioning began a few hours after the distribution of tickets (school graduate Temchin, 1980, waited three hours). The questioning could last for hours (5.5 in the case of the graduate Vegrina; examiners Filimonov and Proshkin, 1980; cited by B.T. Polyak, letter to Pravda, Samizdat, 1980). Parents and teachers of the graduates were not allowed to see the student's papers (letter 05-02/27, 31 July 1988, secretary of the admissions committee L.V. Yakovenko). .An appeal could be lodged only within an hour after an oral examination. The hearing involved in an appeal was extremely hostile (in 1980, A.S. Mishchenko faulted graduate Krichevskii at the hearing for appealing against precisely those remarks of the examiners where he (Krichevskii) was clearly in the right; Kanevskii and Senderov, op. cit.). Procedural Points.
H o w It W a s D o n e : " M u r d e r o u s "
Problems
An important tool (in addition to procedural points and pickiness) was the choice of problems. Readers who are mathematicians can evaluate the level of difficulty of the problems below by themselves. We can assure nonmathematical readers that the level of difficulty of the "murderous" problems is comparable to that of the AllUnion Mathematical Olympiads, and many of them are olympiad problems. (For example, the problem N2 of Smurov and Balsanov turned out to be the most difficult problem of the second round of the All-Union Olympiad in 1985. It was solved by 6 participants, partly solved by 3, and not solved by 91.) For comparison, we adduce first typical ordinary problems (from the mid-1980s). Grades quoted are out of 5. First variant (those who solve both parts get a grade of 5).
1. Show that in a triangle the sum of the altitudes is less than the perimeter. 2. The number p is a prime, p ~ 5. Show that p2 _ 1 is divisible by 24. Second variant (those who solve the first two parts get a grade of 4). 1. Draw the graphs of y = 2x + 1, y = 12x + 11, y = 21xl + 1. 2. Determine the signs of the coefficients of a quadratic trinomial from its graph. 3. x and y are vectors such that x + y and x - y have the same length. Show that x and y are perpendicular, Now the "murderous" problems. The names of the examiners and the years of the examinations are given in parentheses. 1. K is the midpoint of a chord A B . M N and S T are chords that pass through K . M T intersects A K at a point P and N S intersects K B at-a point Q. Sho W that K P =
KQ. 2. A quadrangle in space is tangent to a sphere. Show that the points of tangency are coplanar. (Maksimov, Falunin, 1974) 1. The faces of a triangular pyramid have the same area. Show that they are congruent. 2. The prime decompositions of different integers m and n involve the same primes. The integers m + 1 and n + 1 also have this property. Is the number of such pairs (m, n) finite or infinite? (Nesterenko, 1974) 1. Draw a straight line that halves the area and circumference of a triangle. 2. Show that (1/sin 2 x) G (1/x 2) q- 1 - 4/7r2. 3. Choose a point on each edge of a tetrahedron. Show that the volume of at least one of the resulting tetrahedrons is _< 1/8 of the volume of the initial tetrahedron. (Podkolzin, 1978) We are told that a 2 q- b2 = 4, cd = 4. Show that (a - d) 2 -t- (b - c)2 ~ 1.6. (Sokolov, Gashkov, 1978) We are given a point K on the side A B of a trapezoid A B C D . Find a point M on the side C D that maximizes the area of the quadrangle which is the intersection of the triangles A M B and C D K . (Fedorchuk, 1979; Filimonov, Proshkin, 1980) Can one cut a three-faced angle by a plane so that the intersection is an equilateral triangle? (Pobedrya, Proshkin, 1980) 1. Let//1, H2, H3, H4 be the altitudes of a triangular pyramid. Let O be an interior point of the pyramid and let hi, h2, h3, h4 be the perpendiculars from O to the faces. Show that H 4 + H 4 q-/_/4 + H 4 > 1024 h i . h2- h3. h4. 2. Solve the system of equations y ( x + y)2 = 9, y ( x 3 _ y3) = 7. (Vavilov, Ugol'nikov, 1981) THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
7
Show that if a, b, c are the sides of a triangle and A, B, C are its angles, then
a+b-2c b+c-2a a+c-2b + + >0. sin(C/2) sin(A/2) sin(B/2) (Dranishnikov, Savchenko, 1984) 1. In h o w m a n y ways can one represent a quadrangle as the union of two triangles? 2. Show that the sum of the numbers 1/(n 3 + 3 n 2 + 2 n ) for n from I to 1000 is < 1/4. (Ugol'nikov, Kibkalo, 1984) 1. Solve the equation x 4 - 14x 3 + 66x 2 - 115x + 66.25 = 0. 2. Can a cube be inscribed in a cone so that 7 vertices of the cube lie on the surface of the cone? (Evtushik, Lyubishkin, 1984) 1. The angle bisectors of the exterior angles A and C of a triangle A B C intersect at a point of its circumscribed circle. Given the sides A B and BC, find the radius of the circle. [The condition is incorrect: this doesn't h a p p e n A. Shen.] 2. A regular tetrahedron A B C D with edge a is inscribed in a cone with a vertex angle of 90 ~ in such a w a y that AB is on a generator of the cone. Find the distance from the vertex of the cone to the straight line CD. (Evtushik, Lyubishkin, 1986) 1. Let log(a, b) denote the logarithm of b to base a. C o m p a r e the numbers log(3, 4). log(3, 6 ) . . . . . log(3, 80) a n d 21og(3, 3) 9log(3, 5) . . . . - log(3, 79). 2. A circle is inscribed in a face of a cube of side a. A n o t h e r circle is circumscribed about a neighboring face of the cube. Find the least distance between points of the circles. (Smurov, Balsanov, 1986) Given k segments in a plane, give an u p p e r b o u n d for the n u m b e r of triangles all of whose sides belong to the given set of segments. (Andreev, 1987) [Numerical data w e r e given, but in essence one was asked to p r o v e the estimate 0(k15). A. Shen.]
Let A,B,C be the angles and a,b,c the sides of a triangle. Show that
aA + bB + cC _< 90 ~ a+b+c (Podol'skii, Aliseichik,1989)
60 ~ < -
S t a t i s t i c s - The Mekh-mat at M G U and Other Institutions The most detailed data on graduates of mathematical schools were obtained in 1979 b y Kanevskii and Senderov. T h e y divided the graduates of schools 2, 7, 19, 57, 179, and 444 w h o intended to enter the Mekh-mat into two groups. One group of 47 consisted of students whose parents and grandparents w e r e not Jews. Another group of 40 consisted of students with some Jewish parent or grandparent. The results of olympiads (see table below) s h o w that the graduates were well prepared, but w h e n it comes to admission, the results are noticeably different.
Mekh-mat at M G U
Given the graph of a parabola, to construct the axes. (Krylov E.S., Kozlov K.L., 1989) [These examiners told a graduate that an e x t r e m u m is defined as a point at w h i c h the derivative is zero. They also r e p r o a c h e d another graduate for not saying "the set of ALL points" w h e n he defined a circle as the set of points at a given distance from a given point.[ 8
THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 4,1994
47 14 4 26 40
40 26 11 48 6
Kanevskii and Senderov give figures also for two other institutions: MIFI
Total graduates Admitted
First g r o u p
Second group
54 26
29 3
MFTI
Find all a such that for all x < 0 we have the inequality (Tatarinov, 1988)
Second group
Total graduates Olympiad winners Multiple winners Total o l y m p i a d prizes Admitted
Use ruler and compasses to construct, from the parabola y = x 2, the coordinate axes. (Kiselev, Ocheretyanskii, 1988)
ax 2 -- 2x > 3a -- 1.
First g r o u p
Total graduates Admitted
First g r o u p
Second group
53 39
32 4
Of course, the character of the entrance examinations became k n o w n to school graduates, and those suspected of Jewishness began to apply to other places, for the most part to faculties of applied mathematics where there was no discrimination. (One very w e l l - k n o w n place was the "kerosinka" - - t h e Gubkin Oil and Gas Institute.)
Mathematical Schools and Olympiads When we talk about mathematical schools, we exclude the boarding school #18 at MGU. Proximity to the Mekhmat unavoidably leaves its-imprint. In the remaining schools, discrimination by nationality was mostly insignificant. As a rule, selection of students for a particular class depended largely on the teachers of mathematics and was controlled by the administration to a minor extent. In 1977, in school #91, the administration was presented with a list of~students in the math class and did not make any changes. In 1982, in school #57, the situation was more complicated because the school was subject to district administration, and the class list had to be acceptable to the district committee. So some students favored by the district authorities were accepted outside the competition. In 1987, in school #57, "wartime resourcefulness" was successfully applied: Russian names picked at random were added to the list of students sent for approval to the district committee (which did not check which of the students on the list later attended). It seems that after that there were no problems (perestroika!). One could speculate that discrimination in admissions to the Mekh-mat (very well known to both teachers and students of math classes) and the large percentage of Jews among teachers and students could give rise to a problem of "interethnic relations" (injustice often gives rise to injustice in reverse). I have often heard such speculations, but I am convinced that in most mathematical classes (and the best ones) no such things happened. As for the olympiads, the Moscow city olympiad was for quite a long time relatively independent from official departments. But in the late 1970s, after Mishchenko's letter to the partkom (it is amusing that recently Mishchenko asserted publicly that he was not in the least involved, but he did not challenge the authenticity of his letter), control of the olympiads was given to Mekhm a t - - a n d , to a large extent, to the very same people who controlled the entrance examinations. It seems to me that the result was not so much discrimination as plain incompetence. (For example, in 1989, after my conversation with the people who managed the olympiad, it became clear that a large bundle of papers got lost. Following urgent requests, it was found. I was even permitted to see the papers of the students in the class in which I lecture. A significant portion of these papers were improperly corrected.)
General Remarks, History. It seems that now practically no one denies there was discrimination in entrance examinations (that is, no one except possibly university administrators--but then they are the people least able to shift responsibility). In particular, Shafarevich mentions this kind of discrimination in his article in the collection Does Russia have a Future?
This discrimination causes two kinds of harm. First, many gifted students have been turned away or have not tried to enter the Mekh-mat. In addition to this direct harm, there is also an indirect kind: participation in entrance examinations has become a means of checking the loyalty of graduate students and co-workers, and a criterion for the selection of co-workers. Many distinguished people (regardless of nationality) who refused to be accomplices have not been employed by the Mekh-mat. The situation has brought protests whose form depended on the circumstances and the courage of the protesters. I probably know only some of the incidents. In 1979, document #112 of the Moscow group for implementing the Helsinki agreements, titled "Discrimination against Jews entering the university," was signed by E. Bonner, S. Kallistratova, I. Kovalev, M. Landa, N. Meiman, T. Osipova and Yu. Yarym-Ageev. Included in this document were the statistical data collected by B.I. Kanevskii and V.A. Senderov. On the basis of the 198/1 adtnission figures, Kanevskii and Senderov wrote, and distributed through Samizdat, the paper "Intellectual Genocide: examinations for Jews at MGU, MFTI and MIFI." I well remember my reaction, at that time, to the activities of Kanevskii and Senderov (which I now realize was largely a form of cowardice): the result of their collecting data will be that students of math schools will be rejected just like Jews. (This did not happen, although there were such attempts.) Also, Kanevskii, Senderov, mathematics teachers in math schools, former graduates of math schools, and others, helped students and their parents to write appeals and complaints. Incidentally, this activity was sometimes criticized in the following terms: "By inciting students to fight injustice you are using others to fight your war with the Soviet authorities, and you are subjecting children and their parents to nervous stress." In some cases, the plaintiffs succeeded (by threatening to cause an international scandal or by taking advantage of a blunder of an examiner), but an overwhelming majority of complaints were without effect. There were attempts to help some very capable students (Jews or those who could be taken for Jews) by undercover negotiations. I myself took part in such attempts twice, in 1980 and in 1984. In one case it was possible to convince the admissions commission that the graduate was not a Jew, that his name just sounded Jewish; and in the second case they closed their eyes to the Jewishness of the graduate's father. It was not a simple matter to find a chain of people, ending with a person who was a member of the admissions committee, each of whom could talk to the next one about such a delicate topic. (In one case I know of, one member of such a chain was A.N. Kolmogorov.) To this day I have two minds about the morality of these activities of ours. In 1979-1982, on the initiative of B.A. Subbotovskaya and with the active support of B.I. Kanevskii, matheTHEMATHEMATICAL INTELLIGENCERVOL.16,NO.4,1994 9
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matics instruction was organized for those not going to the Mekh-mat: once a week, every Saturday afternoon, lectdres on basic mathematical subjects were presented to interested students. These sessions took place at the "kerosinka" or at the humanities building of MGU (of course, without the knowledge of the administration-we simply took advantage of the available empty rooms). Xerox copies of the lectures were given out to the students. These studies were referred to as "courses for improving the qualifications of lecturers in evening mathematical schools," but the participants usually called them "the Jewish national university." This went on for a number of years, until one of the participants, and Kanevskii and Senderov, were arrested for anti-Soviet activity; after an interrogation at the KGB, B.A. Subbotovskaya died in a car accident in unclear circumstances. It should be noted that some of the participants in these studies who were not Mekh-mat students (some were Mekh-mat students) were very gifted, but very few of them became professional mathematicians. I remember my reaction, at the time, to the arrest of Senderov and others: well, instead of teaching mathematics they engaged in anti-Soviet agitation, and because of them (!) now everyone has been caught in the act. Other attempted protests: in 1980 and 1981 B.T. Polyak 10 THEMATHEMATICALNTELLIGENCER VOL.16,NO.4,1994
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wrote to Pravda about scandalous practices (without bringing in the issue of anti-Semitism--he must have hoped that he could influence the Mekh-mat within the existing system). Perestroika began in 1988 and one could openly and safely write about anti-Semitism (even to the Committee of Party Control, then still in existence). Some people, including Senderov, then released from prison, went to various departments, including the city partkom and the city Department of Education, trying in some way to influence the Mekh-mat. "The dialogue with the opposition" took more concrete forms and there were no accusations of anti-Soviet agitation, but the only positive result was that one of the graduates involved was allowed a special examination. After that, the discussion continued inside the university (at meetings of the scientific council of the Mekh-mat, in wall newspapers, and so on). It died down gradually, because discrimination in entrance examinations ceased, and many of the participants in the discussion scattered all over the world. Institute for Problems of Information Transmission Ermolovoi 19 101447 Moscow Russia
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 editorin-chief, Chandler Davis.
Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture Doron Zeilberger 1
Today The most fundamental precept of the mathematical faith is thou shalt prove everything rigorously. While the practitioners of mathematics differ in their views of w h a t con,~titutes a rigorous proof, a n d there are fundamentalists w h o insist on even a m o r e rigorous rigor than the one practiced by the mainstream, the belief in this principle could be taken as the defining property of mathematician.
exciting new facts to discover: mathematical pulsars and quasars that will make the Mandelbrot set seem like a mere Galilean moon. We will have (both h u m a n and machine 2) professional theoretical mathematicians, w h o will develop conceptual paradigms to make sense out of the empirical data and w h o will reap Fields medals along
The Day After Tomorrow There are writings on the wall that, n o w that the silicon savior has arrived, a n e w testament is going to be written. Although there will always be a small g r o u p of "rigorous" old-style mathematicians (e.g., [Ref. 1]) w h o will insist that the true religion is theirs and that the comp u t e r is a false Messiah, they m a y be viewed b y future mainstream mathematicians as a fringe sect of harmless eccentrics, as mathematical physicists are viewed b y regular physicists today. The c o m p u t e r has already started doing to mathematics w h a t the telescope and microscope did to a s t r o n o m y a n d biology. In the future not all mathematicians will care about absolute certainty, since there will be so m a n y
1Supported in part by the NSE Based in part on a Colloquium talk given at Rutgers University.Reprinted from "Theorems for a Price: Tomorrow's Semi-RigorousMathematical Culture", by Doron Zeilberger, Notices of the American Mathematical Society, Volume40, Number 8, October 1993,pp. 978-981, by permission of the American Mathematical Society. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
11
with (human and machine) experimental mathematicians. Will there still be a place for mathematical mathematicians? This will happen after a transitory age of semi-rigorous mathematics in which identities (and perhaps other kinds of theorems) will carry price tags.
in roughly an increasing order of sophistication. 1. 2. 3. 4.
2+2=4. (a + b)3 = a 3 + 3a2b + 3ab2 + b3. sin(x + y) = sin(x) cos(y) + cos(x) sin(y). F,+IFn-1 - F 2 = (-1) n.
5. (a + b)'~ = Y~'2=o(~) akb~-k" A Taste o f T h i n g s to C o m e
6. E k = _ n ( - - 1 ) k ( : : k ) 3 = (3n).
7. Let (q)~ := (1 - q)(l - q2)... (1 - qr);then To get a glimpse of how mathematics will be practiced in the not-too-distant future, I will describe the case of algorithmic proof theory for hypergeometric identities (Refs. [11], [13], [WZ11, [WZ2], [Z1], [Z21, [Z3], [Z4], [Ca]). In this theory one may rigorously prove, or refute, any conjectured identity belonging to a wide class of identities, which includes most of the identities between the classical special functions of mathematical physics. Any such identity is proved by exhibiting a proof certificate that reduces the proof of the given identity to that of a finite identity among rational functions, and hence, by clearing denominators, to one between specific polynomials. This algorithm can be performed successfully on all "natural identities" of which we are now aware. It is easy, however, to concoct artificial examples for which the running time and memory are prohibitive. Undoubtedly, in the future, "natural" identities will be encountered whose complete proof will turn out to be not worth the money. We will see later how, in such cases, one can get "almost certainty" with a tiny fraction of the price along with the assurance that, if we robbed a bank, we would be able to know for sure. This is vaguely reminiscent of transparent proofs introduced recently in theoretical computer science [4-6]. The result that there exist short theorems having arbitrarily long proofs, a consequence of G6del's incompleteness theorem, also comes to mind [7].3 1 speculate that similar developments will occur elsewhere in mathematics and will "trivialize" large parts of mathematics by reducing mathematical truths to routine, albeit possibly very long and exorbitantly expensive to check, "proof certificates." These proof certificates would also enable us, by plugging in random values, to assert "probable truth" very cheaply.
Identities
Many mathematical theorems are identities, statements of type " = ' , which take the form A = B. Here is a sample,
2 For example, my computer Shalosh B. Ekhad and its friend Sol Tre already have a nontrivial publication list, e.g., Refs. 2 and 3. 3 Namely, the ratio (proof length)/(theorem length) grows fast enough to be nonrecursive. Adding an axiom can shorten proofs by recursive amounts [8, 9].
12
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
L
q~
-L
7-=0 (q)~(q)n-~
(-1)~q(5~-~)/2
~=-n (q)~-~(q)n+r
7'. Let (q)~ be as in 7; then q~2
q5i+l)-l(1 -
II(1
-
q5i+4)-1 -
i=0
8. Let H,~ be given by Hn : Hn(q) = (1 + q) (1 + q2) (1 - q) (1 - q2)
(1 + (1 - qn),
then
)4
(L2(_qn+1) k k=0
~-~-s
Hk
4(__q)kHn+kHn_k
k=-,~L(l+qk)2
Hn
Hn
z
k ~
81.
) 4
qk2 k=--or
or
= 1 +8 E
qk
(1 + (_q)k)2"
k=l
9. Analytic Index = Topological Index. 10. Re(s) = 89for every nonreal s such that ((s) = 0. All the identities are trivial, except possibly the last two, which I think quite likely will be considered trivial in 200 years. I will now explain.
W h y A r e t h e First Eight I d e n t i t i e s Trivial?
The first identity, while trivial nowadays, was very deep when it was first discovered, independently, by several anonymous cave dwellers. It is a general abstract theorem that contains, as special cases, many apparently unrelated theoremswTwo Bears and Two Bears Make Four Bears, Two Apples and Two Apples Make Four Apples, etc. It was also realized that, in order to prove it rigorously, it suffices to prove it for any one special case, say, marks on the cave's wall.
The second identity, (a + b)3 = a 3 + 3a2b + 3ab2 + b3, is one level of generality higher. Taken literally (in the semantic sense of the word literally), it is a fact about numbers. For any specialization of a and b we get yet another correct numerical fact, and as such it requires a "proof," invoking the commutative, distributive, and associative "laws." However, it is completely routine w h e n viewed literally, in the syntactic sense, i.e., in which a and b are no longer symbols denoting numbers but rather represent themselves, qua (commuting) literals. This shift in emphasis roughly corresponds to the transition from Fortran to Maple, i.e., from numeric computation to symbolic compatation. Identities 3 and 4 can be easily embedded in classes of routinely verifiable identities in several ways. One w a y is by defining cos(x) and sin(x) by (e ix + e-iX)~2 and (e i~ - e-i~)/(2i) and the Fibonacci numbers Fn by Binet's formula. Identities 5-8 were, until recently, considered genuine nontrivial identities, requiring a h u m a n demonstration. One particularly nice h u m a n proof of 6 was given by Cartier and Foata [10]. A one-line computer-generated proof of identity 6 is given in [2]. Identities 7 and 8 are examples of so-called q-binomial coefficient identities (a.k.a. terminating q-hypergeometric series). All such identities are n o w routinely provable [11] (see below). The machinegenerated proofs of 7 and 8 appear in [3] and [12], respectively. Identities 7 and 8 immediately imply, by taking the limit n ---* 0% identities 7' and 8', which in turn are equiv,alent to two famous number-theoretic statements: The first Rogers-Ramanujan identity, which asserts that the n u m b e r of partitions of an integer into parts that leave remainder 1 or 4 w h e n divided by 5 equals the n u m b e r of partitions of that integer into parts that differ from each other by at least 2; and Jacobi's theorem which asserts that the number of representations of an integer as a s u m of 4 squares equals 8 times the sum of its divisors that are not multiples of 4.
T h e WZ Proof Theory Identities 5-8 involve sums of the form
~-'~ F(n, k),
(sum)
k=O
where the summand, F(n, k), is a hypergeometric term (in 5 and 6) or a q-hypergeometric term (in 7 and 8) in both n and k, which means that both quotients, F(n + 1, k)/F(n, k) and F(n,?~ + 1)/F(n, k), are rational functions of (n, k) [(qn qk, q), respectively]. For such sums and m u l t i s u m s we have [11] the following result. THE FUNDAMENTAL THEOREM OF ALGORITHMIC HYPERGEOMETRIC PROOF THEORY. Let F (n; kl, 99 9 kr) be a proper (see [11]) hypergeometric term
in all of (n; k l , . . . ,kr). Then there exist polynomials
PO(n),..., PL (n) and rational functions Rj (n; kl,. 99 kr) such that Gj := RjF satisfies L
E pi(n)F(n + i; k l , . . , ikr) i=0 /.
=
[Cj(n;
+
j=l
-
Gj (n; k l , . . . , k j , . . . , kr)].
(multiWZ)
Hence, if for every specific n, F(n; -) has compact support in ( k l , . . . , k~), the definite s u m g(n) given by
g(n) :=
E
F(n; k i , . . . , k ~ )
(multisum)
kl,...,k~
satisfies the linear recurrence equation with polynomial coefficients: L
E Pi(n)g(n + i) = 0.
(P-recursive)
i=O
(P-recursive) follows from (multiWZ) by s u m m i n g over {kl,..., k~} and observing that all the sums on the right telescope to zero. If the recurrence happens to be first-order, i.e., L = 1 above, then it can be written in closed form: For example, the solution of the recurrence (n + 1)g(n) - g(n + 1) = 0, g(0) = 1, is g ( n ) = n ! . This "existence" theorem also implies an algorithm for finding the recurrence (i.e., the pi) a n d the accompanying certificates Rj (see below). An analogous theorem holds for q-hypergeometric series [13, 14]. Since we k n o w how to find and prove the recurrence satisfied by a n y given hypergeometric s u m or multisum, we have an effective w a y of proving any equality of two such sums or the equality of a s u m with a conjectured sequence. All we have to do is check whether both sides are solutions of the same recurrence and match the appropriate n u m b e r of initial values. Furthermore, we can also use the algorithm to find n e w identities. If a given sum yields a first-order recurrence, it can be solved, as mentioned above, and the sum in question turns out to be explicitly evaluable. If the recurrence obtained is of higher order, then most likely the s u m is not explicitly evaluable (in closed form), and Petkovsek's algorithm [15], which decides whether a given linear recurrence (with polynomial coefficients) has closed form solutions, can be used to find out for sure.
A l m o s t Certainty for an e of the C o s t Consider identity (multisum) once again, where g(n) is "nice." Dividing through by g(n) and letting F --* F/g, THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
13
we can assume that we have to prove an identity of the form F(n; k l , . . . , kr) = 1. (Nice) kl,...,kr
The WZ theory promises that the left side satisfies some linear recurrence, and if the identity is indeed true, then the sequence g(n) = 1 should be a solution (in other words, po(n) + "" + p i ( n ) -- 1). For the sake of simplicity let us assume that the recurrence is minimal, i.e., g(n + 1) - g(n) = 0. (This is true a n y w a y in the vast majority of the cases.) To prove the identity by this method, we have to find rational functions Rj(n; k l , . . . , kr) such that Gj : = Rj F satisfies
F(n + 1; kl, 999 kr) -- F(n; k l , . . . , k~) r
= ~)-~[Gj(n;kl,...,kjq-
1,...,k~)
j=l
- Gj (n; k l , . . . , k j , . . . , kr)].
(multiWZ')
By dividing (multiWZ') through by F and clearing denominators, We get a certain functional equation for the R 1 , . . . , R~, from which it is possible to determine their denominators Q1, 999 Q~. Writing Rj = Pj/Q3, the proof boils d o w n to finding polynomials Pj ( k l , . . . , k~) with coefficients that are rational functions in n and possibly other (auxiliary) parameters. It is easy to predict upper b o u n d s for the degrees of the Pj in (kl,. 9 k~). We then express each P3 symbolically with " u n d e t e r m i n e d " coefficients and substitute into the above-mentioned functional equation. We then expand and equate coefficients of all monomials k~' ... k~ - and get an (often huge) system of inhomogeneous linear equations with symbolic coefficients. The proof comes d o w n to proving that this inhomogeneous system of linear equations has a solution. It is very time-consuming to solve a system of linear equations with symbolic coefficients. By plugging in specific values for n a n d the other parameters if present, one gets a system with numerical coefficients, which is m u c h faster to handle. Since it is unlikely that a r a n d o m system of inhomogeneous linear equations with more equations than u n k n o w n s can be solved, the solvability of the system for a n u m b e r of special values of n and the other parameters is a very good indication that the identity is indeed true. It is a waste of money to get absolute certainty, unless the conjectured identity in question is k n o w n to imply the Riemann Hypothesis. Semi-Rigorous Mathematics
As wider classes of identities, and perhaps even other kinds of classes of theorems, become routinely provable, we might witness m a n y results for which we would k n o w how to find a proof (or refutation); but we would be unable or unwilling to pay for finding such proofs, since "almost certainty" can be bought so m u c h cheaper. 14
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
I can envision an abstract of a paper, c. 2100, that reads, "We show in a certain precise sense that the Goldbach conjecture is true with probability larger than 0.99999 and that its complete truth could be determined with a budget of $10 billion." It w o u l d then be acceptable to rely on such a priced theorem, provided that the price is stated explicitly. Whenever statement A, whose price is p, and statement B, whose price is q, are used to deduce statement C, the latter becomes a priced theorem priced at p + q. If a whole chain of boring identities would turn out to imply an interesting one, we might be tempted to redeem all these intermediate identities; but we would not be able to b u y out the whole store, and most identities would have to stay unclaimed. As absolute truth becomes more and more expensive, we w o u l d sooner or later come to grips with the fact that few nontrivial results could be known with old-fashioned certainty. Most likely we will wind up abandoning the task of keeping track of price altogether and complete the metamorphosis to nonrigorous mathematics. Note: Maple programs for proving hypergeometric identities are available by a n o n y m o u s ftp to math. temple, edu in directory pub/zeilberger/programs. A Mathematica implementation of the single-summation program can be obtained from Peter Paule at paule9 uni-linz, ac. at. References
1. A. Jaffe and E Quinn, "Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 1-13. 2. S.B. Ekhad, A very short proof of Dixon's theorem, J. Combin. Theory Ser. A 54 (1990), 141-142. 3. S. B. Ekhad and S. Tre, A purely verification proof of the first Rogers-Ramanujan identity, J. Combin. Theory Ser. A 54 (1990), 309-311. 4. B. Cipra, Theoretical computer scientists develop transparent proof techniques, SIAM News 25 (May 1992). 5. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy,
Proof verification and intractability of approximation problems,
6. 7. 8. 9. 10.
Proc. 33rd Syrup. on Foundations of Computer Science (FOCS), IEEE Computer Science Press, Los Alamos, 1992, pp. 14-23. S. Arora and M. Safra, Probabilistic checking of proofs, ibid, pp. 2-13. J. Spencer, Short theorems with long proofs, Amer. Math. Monthly 90 (1983), 365-366. K. G6del, On length of proofs, Ergeb. Math. Colloq. 7 (1936), 23-24, translated in The Undecidable (M. Davis, Ed.), Raven Press, Hewitt, NY, 1965, pp. 82--83. J. Dawson, The GSdel incompleteness theorem from a length of proof perspective, Amer. Math. Monthly 86 (1979), 740-747. P. Cartier and D. Foata, Probl~mescombinatoires de commutation et rdarrangements, Lecture Notes in Math. 85, Springer, 1969.
