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 <-( < c in the coordinate plane. By assuming a helical shape this surface can be given explicitly as r(x, y) = ( - 2 a s e c h ~ sin vG 2c~sech~ cos v~, x - 2a tanh (),
(9)
where u = l / r / a n d c~ = 2 / ( r / + l/r/) [6]. Lamb [10], while studying the motion of curves, found the sine-Gordon equation as a possible evolution equation for curves of constant curvature (~) or of constant torsion; the latter corresponds to one of our coordinate lines, which turns out to be moving normal to itself with velocity s i n ( f ~ dx). The curve then sweeps out a surface of constant negative curvature, such as the one in Figure 4. This point of view provides an explanation of w h y a special equation such as sine--Gordon occurs in the theory of surfaces. The sine-Gordon equation is one of the AKNS hierarchy of integrable partial differential equations [2], and it has been shown in [12] that the SerretFrenet equations (the analogue for space curves of the Gauss-Weingarten equations), are equivalent to a special case of the AKNS scattering problem. Thus, local timeevolution equations can be given to the curve, which result in its curvature and t o r s i o n - - and hence the properties of the surface it sweeps o u t - - satisfying equations from the AKNS hierarchy.
Figure 5.
3. Breather, w = 4 (Fig. 5).
4 tan -1 ( v- ' l -~ w2 sin(w(x - y)) sech(v/1 -
w2(x + y))) O<w
gives solutions that pulse in the x - y direction and are localized in the x + y d i r e c t i o n - - a breather. For irrational w these surfaces intersect themselves infinitely often, but for rational co's can close neatly, giving cusped surfaces with finite area. 4. Breather, w = a (Fig. 6).
Melko and Sterling [11] have m a d e a detailed s t u d y of the relationship between the geometry of the sine-Gordon equation and that of surfaces of constant negative curvature. They point out that surfaces like the one" here contain planar elastic curves, such as the Euler loop ( ~ ) - - curves with stationary total square curvature, as in a bent plastic ruler. They also suggest an intriguing connection with the nonlinear SchrSdinger equation: the surfaces swept out by curves obeying this equation [6] m a y form the b o u n d a r y of the space of constant-negative-curvature surfaces.
Figure 6. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
35
5. Periodic Breather. There are w h i c h are also periodic in x + y,
similar
breathers
w = 4 t a n -1 [a sn(b(x - y) ] k 2) d n ( c ( x + y) ] 1 - m2)],
Figure 7. (from Mathematische Modelle, hrsg. Yon Gerd Fischer. Vieweg, Braunschweig/Wiesbaden, 1986.)
Figure 8. 36 THEMATHEMATICALINTELLIGENCERVOL.16,NO.4,1994
w h e r e a 2 = k / m a n d l/kb 2 = 1/mc 2 = (1 + k2)/k + (1 + m2)/m. This contains m a n y s i m p l e r solutions as special cases: k, m ~ 0 w i t h m / k = 0;2/(1 - w 2) held c o n s t a n t r e c o v e r s the r e g u l a r b r e a t h e r w i t h p a r a m e t e r w; k, m --* 1 gives ~; ---* 4 tan -1 t a n h [ ( x - y)/2], w h i c h is a n o t h e r w a y of w r i t i n g the p s e u d o s p h e r e ; for m ~ 0 w i t h k fixed, w --* 4 tan -1 [(x - y) s e c h ( x + y)], the k i n k a n t i k i n k s o l u t i o n [for w h i c h 0;(x, y) is a clockwise helix as x - y ---, 0% zero at x - y = 0, a n d a c o u n t e r c l o c k w i s e helix as x - y ---* - c o ] . The k i n k - a n t i k i n k solution g i v e s Kuen's surface (Fig. 7). This s u r f a c e w a s f o u n d in t e r m s of e l e m e n t a r y functions in 1884 a n d a plaster m o d e l constructed, b y w h a t p a i n s t a k i n g p r o c e s s o n e can o n l y i m a g i n e , as p a r t of the collection of m a t h e m a t i c a l m o d e l s at the U n i v e r s i t y of G6ttingen. T h e r e ' s a p h o t o of this collection in the Winter 1993 i s s u e of Mathematical Intelligencer, p. 61, a n d the w h o l e set is r e v i e w e d in Mathematische Modelle b y G e r d Fischer [5]. Forget c o m p u t e r g r a p h i c s - - w h e r e else can y o u see a p l a s t e r m o d e l of the i m a g i n a r y p a r t of the W e i e r s t r a s s - p function? In general, the periodic b r e a t h e r surfaces are rather c o m p l i c a t e d , b u t they are p r e t t y in s m a l l pieces. In Figures 8-10 are three f r a m e s of a m o v i e s h o w i n g the p s e u d o s p h e r e a w a k e n i n g : k = 0.9 is fixed a n d m = 0.43, 0.11, a n d 0.01. It p a s s e s t h r o u g h a m a n t a r a y state a n d e n d s as a Calla lily. E d w a r d Weston, w h e r e are y o u n o w ?