Continued on page 76
not enough of others. I was pleased with the considerable thusiastically use it again when I next teach the history attention paid to probability and statistics, in compari- of mathematics. son with other general histories. But Katz's treatment of twentieth-century mathematics is sketchy, emphasizing References only set theory, its problems and paradoxes; topology; new ideas in algebra; and computers and applications. 1. Carl B. Boyer and Uta C. Merzbach, A History of Mathematics. New York, Wiley,1989. And some will think that, though three good chapters treat the nineteenth century, the importance of the cen- 2. Morris Kline, Mathematical Thought from Ancient to Modern Times. New York, Oxford, 1972. tury and the sheer amount of its mathematics are under- 3. Dirk J. Struik, A Concise History of Mathematics. 4th Edition. represented. There are some minor errors, some typoNew York, Dover, 1967. graphical, some of emphasis. One supposedly useful feature of the book is the breaking up of each chronolog- Pitzer College ical chapter into topics, so that a teacher can emphasize, Claremont, CA 91711-6110 say, the history of equation-solving from ancient Egypt USA and Babylonia, Greece, China, Islam, up to Abel, Gauss, and Galois. These divisions sometimes make the narrative seem choppy. Students found the book "challenging" (that means not easy); they also found it interesting to read. Readers Zeilberger Continueafrom page 14 m a y agree with some of m y students who found the book 11. H. S. Wilf and D. Zeilberger, An algorithmic proof theory for too long and felt that often one couldn't see the forest for hypergeometric (ordinary and "q') multisum/integral identithe trees. Here one must remember that Katz is writing ties, Invent. Math. 108 (1992), 575-633. a textbook. The mathematical demands on the student 12. G.E. Andrews, S. B. Ekhad, and D. Zeilberger, A short proof ofJacobi'sformula for the number of representations of an integer reader must remain finite. An excellent, much briefer as a sum of four squares, Amer. Math. Monthly 100 (1993), work is Struik's Concise History [3]. Still, the history of 274-276. mathematics is sufficiently tangled that one welcomes 13. H.S. Wilf and D. Zeilberger, Rational functions certify comKatz's attention to specifics. Readers wanting a more debinatorial identities, J. Amer. Math. Soc. 3 (1990), 147-158. tailed account of nineteenth- and twentieth-century top14. T.H. Koornwinder, Zeilberger's algorithm and its q-analogue, Univ. of Amsterdam, preprint. ics can consult the general works by Carl Boyer (in the edition updated by Uta C. Merzbach) and Morris Kline, 15. M. Petkovsek, Hypergeometric solutions of linear recurrence equations with polynomial coefficients, J. Symbolic Comput. or the many items in Katz's full bibliography on specific 14 (1992), 243-264. topics. [AZ] G. Almkvist and D. Zeilberger, The method of differenThe most serious criticism one can make is that Katz's coverage reflects the limitations of twentieth-century tiating under the integral sign, J. Symbolic Comput. 10 (1990), 571-591. scholarship. One might think this is good in that Katz's [Ca] P. Cartier, Ddmonstration "automatique" d'identitds et scholarship is up-to-date and the materials this schol- fonctions hypergdometriques [d'apres D. Zeilberger], S6minaire arship addresses are important. However, because the Bourbaki, expos4 no. 746, Ast6risque 206 (1992), 41-91. [WZ1] H. S. Wilf and D. Zeilberger, Towards computerized book is not itself one of path-breaking scholarship, it shares many of the emphases and the omissions of the proofs of identities, Bull.Amen Math. Soc. (N.S.) 23 (1990), 77-83. [WZ2] - - - , Rational function certification of hypergeometric existing literature. Much remains to be studied. Impor- multi-integral/sum/"q" identities, Bull. Amer. Math. Soc. (N.S.) 27 tant questions like whether ibn al-Haytham's formulas (1992) 148-153. or the Islamic and Jewish work on induction influenced [Z1] D. Zeilberger, A holonomic systems approach to special their (re)discoverers in Europe, whether the medieval functions identities, J. Comput. and Appl. Math. 32 (1990), 321Chinese or Indian "Pascal" triangles influenced Pascal, 368. [Z2] ~ - , A fast algorithm for proving terminating hypergewhether seventeenth-century mathematicians knew, di- ometric identities, Discrete Math. 80 (1990), 207-211. rectly or indirectly, the Indian work on trigonometric se[Z3] - - , The method of creative telescoping, J. Symbolic ries (such as the arctangent series above), have recently Comput. 11 (1991), 195-204. [Z4] - - - , Closed form (pun intended!), Special volume in been the subject of much speculation. Equally important questions about Cauchy's use of infinitesimals or Leib- memory of Emil Grosswald, (M. Knopp and M. Sheingorn, eds.), Contemp. Math. vol. 143, Amen Math. Soc., Providence, niz's philosophy are not yet settled. Readers with unan- RI, 1993, pp. 579-607. swered queries must await another decade of research. In the meantime, Victor J. Katz should be congratulated on having produced an excellent and readable text, based on sound scholarship and attractively presented. Department of Mathematics A mathematician could appropriately put this book on Temple University the family coffee table, but would be even better advised Philadelphia, PA 19122 USA to read the many fascinating things it contains. I will en-
[email protected] 76 THEMATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
The Death of Proof? Semi-Rigorous Mathematics? You've Got to Be Kidding! G e o r g e E. A n d r e w s 1
Introduction
The Evidence?
T h r o u g h the s u m m e r of 1993 I was desperately clinging to the belief that mathematics was i m m u n e from the g i d d y relativism that has pretty well d e s t r o y e d a number of disciplines in the university. Then came the October Scientific American and John Horgan's article, "The death of proof" [HI. The theme of this article is that computers have changed the world of mathematics forever, in the process making proof an anachronism. O h well, all m y friends said, H o r g a n is a nonmathematician w h o got in w a y over his head. Apart from his irritating comments and obvious slanting of the material, "The death of proof" actually contains interesting descriptions of a n u m b e r of i m p o r t a n t mathematics projects. Indeed, as W. Thurston has said, IT] "A more appropriate title w o u l d have been 'The Life of Proof.'"
Unlike Horgan, Zeilberger is a first-rate mathematician. Thus one expects that his futurology is based on firm ground. So w h a t is his evidence for this paradigm shift? It was at this point that m y irritation t u r n e d to horror. In a list of identities used to back u p his predictions, he lists two intimately related to me, and it is these which turn out to be the star witnesses in his case. To present his argument fairly, let us refer to his 10 identities, of which he says, "All the above identities are
Semi-Rigorous Mathematics Then came the October Notices of the A. M. S. a n d an article [Z2] by m y friend and collaborator Doron Zeilberger: "Theorems for a price: t o m o r r o w ' s semi-rigorous mathematical culture" [reprinted a b o v e - - Editor]. The theme of this article is reasonably s u m m a r i z e d by the following quote: There are writings on the wall that, now that the silicon savior has arrived, a new testament is going to be written. Although there will always be a small group of 'rigorous' old-style mathematicians ... , they may be viewed by future mainstream mathematicians as a fringe sect of harmless eccentrics ... In the future not all mathematicians will care about absolute certainty, since there will be so many exciting new facts to discover... As absolute truth becomes more and more expensive, we would sooner or later come to grips with the fact that few nontrivial results could be known with oldfashioned certainty. Most likely we will wind up abandoning the task of keeping track of price altogether and complete the metamorphosis to nonrigorous mathematics.
1Partially supported by National Science Foundation Grant DMS 8702695-04. 16
THE MATHEMATICAL INTELLIGENCERVOL. 16, NO. 4 (~)1994 Springer-Verlag New York
trivial, except possibly the last two, which I think quite likely will be considered trivial in two hundred years." Your guess is as good as mine why 9 and 10 will be trivial in 200 years. He then focuses on t-8. Identities 1-5 are preeighteenth-century. It is quite true that these theorems are easy to prove once you know h o w - - many theorems are. However, at least for 3-8, their proofs yield insights well beyond the bare statements of the identities. Consequently if we regard them as merely results to be verified or turned up by computer, we are incurring a staggering loss,of insight. Don't worry, Zeilberger assures us: "We will have (both human and machine) professional theoretical mathematicians, who will develop conceptual paradigms to make sense out of the empirical data and w h o will reap Fields medals along with (human and machine) experimental mathematicians." And what is the evidence for this? Zeilberger tells us, "For example, my computer Shalosh B. Ekhad and its friend Sol Tre already have a nontrivial publication list, e.g., [E], [ET]." But there is a problem here. While the computer has indeed generated proofs of 1-8, it discovered none of the identities. The two most recent theorems on the list are 7 ([All, [B], [ET]) and 8 [AEZ]. The actual discovery of 7 [A1] was from an examination of G. N. Watson's massive general identity [Wa] that he used to prove the Rogers-Ramanujan identities (i.e., 7'). Surely one can argue that Watson's proof of his theorem is as trivial as Zeilberger's computer's proof of 7; the main observation used by Watson is that a polynomial with more zeroes than degree is identically zero. However, Watson's identity has spawned both new discoveries and new research that reach way beyond its original purposes. The actual discovery of 8 [AEZ: p. 276] was from an examination of Jackson's q-analog [J] of Dougall's theorem [Do]. Again a result proved originally by the old game of exhibiting too many zeroes of a polynomial. On this account, then, what exactly is the contribution of Zeilberger and his computer? Very simply, he has made a substantial contribution to proving identities, i.e., to rigorous mathematics. He and Herb Wilf [WZ], [Z1] have found an algorithm which can be implemented on the computer and which will produce rigorous proofs of numerous identities of which 7 and 8 are prototypical examples. A natural response is that the computer can be programmed using Zeilberger's algorithms to find new identities also. Indeed, Wilf spoke on this very topic in a talk [Wi] entitled "Billions and billions of combinatorial identities." Therein lies another difficulty. Which among these "billions and billions" are really important? Which are just mild changes of variable in classical results? Which are sterile in their relation to the rest of mathematics? Ira Gessel [G] has undertaken a serious study of the possibilities; but it is not clear that he has produced answers to these questions yet.
The Insight of Proof Ignored completely in Zeilberger's futurology is the insight provided by proof. "In the future," says Zeilberger, "not all mathematicians will care about absolute certainty, since there will be so many exciting new facts to discover." Let us consider an example of an exciting new fact described by J. and P. Borwein and K. Dilcher [BBD; p. 681]: Gregory's series for ~r, truncated at 500,000 terms, gives to forty places 500,000
4E
1_
k=l
= 3.14159_0653589793240462643383269502884197. The number on the right is not 7r to forty places. As one would expect, the 6th digit after the decimal is wrong. The surprise is that the next 10 digits are correct. In fact, only the 4 underlined digits aren't correct. This intriguing observation was sent to us b~ R. D. North... of Colorado Springs with a request for an explanation. Well, there it is: a computer-discovered, exciting, mathematical fact! Who among us would respond to this observation by saying, "Great! N o w let's go discover some other exciting new fact"? Surely anyone who has applied the alternating series test in a calculus class to show that, for example, the above error in Gregory's series occurs at the sixth decimal must indeed be intrigued by the astounding accuracy of 30 of the next 33 terms, and would want to stop and explain it! What can the computer tell us about this phenomenon? Only what it already has! I do not mean to minimize its contribution. No one could make the above evaluation without a computer. But that is it for the computer. Fortunately for us, that was not it for Dilcher and the Borweins. They provide in the remainder of [BBD] absolute certainty about what is going on, and they provide concomitantly great insight and, dare I say it, beauty. Their paper is an almost perfect example of the computer aiding crucially in the discovery of facts but not in their p r o o f - - and not in the perception that they cried out for proof.
Conclusion Zeilberger has proved some breathtaking theorems [ZB], [Z3], and his W - Z method (joint with Wilf [WZ]) has been a godsend to me [A2] and an inspiration [A3]. However, there is not one scintilla of evidence in his accomplishments to support the coming "... metamorphosis to nonrigorous mathematics." Until Zeilberger can provide identities which are (1) discovered by his computer, (2) important to some mathematical work external to pure identity tracking, and (3) too complicated to allow an actual proof using his algorithm, then he has produced exactly no evidence that his Brave N e w World is on its way. THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 4,1994 1 7
cation." I won't give the plot away, but I recall the words of Claude Rains near the end of Casablanca: "Round up the usual suspects!" References
I regret feeling compelled to write this article. Unfortunately articles on why rigorous mathematics is dead create unintended side effects. We live in an age of rampant "educational reform." Many proponents of mathematics education reform impugn the importance of proofs, and question whether there are right answers, etc. A wonderfully sane account of these problems has been given by H.-H. Wu [Wul], [Wu2]. A much more disturbing account "Are proofs in high school geometry obsolete?" concludes Horgan's article [HI. It is a disservice to mathematics inadvertently to provide unfounded ammunition for the epistemological relativists. If anyone reading this believes the last paragraph is rubbish because attempts (unknown to me) are currently underway to insert the Continuum Hypothesis or the Theory of Large Cardinals into the NCTM Standards for School Mathematics, please don't write to tell me about them. I can take only so many shocks to my system. Finally, wisdom suggests that grand predictions of life in 2193 ought to be treated with scepticism. ("Next Wednesday's meeting of the Precognition Society has been postponed due to unforeseen circumstances.") A long-overdue analysis of some of our current prophets has been attempted by Max Dublin [Du]. Especially noteworthy is Dublin's Chapter 5, "Futurehype in Edu18
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
[A1] G. E. Andrews, Problem 74-12, SIAM Review 16 (1974), 390. [A2] G. E. Andrews, Plane partitions V: the T.S.S.C.P.P. conjecture, J. Combin. Theory Ser. A 66 (1994), 28-39. [A3] G. E. Andrews, Schur 's theorem, Capparelli's conjecture and q-trinomial coefficients, Contemp. Math. (in press). [AEZ] G. E. Andrews, S. B. Ekhad and D. Zeilberger, A short proof of Jacobi's formula for the number of representations of an integer as a sum of four squares. Amer. Math. Monthly 100 (1993), 274-276. [BBD] J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers, and asymptotic expansions. Amer. Math. Monthly 96 (1989), 681-687. [B]D. M. Bressoud, Solution to Problem 74-12. SIAM Review 23 (1981), 101-104. [Do] J. DougaU, On Vandermonde's theorem and some moregeneral expansions, Proc. Edinburgh Math. Soc. 25 (1907), 114-132. [Du] M. Dublin, Futurehype. The Tyranny of Prophecy, Viking (the Penguin Group), London and New York, 1989. [El S. B. Ekhad, A very short proof of Dixon's theorem, J. Combin. Theory Ser. A 54 (1990), 141-142. [ET] S. B. Ekhad and S. Tre, A purely verification proof of the first Rogers-Ramanujan identity, J. Combin. Theory Ser. A 54 (1990), 309-311. [G] I. Gessel, Finding identities with the WZ method, talk presented at the ACSyAM Workshop on Symbolic Computation in Combinatorics at MSI/Cornell University, September 21-24, 1993. [HI J. Horgan, Thedeath of proof, ScientificAmerican 269 (1993), no. 4, 74-103. [J] E H. Jackson, Summation of q-hypergeometric Series, Mess. Math. 50 (1921), 101-112. [T] W. Thurston, Letter to the editor, Scientific American 270 (1994), no. 1, 9. [Wa] G. N. Watson, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc. 4 (1930), 4-9. [Wi] H. Wilf, Billions and billions of combinatorial identities, talk presented at Allerton Park in the Conference in Honor of Paul Bateman, April 25-27, 1989. [WZ] H. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990), 147-158. [Wul] H.-H. Wu, The role of open-ended problems in mathematics education, The Journal of Mathematical Behavior (to appear). [Wu2] H.-H. Wu, The role of Euclidean geometry in high school, The Journal of Mathematical Behavior (to appear). [Z1] D. Zeilberger, The method of creative telescoping, J. Symbolic Computing 11 (1991), 195-204. [Z2] D. Zeilberger, Theorems for a price: tomorrow's semi-rigorous mathematical culture, Notices of the A.M.S. 40 (1993), 978981. [Z3] D. Zeilberger, The alternating sign matrix conjecture (to appear). [ZB] D. Zeilberger and D. Bressoud, A proof of Andrews" q-Dyson conjecture, Discrete Math. 54 (1985), 201-224.
Department of Mathematics Pennsylvania State University University Park, PA 16802 USA
H o w Probable Is Fermat's Last Theorem? Manfred Schroeder
The distance between z n, the nth power of z E [q, and (z + 1) n is approximately nz ~-1. Thus, the probability that a randomly chosen n u m b e r s E N near s = z n equals the nth power of an integer is approximately 1 / n z ~- 1 = n -1 s -I+U~. I shall call the latter expression the "density" of the nth powers at s. Similarly, the density of s = x n + yn (x, y E I~, y > x) is given by the convolution integral
fsn
1 dn(s) = ~-$ J0
dt [(s/2 + t)(s/2
- t)] 1 - 1 / n
a r a n d o m integer z, the approximate probability that its square z 2 is equal to the s u m of two squares, z 2 = x 2 q- y2, is 7r/8, provided the pure squares and the sums of two squares are distributed independently. Numerical evidence, as well as the density of Pythagorean triplets, suggests that this m a y indeed be the case. For n = 3, the density according to Eq. (1), with B (89 89 = 5.2999..., is about 0.294s -1/3. Thus, the probability that the cube z 3 of a r a n d o m l y chosen integer z is equal to the s u m s of two cubes, s = z3, is approximately equal to 0.294/z. (Numerical counting gives 0.295/z.)
1 ( ! ) 1 - 2 / n ~01 du = n-2 (1 -- U2) 1-1/n
=
1 s-l+2/nB(1, 2n 2
1)
(1)
Here B(a, b) is the complete beta function (Euler's integral of the first kind):
B(a, b) - r(a)r(b) F(a + b) For a = b = 1/n, one obtains
(1 B
1) ,
p2(1/n) = F(2/n) "
For the exponent n = 2, we have B (89 89 = 7r. Thus, with Eq. (1), the density becomes d2(s) = 7r/8 ~ 11/28, i.e., the density of the sum of two squares is constant: Approximately 11 out of every 28 r a n d o m l y chosen integers are, on average, equal to a s u m of two squares. This result also follows from the observation that the n u m b e r of lattice points of the integer lattice ~2 within a circle of radius v ~ is approximately 7rs. Thus, given
THEMATHEMATICALINTELLIGENCERVOL.16,NO.4 (~)1994Springer-VerlagNewYork 19
Summing over z from 1 to Z, tells us that Fermat's Last Theorem (FLT) for exponent n = 3 would be violated in about 0.294 log g cases for z < Z. The first such violation would be expected to occur below g ~ e 1/0"294 ~ 3 0 , corresponding to s ~ 27, 000. We know, of course, from the proofs of FLT for n = 3 by Euler, Legendre, Sophie Germain, and others, that there are, in fact, no such counterexamples. For n = 4, we have B(88 88 = 2rc/U = 7.4163 . . . . where U = 0.847213... is the so-called "ubiquitous constant," i.e., the c o m m o n (arithmetic-geometric) mean of 1 and 1/v~. Thus, the density according to Eq. (1) is about 0.23176s-U2 = 0.23176/z 2. (Numerical result: 0.23175/z2.) By s u m m i n g over z from 1 to infinity, we obtain the probabilistic estimate of the n u m b e r of violations of FLT for n = 4. With the infinite s u m over 1/z 2 equal to rr2/6, we see that the probability of such a violation is approximately 0.38 or less than 0.50. For n = 5, the density according to Eq. (1) is about 0.1900/z 3. (Numerical counting gives 0.1896/z 3. ) By summing over all z, one obtains, with ~(3) ~ 1.202, a probability of 0.23 for a violation of FLT for n = 5. For large values of n, we use the approximation B(1,
1) =2n+O(1)
and obtain with Eq. (1) dn(8) = n - 1 8 -l+2/n = n - l z 2-n.
(2)
The density s u m m e d over all values of z from I to infinity equals approximately 1/n. Thus, whereas violations of FLT become increasingly unlikely with increasing n, the grand total of expected violations for all values of n is infinite. In other words, assuming x" + pn and z n are independently distributed, the truth of FLT is highly unlikely. Only a completed proof could lay these doubts to rest. For what exponents n would FLT be expected to fail, if it does fail? Suppose FLT was proved for all exponents below some large integer N. Given the fact that FLT has also been proved for all regular primes (Kummer, 1850) [1], the number of cases for which FLT would be violated between N and M (disregarding all other proven cases) is given by M 1 F = ~ ~ ~ (1 - e-1/2)(log log M - log log N),
(3)
N
where the factor I - e -1/2 ~ 0.39 is the asymptotic fraction of irregular primes/5 (Siegel, 1964) [2]. The "failure estimate" F exceeds unity for M > M0, where log M0 = 12.7 log N. For N = 106, say, the "crossover exponent" M0 equals about 1076. The next failure would be expected around 10152 (give or take a few dozen orders of magnitude). 20
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
However, so far we have ignored an important restriction. For prime n, x n = x m o d n, and x ~ + yn = z,~ would, therefore, imply x + p ~ z mod n. Assuming z < n, the congruence can be written as the equation x + p = z, implying x n + y'~ < z ~. Thus, for x n + y'~ = z '~ it is necessary that z > n. S u m m i n g Eq. (2) accordingly yields approximately n 1-~, which, s u m m e d over n > N = 106, gives a negligible probability (< 10 -6~176176for FLT to fail. Probabilistic analysis has s h o w n its value in estimating the density of Mersenne primes. Although there is no proof of the existence of infinitely m a n y Mersenne primes, such methods have given good predictions of undiscovered Mersenne primes [3]. H o w will it be for FLT?
References 1. P. Bachmann, Das Fermatproblem in seiner bisherigen Entwickelung, deGruyter, Berlin & Leipzig, 1919. Reprinted by Springer-Verlag, Berlin, 1976. 2. C. L. Siegel, "Zu zwei Bemerkungen Kummers," Nachr. Akad. d. Wissen. Gf~'ttingen, Math. Phys. Kl., II (1964), 51-62. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan & H. Maat~),Vol. III, 436-442. Springer-Verlag, Berlin, 1966. 3. M. R. Schroeder, Number Theory in Science and Communication, 2nd ed., Springer-Verlag, New York, 1990. Drittes Physikalisches Institut Universita't G6ttingen D-37073 GiJttingen, Germany
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 responsibilityfor them. An Opinion should be submitted to the editorin-chief, Chandler Davis.
Righting the Early History of Computing, or How Sausage Was MadeMichael Davis Mathematicians make the history of mathematics. Occasionally, they read it. But who writes i t - - a n d how? This article offers a partial a n s w e r - - o n e likely to bring to mind Bismarck's famous comment on laws: "Like sausage, it's better not to see them made."
Discovery
Herman Berg shared Van Sinderen's interest in Babbage. Without a college degree, he corresponded with many academics, museums, and libraries concerning common research interests. He also studied mathematics and foreign languages. During the academic year 19711972, he studied Japanese at the University of Kansas. There, at a meeting of the Scuba Club, he met an accountant from Kansas City who, hearing he lived in Detroit
The July 1983 issue of Annals of the History of Computing carried an article entitled "Babbage's Letter to Quetelet, May 1835." The article's heart was a modern translation into English of a letter in French printed in 1835 in the Bulletin of the Royal Academy of Arts and Sciences of Brussels. Historians of computing consider that letter significant because it contains the first mention in print of Babbage's Analytical Engine, a precursor of today's computer. 1 The article's introduction noted, "The exact date of the letter is not clear, and the original is not known to exist." The article's author, Alfred W. Van Sinderen, a long-time collector of Babbage's manuscripts and published works, was unsure even "whether Babbage wrote to Quetelet in French or whether he did so in English and Quetelet translated it." Though Van Sinderen earned his living as chief executive of the Southern New England Telephone Company, his judgments carried weight with Babbage scholars. 1 Actually, the letter's fame is a bit more complicated. Strictly speaking, not the French version but an 1843 retranslation into English is famous. This first appeared in a translation (by the Countess of Lovelace) of Menabrea's paper on the Analytical Engine. See The Works of Charles Babbage, edited by Dr. Martin Campbell-Kelly, N e w York: N e w York University Press (1989), Vol. 1, p. 25. THE MATHEMATICALINTELLIGENCERVOL.16, NO. 4 (~1994 Springer-VerlagNew York 21
and was interested in computers, urged him to look up a brother-in-law, "Buzz" (Bernard) Galler, at the University of Michigan's Computing Center in Ann Arbor (an hour from Detroit). Berg looked up Galler during the winter break. Galler gave him "the grand tour" and urged him to come to Ann Arbor as a student. Berg, indicating an interest, pointed out that he had another year's commitment to Japanese at Kansas. They would not meet again for a decade. By then, Galler would be Editor-in-Chief of the Annals. In August 1983, Berg received permission to sit in on a course in software engineering that Galler was to teamteacK Galler's teammate actually granted the permission (on condition that Berg explain a software project to be assigned), but it was this class that reintroduced them. Although Galler was not often in class, he did see much of Berg's presentation of a topological sorting algorithm. After class a few days later, they had a long conversation. Berg told Galler of his life during the intervening decade, including studies at the University of Wisconsin, and of his interest in the history of mathematics. Berg was then reading the Proceedings of the 4th International Statistical Congress, London, 1860. Like Babbage's letter of 1835, this congress was an important event in the early history of computing. The Scheutz computing machine (a simplified realization of Babbage's Difference Engine) was in use at the General Registry. 2 Babbage invited congress attenders, including Florence Nightingale, to see it work. Quetelet was there too. Seeing how excited Berg was about the Proceedings, Galler told him something of the internal workings of the Annals, the history of Van Sinderen's paper, and gave Berg a copy of the paper, suggesting he see what he could do with it. Why did Galler do that? Berg offers this explanation (based on what Galler told him at the time): Both reviewers had wanted to delay recommending publication until they were sure the original letter could not be found, because if it could be found, the translation would be unnecessary (and, therefore, not worth publishing). Neither reviewer was quick to give up the search. 3 Meanwhile, Van Sinderen gave Galler a good deal of grief for the long time it was taking the journal to make a decision. Eventually, Galler forced a decision from the reviewers, leaving both them and him not quite satisfied. Apparently, Galler saw Berg as an opportunity to put to rest remaining doubts about publishing Van Sinderen's paper. 2 For an accessible explanation of the difference between Babbage's Difference Engine, the Scheutz machine, and the Analytical Engine, see Donon D. Swade, Redeeming Charles Babbage's mechanical computer, Scientific American (February 1993), 86-91; or Michael Lingren, Glory
and Failure: The Difference Engines of Johann Muller, Charles Babbage and Georg and Eduard Scheutz, Cambridge, MA: MIT Press, (1990). 3 Van Sinderen (1983) himself noted, "The importance of original source documents has already been pointed out by N. Metropolis and J. Worlton in the Annals of the History of Computing, vol. 2, No. 1, January 1980, and this particular letter [of Babbage] is a case in point."
22 THEMATHEMATICALINTELLIGENCERVOL.16,NO. 4,1994
Herman Berg Early in January 1984, Berg "tackled" the paper. His method was straightforward: First, he examined Van Sinderen's sources (using the University of Michigan's libraries and the Detroit Public Library). Next, he looked up the names mentioned in Van Sinderen's paper in standard reference works - - Encyclopaedia Britannica, 11th edition, Dictionary of Scientific Biography, and International Encyclopedia of the Social Sciences. In the last of these, he discovered an entry for some of Quetelet's correspondence (in the Archives et Biblioth~que de Belgique) that Van Sinderen had not mentioned. A search of the Library of Congress's National Union Catalog yielded another reference to Quetelet's correspondence, this one at the Belgian Royal Academy. Berg wrote for a copy of the Academy's file on 22 February. By 26 March 1984, he had before him a photocopy of the missing version of the letter in Babbage's own hand, in English, and dated 27 April 1835. Even a cursory examination revealed significant differences between this original and the French translation. First, of course, was the exact date (27 April rather than some time in May). Second, there were several additional paragraphs. Third, there were numerous errors in the engine's specifications (for example, Babbage's original has "120" (later crossed out) whereas the Quetelet version has "100").
Complications Pleased with what he had found, Berg called Galler the next day (while in Ann Arbor on other business). Berg expected warm congratulations. He got something else. As Berg remembers their talk, Galler almost immediately
changed the subject to a letter Berg had written to Van Sinderen in February. It was, Berg recalls, a long letter in which he praised Van Sinderen's translation, explained h o w he came to examine Van Sinderen's paper, and told Van Sinderen about some sources he had discovered. Berg also mentioned the delay in publication, sketched what he knew, and concluded that Van Sinderen was o w e d an apology (which, apparently, Van Sinderen took as an apology). Galler said Van Sinderen had "laid it in to him" for telling tales out of class. Why had Berg, of all people, been offering an apology for something the Annals had done? Galler seemed to view the letter both as a breach of his confidence and as hurting Berg's relationship with Van Sinderen. His tone was severe: Berg had no business offering an apology to Van Sinderen, no business repeating what Galler told him about the workings of the Annals. 4 Badly shaken by this exchange, Berg did what he could to repair the damage. As soon as he had hung up the phone, he sent Galler a copy of the Babbage letter (through campus mail), hoping that seeing the document might help Galler regain perspective. Berg then went to the office of the University's Vice President for Academic Affairs, looking for an explanation of what he had done wrong and advice about what to do next. A secretary made an appointment for him with Robert Holbrook, an economist then serving as Associate Vice President for Academic Affairs (and as a member of the University's Joint Task Force on Integrity in Scholarship). The appointment was for a few days later. When they met at the appointed time, Holbrook treated Berg cordially, heard him out, and then declared that Berg's ignorance should excuse the breach of editorial confidentiality. He added that Berg's discovery was in any case significant enough to outweigh such a small sin. Berg left with the impression that Holbrook might "straighten Galler out." Berg also wrote letters of apology to Van Sinderen and to the two outside reviewers (whom he had referred to by name). Berg's letter to Van Sinderen seems to have worked. In a letter dated 25 June ("cc-- Galler'), Van Sinderen thanked Berg for "your 'peace offering'," adding that it was "not really necessary, as I always have positive thoughts about people who are interested in Charles Babbage." (The "peace offering" had been a copy of the original Babbage letter.) 4 Berg's explanation of his conduct is that he had not supposed there was any longer any issue of confidentiality. On the one hand, Van Sinderen's paper had been published. A breach of confidentiality could not affect the review of that paper. On the other hand, Berg viewed what he was doing for Galler as analogous to his work as judge at various Detroit science fairs. He was collecting data, telling those he sought help from enough for them to know what help to give (Letter, 21 June 1993). Our only, knowledge of the contents of Van Sinderen's letter to Galler comes from Galler (as remembered by Berg). We might well give the actual text a different reading.