Figure 9.
Figure 10. Acknowledgments I'd like to thank Jim C u r r y and H a r v e y Segur for their encouragement, and Stuart Levy at the G e o m e t r y Center, Minneapolis for his help with the final images.
References 1. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett. 30 (1973), 191-193. 2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Philadelphia: SIAM, (1981). 3. Visualizations were done in geomview, available from the Geometry Center, Minneapolis, by anonymous ftp to geom.umn.edu. 4. L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, New York: Dover (1960) (originally published 1909). 5. Gerd Fischer, Mathematische Modelle, Braunschweig Wiesbaden Vieweg, 2 vols. (1986). 6. H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485. 7. J. N. Hazzidakis, Ueber einige Eigenschaften der F1/ichen mit constantem Kr/immungsmaass, J. reine angew. Math. ("Crelle's Journal") 88 (1880), 68-73. 8. D. Hilbert, Ueber F1/ichen yon constanter Gaussscher Kriimmung, Trans. Amer. Math. Soc. 2 (1901), 87-99. 9. S.B. Kadomtsev, Surfaces ~vith constant exterior geometry of negative curvature, Math. Notes Acad. Sci. USSR 47(4) (I990), 339-341. 10. G. L. Lamb, Jr., Solitons on moving space curves, J. Math. Phys. 18(8) (1977), 1654-1661. 11. M. Melko and I. Sterling, Application of soliton theory to the constructiori of pseudospherical surfaces in ~3, Ann. Global Anal. Geom. 11(1) (1993), 65-107.
12. K. Nakayama, H. Segur, and M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett. 69 (1992), 2603-2606. 13. A. Seeger, H. Donth, and A. Kochend6rfer, Theorie der Versetzungen in eindimensionalen Atomreihen. IIh Versetzungen, Eigenbewegungen und ihre Wechselwirkung, Z. Phys. 134 (1953), 173-193. 14. H. Segur, Who cares about integrability?, Physica D 51 (1991), 343-359. 15. H. Segur, personal communication. 16. J. J. Stoker, Differential Geometry, New York: WileyInterscience (1969). 17. M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume III, Boston: Publish or Perish, Inc. (1975). 18. P. L. Tchebychev, Sur la coupe des v~tements (1878), (CEuvres, vol. II, New York: Chelsea (1962), 708. University of Colorado at Boulder Boulder, CO 80309-0526, USA Current Address: Massey University, Palmerston North, New Zealand THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
37
David Gale* For the general philosophy of this section see VoI. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.
More Paradoxes. Knowledge Games In a column on the subject of paradoxes 3 years ago I presented the following example. You and I are assigned numbers, a and b, w h i c h we are told are consecutive positive integers, say b = a + 1. Each of us knows our o w n n u m b e r but not the other's. Every 10 seconds a beeper s o u n d s and if either of us k n o w s the other's number, w e m u s t announce it immediately after the beep, and the g a m e is over. This seems like a rather crazy game. What possible help can the intermittent beeping be to the players? Suppose, for example, m y n u m b e r is 10 and yours is ! 1 . Then w e both k n o w that after the first beep neither of us will announce, so w h e n this comes to pass it w o u l d seem we are no better off than w h e n we started. Nevertheless: T H E O R E M . With perfect play, the person holding n will announce that the opponent's number is n + 1 after the nth beep. 9 Induction on n: If m y n u m b e r is 1, I will k n o w yours is 2 and I will announce after the first beep. N o w suppose m y n u m b e r is n. If y o u r s is n - 1, then b y induction hypothesis y o u will a n n o u n c e after the (n - 1)st beep, so w h e n y o u d o n ' t do this I k n o w y o u r n u m b e r is n + 1. 9 Although the proof is correct, a puzzle still remains. H o w does a n o n a n n o u n c e m e n t that w e both k n e w w o u l d h a p p e n change the game? To understand this, suppose m y n u m b e r is 3 and y o u r s is 4. Then we both k n o w in advance that there will be no a n n o u n c e m e n t after the first beep. However, I d o n ' t k n o w that y o u k n o w this because,
* C o l u m n e d i t o r ' s a d d r e s s : D e p a r t m e n t of Mathematics, U n i v e r s i t y of California, Berkeley, C A 94720 USA.
as far as I know, y o u r n u m b e r could be 2, in which case you w o u l d have to allow for the possibility that mine was 1; so an a n n o u n c e m e n t would be forthcoming. The beep, therefore, gives me new information. If our numbers are 4 and 5, then I do k n o w that y o u k n o w that there will be no a n n o u n c e m e n t after the first beep, but y o u don't k n o w that I k n o w this, and so on. This is an example of what h a v e come to be called common knowledge games or puzzles. Here is another. In a school for bright children, one morning, in a class of 20 kids, 14 c o m e to school with dirty faces. The teacher says, "We are going to play a game. You can all see each other's faces. This b e e p e r is going to s o u n d every 10 seconds and if y o u k n o w that y o u r face is dirty, please raise y o u r hand after the beep." The game commences, but after several minutes nothing has happened, so the teacher says, "I see I'm going to have to give y o u a hint. At least one of y o u has a dirty face." As e v e r y child is able to see at least 13 dirty faces, this would not seem to be big news. Nevertheless, w h e n the game starts again, all the dirtyfaced children raise their hands on the 14th beep. The proof, left to the reader, is again b y induction, this time on the n u m b e r of dirty faces. What n e w information was gained from the "hint"? Let's consider the case where only y o u and I have dirty faces. Before the hint I know that y o u will not announce after the beep, but after the hint there is the possibility that y o u might, namely, if m y face is clean. If there is a third dirty face, then we all k n o w that there will be no a n n o u n c e m e n t after the first beep, but I don't k n o w that y o u k n o w t h i s - - a n d so on. The most mathematical variant on this theme is a game invented by John C o n w a y a n d Michael Paterson [1]. There are N people in a room, and player k has a non-negative n u m b e r ak written o n his forehead. In addition, there are N or fewer distinct positive numbers, Ak, written on a blackboard, one of which is the sum of the n u m b e r s ak. The numbers need not be integers. Each
38 THE MATHEMATICALINTELLIGENCERVOL.16, NO. 4 (~)1994Springer-VerlagNew York
player sees all numbers except his own. Again we make use of the 10-second beeper, and the idea is to see how many beeps it takes before someone knows his number, or, what is the same thing, knows which Ak is the true total. The Conway-Paterson theorem asserts that this game will always terminate. Again we have a paradox. Suppose, for example, in a three-player game all the lowercase a's are 2 and the capital A's are 6, 7, and 8. Then everyone knows that her number is at most 4; hence, her opponents are looking at two numbers whose sum is at most 6. Hence, any of the threeA's could be the true total, so there will be no responses after the first beep. Perhaps the most surprising thing about the ConwayPaterson theorem is that the proof is extremely simple, provided one goes after it in the right way. Before presenting it, however, let us look at the two-player case with blackboard numbers A < B. We will say a pair of forehead numbers (a, b) is possiblefor the (A, B) game if and only if either a + b = A or a + b = B, so the set of possible (A, B) games consists of a pair of diagonal lines in the plane. See Figure 1. Now the games that terminate after one beep consist exactly of those pairs one of whose members, x, is greater than A, since the player who sees such an x will know that the true total is B. These are the two segments of the B line labeled I in Figure 1. Having eliminated all pairs containing a number greater than A, the game will end on the second beep if any pair contains a number less than B - A, since a player seeing such a number will know that the true total is A. These are the segments labeled 2 on the A line. In this w a ~ each beep allows one to lop off end segments of one of the two lines. From Figure I
8 A
I
3
Figure 1.
3
I
one sees that the entire A line will be eliminated after 4 beeps, so, for example, if a = b = B/2, then the players will know this after the fourth beep. Before giving the general proof, we take note of another surprising fact discovered by Lasry, Morel, and Solimini [2]. The Conway-Paterson game may terminate even when there are more blackboard numbers than players. For example, we will show that the two-player game with blackboard numbers 5, 8, and 15 will terminate after at most 10 beeps. The display below is the possible pairs of numbers for this game.