Soon after mailing these letters, Berg dropped by Galler's office. This visit went no better than the phone conversation. Galler tried to convince Berg that the discovery was not important enough to warrant publication in the Annals, certainly not worth a note, no, not even a letter to the editor to correct the historical record. After all, Galler argued, the letter differed in only small ways from the translation Van Sinderen had made from the French. The differences were not important to the later history of computing. Berg could not understand Galler's response. Had Berg not found the lost "Ur-letter" of computing? Had he not shown that it still existed, dated it, and provided the full text? Until his discovery, who could say how long the original letter was or how well Quetelet had translated it? Scholars would hereafter know that Quetelet had omitted three paragraphs at the beginning and two at the end (and what those paragraphs said). They would have Babbage's exact words. If Van Sinderen's now-unnecessary translation had been worth publishing, w h y not Berg's original? These questions led to another. Programs in the history of science are rare; programs in the history'of mathematics, rarer still. The University of Michigan had neither. Galler's own background was in mathematics (Ph.D., University of Chicago), not history of any sort. His work was far from the literary or industrial "archaeology" to which Berg's discovery belongs. Could it be that Galler's editorial judgment in this area was unreliable? To answer that question, Berg wrote to others in the field describing his difficulties with Galler and asking their opinion of his discovery. Galler soon heard of these letters. On 25 April, he wrote Berg asking him to come in to "discuss some of the letters you have written." They met in May. The tone of this meeting was different from the one before. Although urging Berg to stop writing "those letters," Galler no longer dismissed Berg's discovery altogether. Instead, he urged Berg to do "more" with the Babbage letter. Berg mentioned a number of archives he could check. The meeting ended. Berg left dissatisfied. The "more" Galler was asking seemed more or less what Van Sinderen had already done. Berg might turn up something new (as he had just done). But, without a clear idea of what he was looking for, he was unlikely to turn up anything relevant to the Babbage letter. What was more likely was that Berg would simply be turning up interesting material about other things. Then one of two possibilities might be realized: either Van Sinderen's false claim would remain in print unchallenged, or another scholar would do what Berg had done. If that scholar made the same discovery independently and published it, Berg would get no credit for what he had done. Berg felt he could not just do as Galler asked (though he did try to do that, keeping Galler informed of efforts to get access to various archives). THE MATHEMATICALINTELLIGENCERVOL.16,NO. 4, 1994 23
A b o u t this time, the University of Michigan issued its first-ever Guidelines for Maintaining Academic Integrity. (This was the w o r k of the Joint Task Force of w h i c h Holb r o o k was a member.) The Guidelines included advice on m a i n t a i n i n g priority for a discovery w h e n publication h a d been blocked. Berg u s e d the Guidelines as a checklist. So, for example, he d o n a t e d a c o p y of the B a b b a g e letter (and related d o c u m e n t s ) to the University of Michigan libraries. H e also w r o t e a n y o n e active in the field w h o m he had not already told, sending each an " u n p r i n t " (that is, a copy of the original Babbage letter, a brief s u m m a r y of w h a t Berg had done, a n d a c o p y of Van Sinderen's article). M u c h of this m u s t h a v e m a d e Galler u n h a p p y . As Berg recalls their next m e e t i n g (early July), Galler told Berg he h a d been receiving p h o n e calls advising h i m to publish Berg's discovery as a letter. As Galler recalled the meeting (letter of 10 August), he told Berg he w o u l d continue to help with Berg's history activities provided Berg " d r o p p e d the extraneous correspondence dealing with personalities a n d past events which were really n o n e of y o u r business." If Berg d i d not d r o p the correspondence, Galler "would h a v e n o t h i n g further to do w i t h you." Berg did not do as Galler asked. For example, on 24 July he wrote the History of Science Society, s e n d i n g t h e m a c o p y of the Babbage letter, a n d - - b y w a y of explanation - - stating: Dr. Bernard Galler is in no hurry to publish it even as a letter to the editor announcement to correct the historical record. I find it difficult to separate the mind games he has been playing with me from his editorial judgment. Dr. Galler has backed off from a position giving me no credit to allowing me to publish at some later date when I have an unspecified 'more'. Feeling initially blocked by him I sent copies to all of those I was aware of [being] actively involved in Charles Babbage studies. Thus, even if I was never published, I would have in some form fulfilled a scholarly obligation to communicate my results to others. Currently, it seems like Dr. Galler is still dissembling with me as he scrambles to cover himself with his reviewers and editorial board members. 5
These letters did not necessarily h a v e the effect Berg intended. For example, Van Sinderen r e s p o n d e d to Berg's letter of 30 July with a two-and-half-page s y n o p s i s of their correspondence ( " c c - - B . G a l l e r ' ) . A l t h o u g h he e n d e d b y urging Berg to forget the past, he clearly w a s u p s e t that Berg should "write m e again, p a g e after p a g e of concerns a n d speculations about w h o did w h a t to w h o m containing, a m o n g other things, u n f o u n d e d suspicions that it w a s [one of the two reviewers], a close friend of mine, w h o d e l a y e d publication of m y article in the Annals."
5 According to Berg,the purpose of this letter was not so much to inform HSS of his discovery as to inquire whether the discovery merited a prize or, at least, a letter of praise he might use to strengthen his claim that the Annals should publish a report of it. 24
THE MATHEMATICAL INTELL|GENCER VOL. 16, NO. 4, 1994
On 10 A u g u s t , after receiving a c o p y of Van Sinderen's letter to Berg (dated 31 July), Galler w r o t e Berg again: "[You] did not take m y advice [but] continued to participate in the k i n d of activity which can only be destructive to y o u r relationships with other historians." Therefore (with "great reluctance"), Galler h a d to "terminate" his relationship with Berg. H e did, however, a d d that Berg could continue to submit w o r k to the Annals. A n y submissions w o u l d be sent out to r e v i e w e r s in the usual way: "There will be no bias against y o u . " This letter did, indeed, end their relationship. Doubting Galler w o u l d treat him better t h a n he already had, Berg s u b m i t t e d nothing to him again. Until Galler retired as Editor-in-Chief in 1987, Berg's contact with the Annals only concerned other matters a n d these contacts w e r e always w i t h other editors. Berg's t w o - p a g e note on the missing letter did not a p p e a r in the Annals until J a n u a r y 1992. 6
Plagiarism? 7 In 1989, N e w York University Press published The Works of Charles Babbage in 11 volumes. Volume 3 contains, a m o n g other things, Babbage's p a p e r s on the Analytical Engine. Minutes of the general meeting of the Belgian Royal A c a d e m y of Science (in English translation) in which Quetelet read the letter he h a d received f r o m Babbage a p p e a r on pp. 12-14. A n asterisk beside the title signaled a footnote. The footnote began, "This article is an English version (not strictly a translation) of [the f a m o u s 1835 letter] which i m m e d i a t e l y precedes it [in its French version] in this v o l u m e . " After giving credit to Lovelace's partial translation (1843) and to Van Sinderen's c o m p l e t e translation, the editor indicates that [in] preparing this English version...use has been made of a letter from Babbage to Quetelet, preserved in the Quetelet Collection in the Biblioth6que Royale de Belgique, Brussels. This letter, which is written in English and dated 27 April 1835, is believed to be the same that Quetelet read to the general meeting of the Academie, 7-8 May, 1835. The text of this letter has been used, lightly edited for readability, in the version below. Van Sinderen's translation has been used for the French text which did not form part of Babbage's letter.
6 "On Locating the Babbage-Quetelet Letter," Annals of the History of Computing 14(1) (1992), 7-9. Why did Berg not simply publish his discovery elsewhere? He tried, but there are not many journals in the history of computing. Those he wrote advised him that the Annals was the appropriate place for a note correcting a claim made in Van Sinderen's article. [Editor's note. In a rare lapse of editorial judgment, the MathematicalIntelligencerrejected Berg's paper in 1986.] 7The question mark suggests doubt, two doubts. One doubt concerns the wrong charged. Is the wrong plagiarism, strictly speaking, or failure to attribute? (The American Historical Association recently revised its rules to distinguish these two offenses, while making clear the distinction was not between more serious and less serious.) The other doubt concerns whether any wrong was done at all. As things stand now, we have only part of the story. Until we have the rest, we can reach only a preliminary judgment (however damning the evidence now seems).
The note gives no credit to anyone for finding the missing original. It just says that the letter is in the Royal Library. Van Sinderen receives two mentions for his translation; Berg receives nothing for finding the original. Berg first read this note early in December 1989. The more he looked at it, the more disturbing he found it. There was, first, a shortening of the title of Van Sinderen's article. The date, and only the date ("of May, 1835"), had been omitted from the reference (replaced by the usual mark of elision). H a d the date not been omitted, Berg thought, it would have been obvious that Van Sinderen did notJl~now of the letter's actual date (now indicated in print for the first time). Second, there was that reference to the "Biblioth6que Royale." The Royal Library had transferred its Quetelet collection to the Royal A c a d e m y several decades earlier. The scholar who found the letter should not have m a d e that mistake. 8 Last, as far as Berg could see, the letter was (except for light editing and the omission of the first three and last two paragraphs) the one he had discovered. 9 If there was any reason to credit either the French translation or Van Sinderen's retranslation, was there not more reason to credit Berg for finding the original? The original preempted all translations. There was also a good reason to get the Academy's permission to publish the letter: scholarly custom. Berg had sought, and received, that p e r m i s s i o n - - w h i c h was granted on condition that the A c a d e m y receive proper credit in print. The Works of Babbage neither credited the A c a d e m y for the letter (though apparently relying on it) nor indicated receiving anyone's permission to publish it. Berg held in his h a n d w h a t purported to be the definitive edition of Babbage's work, opened to the page supposedly containing the text of the most famous letter in the history of computing. Yet, what that footnote told readers is that they had before them neither the original letter Berg had found nor Van Sinderen's translation of the French version of Quetelet, but something new, a mix of the two "edited for readability." The rest of the original, though available for inclusion, was omitted from Babbage's Works. What could explain this? The explanation could not be that the editors had confined themselves to previously published work. They did not claim to have such a policy. In fact, they had not followed such a policy. (They had, for example, included w h a t seemed to be a previously unpublished "Statem e n t to the Duke of Wellington.") Berg supposed the worst. Someone was trying to slip by without recognizing Berg's contribution to Babbage scholarship.
8 Bergadmits that it is possiblethat between March1984and early 1989 the Quetelet letter was moved to the RoyalLibrary (and then back to the Royal Academy).He has, however, found no one who knows of such a move. 9Only 3 of the 11 paragraphs Berg had before him--that is, the minute's one introductory paragraph and two concluding paragraphs- were independent of his discovery.See "Checkit Out".
Martin Campbell-Kelly, University of Warwick, England, was the Editor-in-Chief of the Works. He was also a member of the editorial board of the Annals. However, he did not seem to be personally responsible for the footnote. An editorial undertaking on the scale of the Works requires considerable delegation. There were four "consulting editors." One of these, Allan Bromley, University of Sydney, Australia, seemed to have responsibility for the part of Volume 3 relevant here3 ~ He was also one of those to w h o m Berg had announced his discovery. Indeed, Bromley had written a friendly ("Dear Herman") note of acknowledgment (19 June 1984): Thank you for your letter of 22 May and the information enclosed. I was particularly interested to read Babbage's letter to Quetelet, especially the comment "but it will take many months to work out all the details". How true that proved to be!...11 There could, then, be no doubt that, if Bromley was responsible for that section, he h a d used Berg's unpublished discovery without giving credit. Berg had done a scholar's work; he had not received a scholar's pay. What could he do?
Conclusion The short answer to the question just posed seems to be, "Not much." Since I have given "the long answer" elsewhere, 12 I can summarize it here. Berg could see no point in writing Bromley. What could he write to someone he believed guilty of plagiarism? What could such a letter accomplish? He did, however, write to New York University Press; to all the universities involved, and to the Works" English publisher (Pickering and Chatto), w h o said they passed the letter on to Campbell-Kelly (30 June 1990); to a great m a n y professional societies in Australia, England, and the United States; to a great m a n y governmental agencies and some politicians in those countries; to some publications, both academic and popular; to the Pope and several cardinals; and to a miscellany of other individuals. Generally, those in the best position to do s o m e t h i n g - - f o r example, the three universities inv o l v e d - did not even answer Berg's letter. Others often did answer, but their answer was generally that they were in no position to do anything. That was h o w matters stood w h e n I published m y first article on "the Berg Affair". 12 Its publication finally roused those best positioned to answer. Late in 1993, GaUer, Bromley, and Campbell-Kelly wrote letters to the 10See Worksof Babbage,Vol.1, pp. 22-27. 11Berg considers the phrase in quotes to be a smoking gun. The exact words appear in the version of Babbage's letter Berg discovered but not in any of the others. BecauseBromley does not suggest that he had already discovered the original letter on his own, that phrase demonstrates that his first knowledgeof it must have comefrom Berg. 12MichaelDavis, "Of Babbageand kings:A study of a plagiarismcomplaint," Accountability in Research2 (1993), 273-286, esp. pp. 279-282. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
25
editor of Accountability in Research criticizing me for not getting their side of the story before I published Berg's. Campbell-Kelly threatened the journal's publisher with a lawsuit if I (or it) did not retract. The three also provided some insight into what their explanation of events might be. Bromley, though listed prominently in ads for the Works, claimed to have had only a small part, merely advising Campbell-Kelly on selection and arrangement of the papers printed in Volumes 2 and 3. CampbellKelly confirmed that Bromley took no part in the detailed editing or in the provision of documents. That work was performed by one C.J.D. ("Jim") Roberts, a "London-based independent scholar" who was "editorial consultant to the Works" (and, apparently, worked directly under Campbell-Kelly). Roberts seems to deserve more public credit than he has so far received. According to Campbell-Kelly, it was Roberts who, making a systematic search for unknown holdings of Babbage, turned up the original of the letter to Quetelet by writing the Royal Library (one "tiny triumph" among many). CampbellKelly also claimed that neither he nor Roberts knew of Berg's prior discovery. While the letters of Galler, Bromley, and CampbellKelly answered some questions, they raised others: Did Berg actually fail to write Campbell-Kelly about his discovery, or did Campbell-Kelly forget, or did the letter go astray? Why was Roberts able to obtain Babbage's letter by writing the Royal Library (when, as everyone now seems to agree, the letter was in the Royal Academy)? Did the Royal Library have its own copy, the one Berg sent it, 26
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
or did some royal librarian check the Academy's collection, discover the letter, and photocopy it without giving any indication of its provenance (poor scholarly practice, one would think)? And, of course, they left the most important question unanswered: If reluctance to give Berg credit for discovery of the original was not what motivated those involved, what could have motivated them to mix the original with a mongrel retranslation? The persistence of such questions tells much about how the history of mathematics is (sometimes) made. If our first impulse is to turn away (as Bismarck advised), should we not resist and seek reform instead?
Acknowledgment I should like to thank Herman Berg for calling my attention to his case, providing all the documents cited here, telling me what he remembered of events where no documents existed or where they gave only an incomplete picture, reading and commenting on the first draft of this paper, and correcting my errors. While he has done enough to be listed as co-author, I have (with his permission) taken full responsibility for what is written here; I have been free of the usual constraints of dual authorship. Center for the Study of Ethics in the Professions Illinois Institute of Technology Chicago, IL 60616-3793 USA
Letter from Martin Campbell-Kelly Thank you for giving me the opportunity to make a rejoinder to Michael Davis~s article "Righting the Early History of Computing, or How Sausage Was Made." This article is a rehash of Davis's paper "Of Babbage and Kings: A Study of a Plagiarism Complaint" which was published last year in Accountability in Research (Volume 2, pp. 273-286). I have written a long and detailed refutation of Davis's allegations which is currently in press and will, I understand, be published in the next issue of Accountability in Research. While in the present article Davis has wisely not repeated most of his allegations, it is unfortunate that he has not had the good grace to apologize for the numerous factual inaccuracies in the original article, and the several professional reputations he has discredited by rushing into print without first checking his facts. Davis has sought to justify his article largely on the grounds that Warwick University and other organizations failed to respond to Berg's letters. I can only answer for myself. If either Berg or Davis had written to me directly and courteously then I would have replied in like manner, exactly as I reply to all the letters I receive from the general public and other scholars. I am certain the same applies to all the senior officers of Warwick University to whom Berg has written. Since Davis's original article was published in Accountability in Research, Berg has written on several occasions to officers of Warwick University. For the r e c o r d - and hopefully to avoid yet another time-wasting rehashing of this s t o r y - - I will take this opportunity to describe the contents of the letters. Since Davis's articles do not convey the tone of Berg's letters, and I believe that the tone of his letters is germane, I shall quote from them. Shortly after the original article in Accountability in Research was published, Berg sent a letter dated 10 November 1993 to the Vice-Chancellor of Warwick University, Professor Sir Brian Follett FRS. By any standards this letter was incoherent, and the concluding two sentences stated: "No government agency, no professional organization, and no university administration anywhere in the world has shown the political courage to handle my plagiarism complaint. This has resulted in the total loss of standards of academic integrity in the research scholarship of the entire (global) scientific / research / education establishment." (This, one notes, was written after Davis's article had been published. I thought that seemed a little unappreciative of Davis's efforts.) I gave my Vice-Chancellor a full explanation of the plagiarism allegation, and passed him copies of the relevant papers and correspondence to place on file. He said that he had encountered similar cases as a Secretary of the Royal Society; and added that he was appalled at Davis's article "and perhaps worse that the editor of the journal should have accepted the material
for publication." I did not ask him what further action he took with the letter, nor has he told me. Subsequently, Berg sent packages to the Registrar of the University, to the Chairs of two University Faculties, as well as to the Vice-Chancellor again. In all four cases, the package consisted of extracts from the Accountability in Research article, and photocopies of various letters-there was no covering letter, or any kind of explanation. In all four cases the mystified recipients passed the communications on to me and - - in the absence of a covering letter--there seemed no call for a reply. Some weeks ago I received a postcard from Berg from (unaccountably) the World of Ford Museum, Detroit, postmarked 24 February 1994. I quote the text in full below, exactly as I received it. The capitalization and underscores are all Berg's, although I have used bold font for the word that Berg wrote in red ink for emphasis. IF YOU WOULD PLEASE BE SO KIND AS TO INFORM ME OF THE NAME OF THE ACADEMIC INSTITUTION(S) THAT AWARDED YOU A COLLEGE DEGREE, I WILL WRITE TO THEM AND REQUEST THAT YOUR COLLEGE DEGREE(S) BE WITHDRAWN/K2ANCELLED. IN CALENDARYEAR1984,ONE OF THE FIRSTPEOPLE IN THE UNITED KINGDOM THAT I WROTE TO ABOUT MY DISCOVERY OF THE BABBAGE-QUETELETLETTER OF 27 APRIL 1835 ON THE ANALYTICALENGINE, WAS HIS ROYAL HIGHNESS, PRINCE PHILLIP, THE DUKE OF EDINBURGH, CHANCELLOROF CAMBRIDGE UNIVERSITY FOR CHARLES BABBAGE WAS A LUCASIAN PROFESSOR OF MATHEMATICSTHERE. YOU ARE ACCUSED OF CONTRIBUTING TO THE FURTHERDECLINE OF SCIENCE IN ENGLAND (A CHARLES BABBAGE TITLE) LEADING TO A TOTAL LOSS OF STANDARDS OF ACADEMIC INTEGRITY IN RESEARCH SCHOLARSHIP OF THE ENTIRE GLOBAL SCIENTIFIC / RESEARCH / EDUCATION ESTABLISHMENT. THE WORKS OF CHARLES BABBAGE CONTAINS AN ESSAYABOUT FRAUD AND MISCONDUCT IN SCIENCE, WHICH DESCRIBES THE PRACTICE OF "CUTTING AND TRIMMING", ONE OF THE MANY GAMES THAT HAVE BEEN PLAYEDUNDER YOUR EGREGIOUS EDITORSHIP-IN-CHIEE HERMAN BERG MATHEMATICALSECOND SIGHT 18964 PINEHURST STREET DETROIT, MICHIGAN 48221-1961 UNITED STATESOF AMERICA I do not understand why Berg wrote this letter. The facts he requests are in the public record (for example in the CV accompanying articles and books I have written) and he is a capable enough researcher to find them for himself. I have shown this postcard to two senior scholars in the history of computing, who have been the recipient of Berg's letters, and they both confirm that it is fully representative of Berg's style. More seriously, in spite of my corrections to the catalogue of errors in the original article in Accountability and
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 (~)1994 Springer-Verlag New York
27
MATH INTO TEX A SIMPLE INTRODUCTION TO AMS-LATEX G. Griitzer, Universityof Manitoba George Gr~itzer'sbook provides the beginner with a simple and direct approach to typesettingmathematics withAMS-L^TEx. Usingmany examples, a formula gallery, sample files, and templates,Part I guides the reader through setting up the system,typing simple text and math formulas, and creating an artide template. Part II is a systematicdiscussion of all aspects of AMS-LATEXand contains both examplesand detailed rules. There are dozens of tips on how to interpret obscure ~errormessages,"and how to find and correct errors. Part III and the Appendicestake up more spedalizedtopics, from customizingAMSLATEX to the use of PostScript fonts. Even with no prior experience using any form of TEX,the mathematician, scientist, engineer, or technical typist, can begin preparing articles in a day or two usingAM&LATEX.The experiencedTEXerwill find a wealth of information on macros,complicatedtables, postscriptfonts, and other detailsthat permit customizingthe LATEx program. Thisbook is truly unique in its focus on getting started fast, keeping it simple,and utilizing fully the power of the program.
CONTENTS: Introduction 9 Part I: A Short Course 9 The structure of AMS-LATEX9Typingyour firstarticle 9 Part H: A Leisurely Course 9Typing text 9 Typingmath 9 The Preamble and the Topmatter 9 The Body of the article 9 The Bibliography9 Multiline math displays 9 Displayedtext 9 Part HI: Customizing 9 Customizing A/V/S-LATEX9 TEX macros ~ Appendices (A-G) ~ Bibliography ~ Index 1993 294 PP., PLUSDISKEITE SOFTCOVER $42.50 ISBN0-8176-3637-4
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28
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
Research, Davis has failed yet a g a i n to give m e an o p p o r tunity to r e a d w h a t he has written before rushing into publication. For that I have to t h a n k the editor of Mathematical Intelligencer. Even worse, the m a n u s c r i p t I h a v e seen s h o w s all the signs of h a s t y a n d careless revision s u b s e q u e n t to m y letter to the Editor of Accountability and Research. Thus, the figure titled "Check It Out" - - w h i c h p u r p o r t s to s h o w that I used Berg's c o p y of the BabbageQuetelet letter in the Works of Babbage-- shows nothing m o r e t h a n that I used a copy of the letter (a c o p y which, as I h a v e stated, was obtained c o m p l e t e l y i n d e p e n d e n t l y of Berg). In another place I read that "Bromley took responsibility for that part of the v o l u m e " ; two pages later this is flatly contradicted b y the statement that "BromIey took no p a r t in the detailed editing." This is s l o p p y and self-contradictory, and the history of m a t h e m a t i c s c o m m u n i t y deserves better. Department of Computer Science University of Warwick Coventry, CV4 7AL UK
Michael Davis Responds Let m e m a k e three brief points, lest m y silence be taken as evidence that I h a v e changed m y v i e w on a n y substantial question: 1. Campbell-Kelly gives excuses for ignoring Berg's complaint. T h e y are not such as to s h o w his conduct w a s wise or just. I sought to s h o w it w a s not. 2. I do not k n o w of " n u m e r o u s factual inaccuracies" in m y earlier article. Bromley p o i n t e d out one: the Works of Babbage did not p u r p o r t to be the complete works. C a m p b e l l - K e l l y ' s letter in Accountability in Research gives "refutation" only in this sense: to m y report of events as Berg experienced t h e m he gives his report of w h a t he thought h a d h a p p e n e d . But I m a d e it clear I w a s setting forth a complaint; I did not and do not aspire to adjudicate it. 3. If I d i d tell all sides, necessarily at greater length, the result m i g h t still not please Campbell-Kelly; for even after r e a d i n g his "refutations" I continue to believe he was in the wrong.
Student Questions You Love to Hate Monte J. Zerger
Questions like the following usually induce two different responses in me, one following on the heels of the other. First comes the "revenge response," the desire to repay the student with a terse, annihilating reply. This response wells up from the shadow side of my being, and threatens to erupt in an ugliness not becoming of a college professor. So, I suppress i t - - except on extremely rare occasions-- and feigning joviality, settle for a sensible and harmless one. See if you can place yourself in the following scenarios, and recognize the reaction.
QUESTION: "Did I miss anything important Wednesday?" REACTION: You catch yourself just in time to stop these words, dripping with sarcasm, from rolling off the end of your tongue: "No, it was just like every other day."
1. SITUATION: You have just presented a marvel of abstract mathematics whose beauty does things for you that only Rembrandt and Beethoven do. You are transported far above any thought or concern for possible applications to the imperfect, mundane world below. Then from the back of the room c o m e s - QUESTION: "Where would I ever use this?" REACTION: Valiantly, you fight back the impulse to put the student where he deserves-by replying with something frivolous like "On your wedding night" or "While being held in a Mexican jail on trumped-up charges." 2. SITUATION: You are sitting in your office five minutes before your class is to begin. A student who missed the previous lecture bounces into your office, and inquires with an annoying air of nonchalance and frivolity, THE MATHEMATICALINTELLIGENCERVOL- 16, NO- 4 ~)1994 Springer-VerlagNew York 2 9
3. SITUATION: The lecture was one of y o u r best, but somehow y o u manage once again to misjudge the looks on the faces of the students, and naively believe that a significant transfer of mathematical knowledge has been taking place. You are riding on "cloud nine," eagerly awaiting the chance to field questions about it from the class. W h e n y o u ask for questions, a student in the front row responds, QUESTION: "Will this be on the test?" REACTION: The "little guy" in y o u r head goes berserk. "This is w h a t I get in return!" y o u hear him say. "No excitement, no awe, no nothing, but concern about the next test." It frequently causes you to grip your eraser until your knuckles turn white, while y o u fight the impulse to throw it at him.
A. Lasota, Silesian University, Katowice,Poland; M. C. Maekey, McGillUniversity, Montreal, Que.
4. SITUATION: You are seated at your desk wading through a seemingly endless (and endlessly disappointing) stack of exams. Papers and books from a pet project you've had to push aside are strewn everywhere. Buried in work, y o u don't even hear the student seeking help approaching your open door. But y o u do hear, and all too clearly,
Chaos, Fractals and Noise Stochastic Aspects of Dynamics
QUESTION: "Are you busy?"
2nd ed. 1994. XlV, 472 pp. 48 figs. (Applied MathematicalSciences,Vol.97) Hardcover $ 49.00 ISBN0-387-94049-9 This book treats a variety of mathematicalsystemsgenerating densities, ranging from one-dimeusionaldiscretetime transformations through continuoustime systemsdescribedby integro-partialdifferentialequations. Exampleshave been drawn from a variety of sciencesto illustratethe utility of the techniquespresented. This materialwas organized and written to be accessible to scientists with knowledgeof advanced calculusand differentialequations. Variousconcepts from measure theory, ergodictheory, the geometry of manifolds,partial differentialequations,probability theory and iarkov processes, chasticintegrals and differential equations are introduced. The past few years have witnessed an explosivegrowth in interest in physical,biological,and economicsystemsthat could be profitably studied using densities. Due to the general inaccessibilityof the mathematicalliterature to the non-mathematician,there has been little diffusionof the concepts and techniques from ergodic theory into the study of these "chaotic"systems.This book intends to bridge that gap.
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30 THE MATHEMATICAL INTELL[GENCER VOL. 16, NO. 4, 1994
REACTION: Even though this happens time and time again, y o u are invariably so completely overcome by the incredible nature of the query that you can only smile wanly and reply in defeat, "No." You can probably add to the list. Others I've collected include: 5. " H o w do you do word problems?" I must admit I have in moments of weakness succumbed to temptation and responded, "While seated in the lotus position, meditating before a candle in a bare room." 6. " H o w come I understand it w h e n you're doing it in class, but w h e n I get home I can't do it?" 7. "What w o u l d I need to get on the final to get a B in the course?" This one typically comes a day or two before the final. 8. "What is calculus anyway?" Don't get me wrong. I love teaching, and I love m y students. But recently, I have been contemplating a totally selfish action, that of amending all m y syllabi to include another, innocuous-looking requirement for successful completion of the course. It would simply refer students to an attached page of questions, headed by a blunt instruction: "DO NOT ASK ANY OF THE FOLLOWING QUESTIONS."
School of Science, Mathematics, and Technology Adams State College Alamosa, CO 81102 USA
A Gallery of Constant-Negative-Curvature Surfaces Robert McLachlan
To study surfaces one must first choose how to represent them. Unfortunately, there is no canonical representation; instead, there is a long catalog of possibilities, each useful in different circumstances. Writing the surface as a graph [e.g., z = z(x, y)] leads to messy formulae for intrinsic quantities such as curvature; leaving the coordinates completely arbitrary includes redundant information. But when dealing with surfaces of constant negative curvature, one coordinate system stands out as being particularly apt. It goes by the name of parametrization by Chebyshev nets. Actually, any surface can be locally covered by a Chebyshev net. Before I describe this coordinate system, a quick review of the geometry of surfaces is called for. Although this subject used to be part of every mathematics education, and before that, one of the centerpieces of mathematics, it is not widely taught today. In my case I found myself in the thick of the Einstein tensor and the symplectic 2-form without having laid eyes on any of the equations that go by Gauss's name. Take arbitrary coordinates x and y on the surface, which occupies the points r(x, y) E R 3. Define two tangent vectors and one normal vector:
where F ~ are the Christoffel symbols F ~ = 19)'7" (g..,~ + g"7,. - guv,-~), g"~ = ( g . . ) - ] , and repeated indices are summed on. The six quantities in (2) are not all independent. They are related by three consistency conditions for the PDEs (3) - - the Gauss--Codazzi equations: K(det g) 2 - - l g l l , y y q- gla,xy -- l g22,xx =
g1 g l l , x
gl2,x -- ~1 g l l , y
gl2,y -- 89
gll
g12
lg22,x
gl 2
g22
~1 gu ,y
lg22,x
89
911
g12
189
g]2
g22
0 -
,
h.x,. - r~.h7;, = h..,;, - r~';&7.,
(4)
(5)
T1 X T2
r. =rp,
n-
[Ta~--~IT1 x
(1)
where Greek indices range over # = 1, 2 and the subscripts ",1" and ",2" denote O/Ox and 0/09, respectively. (The notation follows Spivak [17], Chap. 2.) Note that n is a unit vector, but the % are not necessarily unit vectors. We have the metric or first fundamental tensor g ~ and the second fundamental tensor ha~: g.v = % ' r ~ ,
h.~ = n . % , . = n . r , . .
(2)
As one moves along the shrface, the tangent and normal vectors change according to the Gauss-Weingarten equations,
T'~,u = T)~F~u q-nh.u,
(3) THEMATHEMATICALINTELLIGENCERVOL.16, NO. 4 (~)1994Springer-Verlag New York 31
where (5) gives independent information for (~,,),, #) = (1, 1, 2) or (2, 2, 1) only. Of the three remaining degrees of freedom, two are due to the arbitrary coordinates, leaving o n e - - w h i c h is expected because the surface could be written, e.g., r = (x, y, z(x, y)). The principal curvatures ~1, n2 of the surface at a point are the eigenvalues of the matrix g~;~h;~ there; their product n1~2 = det h/det g is the Gaussian curvature K. To construct a Chebyshev net physically, take a piece of nonstretch fabric that is loosely woven so that the angle between the threads can change. N o w drape it over the surface so that the warp and weft of the fabric become coordinate lines on the surface. Because the threads cannot stretch, all coordinate lines are still parametrized by arclength--gll = g22 = 1 - - b u t g12 is arbitrary. The metric, therefore, takes the form ( g.. =
1
cosw)
cos w
1
,
(6)
where w(x, y) is the angle between two coordinate lines. Perhaps it should be called a Chebyshev fishing net to emphasize that the knots don't move. Such a coordinate system can always be constructed locally, starting from any two intersecting curves [16], p. 202. The classic reference for the introduction of Chebyshev nets is [18], which is translated in full on page 37. It sounds like M. Tch6bichef gave a good seminar, but it's a pity that this original treatment was never published in more detail. His rubber ball may have looked something like the one in Figure 1; in the spirit of the 19th century, I omit the details.
Equation (4) may be written in a clumsy form, but it expresses Gauss's theorema egregium ("remarkable theorem"): the Gaussian curvature K = det h/det g, although apparently depending on both g and h, is, in fact, an intrinsic property of a surface; that is, it depends only on the metric g. Substituting the metric (6) into (4) gives the pleasingly simple formula 020d - - -
Ksinw.
Ox Oy
(7)
When K = - 1 , as for a surface of constant negative curvature, this is the famous sine-Gordon equation. In this context, it apparently appeared for the first time in the work of Hazzidakis [7]; he commented that it has solutions of the form w = ~(x + y) (see the pseudosphere, p. 34) but did not write them down. An asymptotic direction v ~ is one satisfying v~ ht,~v ~ = 0. If the Gaussian curvature K, and hence det h, are negative, then this equation has two solutions vt' at each point. Integrating gives two families of asymptotic lines, each tangent to an asymptotic direction everywhere. When these are taken as coordinate lines, h l l = h22 = 0. Asymptotic lines have other nice properties: unless straight, they have normals tangent to the surface, and their torsion is x/ZK, and hence constant when K is constant ([16], p. 100). N o w for the connection: on a surface of constant negative curvature, the asymptotic lines form a Chebyshev net. More precisely, one can choose to parametrize any two intersecting asymptotic lines by arclength; a calculation [17], p. 365 then shows that they all are. N o w solving (5) gives h12 ---- ~ sin w, s o ( h=
0
v / Z K sin w )
x/-ZK sin w
0
.
(8)
The number of functions specifying the surface has been reduced to the minimum possible, namely, one. Given consistent g,v and h , . a unique surface may be reconstructed [16], p. 146; so surfaces of constant negative curvature are locally in a 1-1 correspondence with solutions of the sine-Gordon equation. Unfortunately this correspondence really is only Iocal. A drawback of Chebyshev nets is that they sometimes can't be extended indefinitely over a surface, due to Hazzidakis's formula, a special case of the GaussBonnet theorem [7]. Consider a coordinate rectangle X : Ix1, X2] X [Yl, Y2] corresponding to a piece of the surface r(X). Then
/j o.. dx dy
= -
//.
K sin w dx dy
= - ff KdA--KT. J Jr( x)
Evaluating the left-hand side g i v e s Figure
1.
32 THE MATHEMATICALINTELLIGENCERVOL.16, NO 4,1994
- - K T = W(Xl, Yl) - od(x2, Yl) + 0d(X2, Y2) - W(Xl, Y2).
For a well-defined net, the angles w satisfy 0 < a; < 7r, so the total curvature K T of the piece r(X) m u s t be less than 27r in magnitude. [When K = - 1 , the area of r(X) m u s t be less than 27r.] Stoker ([16], p. 199) comments that "tailors have learned this fact from experience." However, the second example below is of an infinite surface covered by a single Chebyshev net, so perhaps tailors' experience is not sufficiently broad. The sphere has total curvature 47r, and, sure enough, can be covered by exactly two Chebyshev nets, each covering a hemisphere. The s ~ d y of surfaces of constant negative curvature was inaugurated by Ferdinand Minding in a paper published in Crelle's Journal in 1839. The content of the paper is in fact much broader, but it provides in particular the first set of concrete examples of surfaces of negative curvature, including the pseudosphere and the helical surface given below in (9). These can be found directly by making the ansatz of helical shape. But the earliest derivation of nontrivial solutions of sine-Gordon (one and two solitons, periodic solutions in elliptic functions, and some wave packets) is that of Seeger, et al. [13] in 1953. Perhaps the reason that so m a n y exact solutions were able to be found is that the sine-Gordon equation is completely integrable. Such integrable soliton equations have an extremely rich mathematical structure and can be solved by the method of inverse scattering [2]; this was first done for the sine-Gordon equation by Ablowitz et al. in 1973 [1]. Segur [14] has recently written an informal review of the practical importance of integrability. Later on, I'll give a plausible reason for just why such a remarkable equation should crop up in differential geometry. N o w that so m a n y exact solutions of the sine-Gordon equation are around, m o d e r n computer tools make it a pleasure to while away a few hours and see to which surfaces they correspond [3]. Of course they can't be complete, or embed in ~3, but if we don't m i n d a few cusps or self-intersections, then we're in business. It's a famous result of Hilbert [8] that complete surfaces of constant negative curvature can't embed in R 3. In fact, singularities will always form in a surface corresponding to a single smooth solution of sine-Gordon: y o u can see trouble coming, because w h e n a; = nTr the coordinate lines are tangent to one another. Thus, a line in the coordinate plane along which w = nTr corresponds to a curve along which the surface has only one asymptotic line, instead of the two implied by negative curvature; in fact, the surface has only one tangent vector there. If y o u think of the model of sine-Gordon as the continuous limit of a line of pendula coupled by springs, it's clear that there is no solution of sine-Gordon on I~2 that never takes any of the values nTr. The principal curvatures of the surface are cot w + csc w, so that as a; --, nTr, they tend to zero and infinity. The pseudosphere (the first example below) has a y = +xB/2-type cusp at the singularity. However, it turias out that the singularity at a; = nTr in the Gauss-Weingarten equations (3) is removable, so
their solution exists everywhere. The coordinate lines are smooth everywhere and just blast through their points of tangency. So although it w o u l d be possible to piece together different surfaces smoothly along the singular lines, there is a natural continuation through them, which I've used here. As for the self-intersections, "with the sine-Gordon equation, you're living on miracle street" [15], so they might not be too bad. The surface corresponding to a given w m a y be found by integrating the Gauss-Weingarten equations. In the present case, from (3), (6), and (8), they are
-~y
= _ cSC _cotO cscO ino~ ) ( 0 0
,
~
,
--Wy
CSC
-csc w
W
Wy
cot
W
cot w
0
0
In the figures that follow I have integrated these equations numerically, and then integrated (1) to find the surface. The only special care required during the integration is to project the tangent vectors so that 7"1 9 7-2 = Od, 7-1 " n = 7 - 2 " n = 0 , and 17-11 : 17-21 = In ] ~--- 1 exactly; this ensures that the singularities at w = nrr are removable and can be integrated through. You m a y note that the surfaces all look fairly similar locally. In fact, they are all i s o m e t r i c - - t h i s is Minding's theorem of 1839. But they're not identical. A surface of constant negative curvature can be analytically immersed in Hilbert space such that neighborhoods of any two points are congruent (as is t r u e for a plane, sphere, or cylinder in R3); but this cannot be done, even with a C O immersion, in any finite-dimensional Euclidean space [9]. Here are some special solutions of the sine--Gordon equation, and the constant-negative-curvature surfaces to which they correspond:
1. The Pseudosphere. Instead of the general approach using Chebyshev nets, one can look directly for those surfaces of revolution that have constant negative curvature. A special case is the pseudosphere (Fig. 2), which is y = x/1 - x 2 - - cosh -1 (I/x) rotated around the y-axis [17], p. 239. Eisenhart [4] has an early illustration. Interestingly, its area is 27r and its surface area is constant in x, that is, d A = 27r dx (try this out on your calculus s t u d e n t s - - o r the inverse problem!). In the Chebyshev net parametrization, the pseudosphere corresponds to an X-independent solution of the sine-Gordon equation in the form ~ 2 ~ d / I g T 2 - - ~20d/OX2 = sin a; (here X = x - y, T = x + y); in fact, the homoclinic orbit connecting a; = 0 and w = 27r is w = 4 t a n -1 e x+u. THE MATHEMATICAL INTELL1GENCER VOL. 16, NO. 4, 1994
33
Figure 2.
Figure 4.
The coordinate lines are s h o w n in Figure 3; one can see h o w they are parametrized b y arclength and h o w their angle traces out the homoclinic solution of the (cO2w/OT2 = sin ~;) p e n d u l u m , passing through 7r at the cusp. The ",[ " points correspond to o; ~ 0 and 0; --, 27r, i.e., x + y --* + ~ . In this limit the two sets of coordinate lines are asymptotically tangent to one another, but the singularity is never reached; because (from the G a u s s Weingarten equations) "1"1x~ Tly~ "t-2x~ and T2y --~ 0, the spike must tend to infinity. If y o u squint a bit, y o u might even believe that the coordinate lines" normals lie in the surface. 2. 1 - S o l i t o n T r a v e l i n g Wave. The pseudosphere solution is only a special case of the 1-soliton solution of sineGordon: w = 4 tan -1 e ~,
Figure 3.
34 THEMATHEMATICAL INTELLIGENCERVOL.16,NO.4,1994
~ = ~?x + Y/~I.
As ~ changes from 1, the cusp no longer closes on itself and becomes a helix. You can see the solitary wave traveling u p the spiral here for 7/ = 2 (Fig. 4). There is still a self-intersection on the central axis; the part h i d d e n from view is like the long spike on the pseudosphere. At
first sight this spiral violates the bounded area of coordinate rectangles mentioned earlier; but since the cusp lies on ( = 0, a coordinate rectangle not intersecting this line does indeed have area _< 2zr. To get infinite area you need a diagonal strip 0 al such that v e c t o r (a~,b2,... ,bN) is in S, but this yields a sum greater than a n y of the sums already accounted for; thus, an (N + 1)st sum. 9 We r e m a r k that N + 1 is "best possible," as illustrated by the example of the set of all N-vectors all of whose coordinates are either a or b. The s u m then depends only on the n u m b e r of b's, which can v a r y from 0 to N. As an exercise, the reader m a y try to show that the three-player game with forehead n u m b e r s 2, 2, 2 and blackboard n u m b e r s 6, 7, 8 terminates after 15 beeps.
References 1. J. H. Conway and M. S. Paterson, A Headache-Causing Problem, in privately published papers presented to H. W. Lenstra on the occasion of the publication of his Euclidische Getallenlichamen. 2. J. M. Lasry, J. M. Morel, and S. Solimini, On knowledge games, Revista Matematica de la Universidad Complutense de Madrid 2(2/3) (1989).
"Chaos Unimportant" Claims Topologist Ethan Akin
With blackboard to the left and projector screen to the right, the analyst began her lecture, " W h y Chaos Is Common": "We model a discrete-time dynamical system by iterating a continuous m a p F : X ~ X , where X is a metric space. If there exists a finite, real constant L such that
d(F(x), F(y)) < L d(x, y)
(1)
of picture is w r a p p e d in on itself and enclosed in a compact space. Consider the covering m a p from the reals to the unit circle in the complex plane: z = e ix. With L = 2 the map is transformed from F(x) = 2x to F(z) = z 2. Typically, it is this stretching in at least some directions that leads to the computational problems associated with chaos..."
for all x, y E X, then the smallest such L is called the Lipschitz constant and the m a p is called a Lipschitz map. The
Before she could introduce the Thom torus map and proceed as planned to some delicate c o m p u t i n g , the speaker was rather rudely interrupted by a topologist in the audience:
Lipschitz constant is usually larger than l, the exceptions being easily understood maps, e.g., contractions [L < 1] or isometries like rotations [L = 1 with equality in (1)]. From (1) it easily follows by induction that
"It seems to me that you are distressing these people for no good reason. The problems y o u have described can be dealt with easily. Behold:
d(Ft(x), Ft(y)) 1, the b o u n d s L t tend to infinity exponentially. Thus, even if two distinct points x, y are very dose, their orbits m a y move apart very fast, as d(Ft(x), Ft(y)) moves a w a y from zero at an exponential rate. Furthermore, non-Lipschitz maps, lacking even estimate (1), m a y behave even worse. Of course, (2) is just an inequality bounding the worst that can happen. But the worst is quite common. For example, if X is the real line and F is multiplication by L > 1, then the estimates in (1) and (2) become equations for all x, y E X. This linear case doesn't feel very chaotic. The universe is just expanding outward from zero at a regular rate and points are moving apart correspondingly. The serious examples arise w h e n this kind THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 (~) 1994 Springer-Verlag N e w York
45
THEOREM 1. Let (X, d) be a metric space and F : X ~ X be a continuous map. There exists a metric dF on X , yielding the same topology as d and satisfying: for all K E T and X, B E X
dHx, y) 0 such that d( x, y) < 6 implies dF( x, y) < c. This, together with (9), shows that when {F t } is (uniformly) equicontinuous, the metric defined by (8) is topologically (resp., uniformly) equivalent to d. QED The concept of uniform equicontinuity is not altered when we replace d by a uniformly equivalent metric. So if (7) holds for some metric dF, uniformly equivalent to d, then the sequence of iterates is uniformly equicontinuous with respect to d. Thus the converse of Theorem 2 is true in the uniform case. It may already be clear now how Theorem 1 arises. Regarded as a map to C(T; X), OF is continuous only in the special case described by Theorem 2. But there is another, coarser, natural topology on C(T; X), that of uniform convergence on compacta, which is just the product topology when we regard the set of sequences as the product of countably many copies of X. To define a convenient metric we choose a function p : T --* [0, ~ ) satisfying
p(o) = o, tl > t2
in T ~ p(t~), > p(t2), lira p(t) = oo.
t----~oO
(10)
For example, p(t) = t will do. N o w define
b ( a , #) = sup{min[d(a(t), #(t)), 1/p(t)] : t e T } (11) with the convention that'min(a, 1/0) = a. So we have fore > 0
D(a, fl) 0, there are only finitely many t's such that p(t) < ~-1 because p(t) tends to cc with t. Thus, given e > 0 and x E X, we can choose 6 > 0 so that d(x, y) < 6 implies d(Ft(x), Ft(y)) < for all the t's satisfying p(t) < e -1. Furthermore, if F, and so also each iterate, is uniformly continuous, we can choose 6 independent of x. Thus, the continuity of F implies that dE is topologically equivalent to d, and uniform continuity yields uniform equivalence.
Proof of Theorem 1. Choose p(t) = log2 (1 + t) and define dF via (13). We have already shown that dF is topologically equivalent to d and is uniformly equivalent when F is uniformly continuous. We prove implication (3). From (12), dF(X, y) < 1 / ( K + 1) implies 1
for all t < 2 K+I - 1.
(15)
Let s _< 2 K. From (15) we have 1 d(Ft(F~(x)), Ft(F~(y)) < -~
Let us compute an example to illustrate the relationship between dE and d. We will use multiplication by L = e o n X = ~ . T h u s , Ft(x) = e t x a n d w i t h d = d(x, y) = tx - Yl, d(Ft(x), Ft(y)) = etd. It is a bit easier to use (13) when t is allowed to vary over the positive reals. In effect, we are doing the computation for the real flow in which the map is embedded. With d > 0, the function etd is increasing while 1/log2(1 + t) is decreasing. Denote by t* the time when the two graphs cross. By (13) the common value is then dR. So we have
dR = et'd = 1/Iog2(1 + t*) , or, equivalently,
dR(X, y) > dR(x, y) > d(x, y).
d(Ft(x)' Ft(y)) < K-4- 1
Remark. Given an arbitrary L > 1 we can get a metric uniformly equivalent to dF with respect to which F has a Lipschitz constant less than L. This easily follows from (3) when we replace dF by min(dF, c) with e > 0 sufficiently small.
f o r a l l t < 2K - 1 (16)
d = dEe -t*
with
t* = 2 (1/d~) - 1.
(18)
Regarding d as a function of dR we see that as dR tends to zero, d moves rapidly toward zero. Compare this with the much slower approach of the classical function obtained by using t* = (1/dR) 2 instead. Thus, from the topological viewpoint, the problems of chaos are only asymptotic, or long-run, concerns. Our rude topologist was technically correct that difficulties due to diverging orbits can be deferred arbitrarily far out toward infinity by replacing the metric. However, it is the analyst who is right about real problems. Furthermore, her reply directed attention to the key issue, the appropriate notion of equivalent metric. By the very fact of being "computational" our problems tend to be built using real numbers and arrive equipped with a metric or class of metrics naturally related to the underlying algebraic structure. The appropriate category of equivalence is probably local Lipschitz (or local H61der) equivalence. Perhaps this is w h y Hausdorff dimension, though it is not a topological invariant, has proved to be a useful tool. Metric problems require correspondingly delicate invariants. Our analyst gets the last word: "Your theorem may be true but you are nonetheless categorically incorrect."
because 1 / K > 1 / ( K + 1) and t + s < 2 K+I " 1. Using (12) again we see that (16) implies 1 dF(FS(x), FS(y)) n, then f is not one-to-one. Tvloregenerally, if m > kn for some positive integer k, then there is some element b E B that has at least k + 1 preimages under f, that is, IfM ((b})l > k + 1.
Come, come, second author! A box is only a box, and the bedraggled creatures strutting in the campus quad outside my window are not more elegant for being called Columba livia. A combinatoricist of my acquaintance recently visited a prestigious Eastern institution. He encountered in the hall an eminent Fields medalist, not known for his humility. After the initial de rigeur declarations of research interests, the conversation took a combative turn. "Combinatorics!" jeered the medalist, who himself worked in the far reaches of differentiable manifolds. "So much fuss over it, and it's all just counting. It's all trivial." "I don't think that's fair," was the retort. "But it's true! Let's bet dinner. Tell me something in combinatorics I can't do in one hour and I'll buy dinner." "Let's see," mused the combinatoricist. "There's the Erd6s-Szekeres result: any sequence of n 2 -}- 1 distinct real numbers must contain a subsequence of n + 1 terms that is either increasing or decreasing." The combinatoricist told me that the subsequent dinn e r - purchased by the medalist and held at a famous waterfront restaurant--was altogether exceptional. The proof of the result uses the pigeonhole principle. 3 After the aforementioned lecture, during which the speaker had given Niven's justly celebrated 4 proof of
3 Kenneth A. Ross and Charles R. B. Wright, DiscreteMathematics, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ, p. 265. To be fair to Author #2, I admit that the proof requires that author's more general form of the principle. 4 The half-page proof was recently reprinted in The CollegeMathematics Journal (November 1991). THE MATHEMATICALINTELLIGENCERVOL.16,NO. 4, 1994 69
the irrationality of 7r, I had a dispute with a colleague that revealed just how much the aesthetic bearings of mathematicians can differ. The proof, I maintained, was one of the prettiest ever concocted. My colleague strongly disagreed. "It's a shameless tricK" he scoffed, "and it's totally unmotivated." "Exactly!" I countered. "And that's why I love it." It flatters our passion for security and our desire for control to believe that brute endurance and a hefty bag of theorems will help us scale any summit. Mathematics, though, is a devious business, and to think we can make progress without the sporadic flashes of totally unmotivated imagination of some of our colleagues is to believe that good will and an open heart will procure our survival on the streets of N e w York City. In their introduction, the authors state that what separates this book from the usual problem books is the difficulty (I would put it, sophistication) of the problems. It's true that among the results presented are well-known results-- the strong law of large numbers for orthogonal sequences and the density of trajectories of ergodic mappings, for instance. What really distinguishes the book is the allure of the problems and the satisfaction their solutions provide. The authors give solutions, though sometimes abbreviated, for nearly all the problems. Most of the solutions are elegant and very brief, one or two sentences frequently. When no solution is provided, it generally means the solution is a trivial consequence of solutions immediately preceding. The following is one of a set of related problems: Let an be a positive sequence, An = al + a2 +.. 9+ an, c~ = -1 -1 a-~1 q- an+ 1 -b an+ 2 q- . . . . Show that An = O(an) if and only i/as = O(a~).
Suppose the function f : N --, I~ satisfies f ( n + 1) > f ( f ( n ) ). Show that f ( n ) = n.
The book offers substantial support for our classroom coverage of these subjects, as most current textbooks treating these ideas contain insufficient exercises. I don't usually provide a detailed description of the contents of a book I review, but I want to give the reader a sense of this book's astonishing breadth. Chapter I. Introduction: sets, inequalities, irrationality Chapter II. Sequences: computation of limits, averaging of sequences, recursive sequences Chapter III. Functions: continuity, semicontinuity, differentiable functions, functional equations Chapter IV. Series: convergence, monotonicity, computations of sums, series of functions, trigonometric series Chapter V. Integrals: improper integrals, computation of multiple integrals Chapter VI. Asymptotics: asymptotics of integrals, the Laplace method, asymptotics of sums, asymptotics of implicit functions and recursive sequences Chapter VII. Functions: convexity, smooth functions, Bernstein polynomials, almost periodic functions and sequences Chapter VIII. Lebesgue measure and the Lebesgue integral: Lebesgue measure, measurable functions, integrable functions, the Stieltjes integral, e-entropy and Hausdorff measures, asymptotics of higher integrals Chapter IX. Sequences of measurable functions: convergence in measure and almost everywhere, convergence in the mean, the law of large numbers, the Rademacher functions, Khintchine's inequality, Fourier series and the Fourier transform Chapter X. Iterates of transformations of an interval: topological dynamics, transformations with an invariant measure
The idea is to get rid of the sums, to convert them into something tractable. It's often a good tactic to replace an "O" sign with a generic bounded sequence or an "o" sign with a generic null sequence, quantities that are more easily exploited. Doing so here and then differencing gets rid of the sums in An and ~n. A host of important results Browsing through the b o o k one finds results of strikin summability theory is amenable to the same approach. ing simplicity, The point is: Order signs are a notational convenience but, in proofs, a technical obstacle. Let ak be the first digit in the decimal expansion of 2 k, k = Almost no background is necessary to solve the prob- 0 , 1 , 2 , . . . , i.e.,ao = 1,al = 2, a2 = 4, a3 = 8 , a 4 = 1, etc. lems in the first seven chapters, and Rudin's Principles of What is the frequency of the digit p c {1,2, 3 , . . . , 9} in this Mathematical Analysis (McGraw-Hill) and Kolmogorov sequence? and Fomin's Introductory Real Analysis (Dover) provide most of the information necessary to do other problems. that require heavy-duty mathematics (ergodic theory) to Even to understand the formulation of some problems solve. (ANS: log10(1 + l/p).) I once told my students that in Chapter X requires a secure grounding in ideas such mathematical questions should come attached with tags, as the Lebesgue convergence theorems, the Riesz rep- like those that counsel us on the proper care of garments: resentation theorems, convergence in measure, and con- Warning! For best results, this statement should be gently vergence almost everywhere, even when these tools may tumbled with ergodic theory. not be used explicitly in the solutions. We must continually relearn that our intuitive percepThe book is quite current in its concerns and has ma- tion of a problem may have little to d o with planning an terial on topological dynamics, entropy, Hausdorff mea- economical method of attack, as the following example sures, iterates of functions: shows: 70
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO, 4, 1994
Show that for almost all numbers in (0, 1) O's and l's occur equally often in their binary expansions. Let e~(x) be the nth digit in the binary expansion of x C (0, 1] and let aN(x) = (l/n))-~=1 ~k(x). Using a mean convergence argument, one can easily show that an(X) ~ 1/2 a.e., and this implies the stated result. At first glance, mean convergence has nothing to do with things, s The translation in the book, provided by H. H. McFadden, is vibrant and supple, one of the best I have ever rea d It is particularly invigorating when it allows the occasional whimsy of the Russian authors to shine through:
After falling into a 1000 dimensional space, Alice was asked to make thin glass hoops (spherical zones) in such a way that their width is 1/10 of their diameter. "It couldn't be simpler," said Alice. "You need to blow up spheres, and then cut off the excess." What percentage of glass is wasted with this technique? I emphasize that with the series of AMS translations of mathematical monographs of which this volume is a member, the buyer is very much at risk. The quality of translation ranges from the nearly flawless (this book) to the sheer underedited horror of Theory of Entire and Meromorphic Functions, Vol. 122, by Zhang Guan-Hou. Some passages from that book are still causing me to shake my head: "We may also differentiate further correspondingly the lower order #," or "but probably with the exception of at most two values." I am not a purist. I believe that communication must be the numen of mathematical exposition, but in Vol. 122 the brutalities visited on language sometimes make a hash of the mathematics too. Many Western mathematical writers seem to believe that problem books are inadequate instructional devices, that students simply can't learn substantial mathematics from them. Having just taught an undergraduate course in combinatorics, I disagree. It's true that in combinatorics there are significant general principles, such as the inclusion-exclusion principle or the previously mentioned pigeonhole principle, but each problem generally has its own gestalt. Considerable ingenuity and ad hoc manipulation may be required to set up the problem so one of these principles can be used, and the problems are as heterogeneous as those in any problem book. It surprised me to find that, through working problems, my students somehow developed a feel for how to attack unrelated problems. Like swimming, counting skills can't be taught through exposition alone.
5 This is a well-known problem; there are purely number-theoretical arguments, or measure-theoretic arguments which do not require mean convergence (see The Theory of Numbers, by G. H. Hardy and E. M. Wright, Chap. 9) but those proofs are considerably more complicated.
The greatest Russian textbook writers always have been eminent mathematical stylists. Determined not only to teach mathematics but to teach students to enjoy it, they believe in the utility of problems. Problem books like the one being reviewed are a Russian institution, and a natural outgrowth of the long-standing Russian regard for mathematical pedagogy. Besides the complex variables book mentioned earlier, I have three others in my collection, I. V. Proskuryakov's Problems in Linear Algebra, A. Kutepov and A. Rubanov's Problems in Geometry, and B. Demidovich's Problems in Mathematical Analysis, all from MIR publishers and all in English. The first is an upper-level b o o k suitable for advanced seniors and graduate students. The second is a lower-level book, suitable for high school students and college firstyear students, and the Demidovich book is appropriate for college calculus students. This book is innovational; it incorporates numerical analysis in an intrinsic w a y into the calculus regime. Usually, students are prevented from coming to grips w~th the slippery abstraction that is infinite series because they do not understand the nature of a remainder. Only numerical analysis, it seems to me, can help the student overcome this impediment. The book's 3200-plus exercises deal with the approximation of Fourier coefficients, finding roots of equations, and numerical integration. Western publishers have not been inattentive. In 1975 the Mathematical Association of America published the first volume in the Dolciani Mathematical Expositions Series, Mathematical Gems I, by Ross Honsberger. Other volumes followed, including five volumes by Honsberger, Old and New Unsolved Problems in Plane Geometry and Number Theory, by Victor Klee and Stan Wagon, Problems for Mathematicians, Young and Old, by Paul Halmos, and the charming The Wohascum County Problem Book, by George Gilbert, Mark Krusemeyer, and Loren Larson. These books are short and modest in scope, in the sense that the problems are not unduly technical. This does not mean that the problems are easy. The last volume contains problems that proved substantial enough to bedevil my intermediate analysis class, among them:
Does there exist a positive sequence an such that both y~ a,~ and y~ 1/n2an are convergent? (The current book lists this problem also [3].) More recently, under the editorship of Paul Halmos, Springer-Verlag has initiated a splendid and ambitious series of problem books in mathematics. Among the titles are Problems in Analysis and Problems in Real and Complex Analysis (both by Bernard R. Gelbaum): Unsolved Problems in Number Theory, by Richard K. Guy; Exercises in Probability by T. Cacoullos; and Theorems and Problems in Functional Analysis, by A. A. Kirillov and A. D. Gvishiani. I think publishing such books is a very healthy trend; it is a move away from eclecticism and specialization and toward the impartial sharing of the immense cultural resource that is mathematics. The Gelbaum book, THEMATHEMATICALINTELLIGENCERVOL.16, NO.4, 1994 71
Problems in Real and Complex Analysis, is very enterprising (488 pages) but it is not really comparable to the present book. The problems are much more technical, more methodology-oriented, like mini-theorems or exercises in a very advanced graduate text. The present book is physically beautiful and elegantly printed. I have an old-fashioned partiality for handset mathematical type, but I must admit the AMS TEX macro system produces a very comely result. It would be a shame for this book to be buried in a series of books so clearly aimed at a limited readership. And how many readers are willing to fork over $112 to buy it? The book demands a paperback edition. The American Mathematicai Society should give us one, and maybe see about translating into popular editions some of the other great Russian problem books, too.
Department of Mathematics and Computer Sciences Drexel University Philadelphia, PA 19104 USA
References 1. Pick )~ > 1 and write
an>_~ -n
z
an