Table 1. Possible Pairs of Numbers 5
8
15
{0,5} s
{0,8} 6
{0,15} 1
{1,4} 9
{1,7} 8
{1,14} 1
{2,3} 3
"~2,6} 2
{'~, 13} 1
{3,5} 4
~3, 12} 1
{4,4} 10
{4, 11} 1 {5, 10} 1 {6,9} 1 {7,8} 7
As usual, the analysis proceeds by first finding all pairs which give a one-beep game, then those which give a two-beep game, and so on. The number after the braces in the above display indicates for each pair the beep on which the game will terminate. First, if either number is greater than 8, then the player who sees it will know the true total is 15 and will announce it after the first beep. Note that these numbebs, 9 through 15, are exactly those that appear in only one of the possible pairs. After deleting these pairs, we see that there is only one pair, {2, 6}, containing a 6, so that if a player sees a 6, she will know the true total is 8 and announce after the second beep. Eliminating this pair, we find that {2, 3} is the only remaining pair containing 2, so it is eliminated next, and so on. The rule could hardly be simpler. At each stage, look for all numbers contained in only one pair and eliminate those pairs. The 10-beep game is the one in which each player's number is 4. On the other hand, if the 8 of the above example is changed to 9, we will show that the game cannot terminate. As before, we see that the game will terminate after the first beep if and only if one of the numbers is 10 or higher. However, we are now stuck, because, as shown by the display below, every number occurs in two of the remaining pairs; so no matter what number a player sees, THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
43
h e / s h e will not be able to d r a w any conclusions as to the true total. Table 2. Possible Pairs of N u m b e r s 5
9
15
{0,5}
{0, 9}
{0, 15} 1
{1,4}
{1,8}
{1, 14} 1
{2,3}
{2,7}
{2, 13} 1
{3,6}
{3, 12} 1
{4,5}
{4,11} 1 {5, 10} 1 {6,9} {7,8}
In [2], the authors give a complete analysis of twoplayer, three-number games. Suppose A1 < A2 < A3. Then the necessary and sufficient condition for such games to terminate is that there exist integers p and q such that A1 < p(A2 - A1) + q(A3 - A2) < A3 - A2. With the second example in mind, one can n o w give the surprisingly simple proof of the C o n w a y - P a t e r s o n theorem for the N-player, N - n u m b e r game. The key idea is to argue by contradiction. Instead of proving that the g a m e must terminate, one proves that it is impossible for it not to terminate. Namely, as in the above example, for a g a m e to fail to terminate it must reach a point where after some beep it is impossible to eliminate a n y more possible N-tuples. W h e n can this happen? The answer is given by a simple theorem about vector spaces. Let us call a set S of N vectors ambiguous if for any m e m b e r a in S and any index i there is a vector a ' in S that differs from a only in the ith coordinate. N o t e that if a game reaches a point where the remaining set of Ntuples is ambiguous, then no player will learn anything from the next beep: There will always be at least two possible values for his o w n number. Given a vector a, let ao denote the sum of the coordinates of a. LEMMA. If S is a finite ambiguous set of N-vectors, then the set of sums ao must contain at least N + 1 members. 9 Immediate for N = 1. N o w choose a m e m b e r a = (al, 99 9 aN) such that al is a m i n i m u m for all a in S, and let S' be the set of all ( N - 1)-vectors x = ( x 2 , . . . , XN) such that (al, X 2 , . - - , XN) is in S. Then S' is also an ambiguous set of vectors, so b y the induction hypothesis the vectors of S' have at least (N - 1) + 1 = N distinct sums; thus, the vectors (al, x 2 , . . . , XN) have at least N sums. Let (b2,..., bN) be a vector of S' whose s u m is largest. 44
THE MATHEMATICAL INTELLIGffNCER VOL. 16, NO. 4,1994
By the choice of al and the definition of ambiguity, there is a~ > 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)) <_ L t d(x, y)
(2)
for all x, y E X and t E T, the set of non-negative integers. With L > 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) <_ K
1
dF(Ft(x), Ft(y)) <_ -~ 1
(3) for all t < 2 s .
Furthermore, if F is uniformly continuous with respect to d, then dR can be chosen uniformly equivalent to d. So if your two initial points are within 1/1001 of each other, then the two orbits remain 1/1000 close for the first 21~176176 iterates. That many iterates ought to satisfy anybody."
In the same way, OF : X ~ C(T; X ) is a uniformly continuous map if and only if {F t : t E T} is a uniformly equicontinuous subset of C(X; X). From this we obtain: THEOREM 2. Let (X, d) be a metric space and F:X --, X. If the sequence of iterates { Ft:t E T} is equicontinuous, then there exists a metric dF, yielding the same topology as d, and satisfying: for all x, y E X
dF(F(X), F(y)) <_dHx, y);
(7)
that is, with respect to the metric dF, the map F has Lipschitz constant at most 1. Furthermore, if {F t : t E T} is uniformly equicontinuous, then dF can be chosen uniformly equivalent to d. Proof. We can use the orbit map OF to pull back the metric D from C(T; X) to X, defining
The analyst's reply was understandably heated: "Just a (expletive deleted) minute there, fella. All this topological flim-flam won't help one bit to apply Newtoffs Method. It seems to me that you are using a pretty sloppy notion of equivalent metric."
dR(x, y) ~ D(OF(X), OF(y)) = sup(d(Ft(x), F t ( y ) ) : t E T}. dF is always a metric on X and dF(X, y) >_d(x, y)
In order to resolve this dispute, we consider the proof of the topologist's theorem. There are several different ways of viewing the dynamical system induced by F:X ~ X. The flow is the map T x X ---, X taking (t, x) to Ft(x). Alternatively, because each iterate F t is in the set C(X; X ) of continuous maps from X to X , we can consider the sequence of iterates {Ft: t E T} in C(X; X). In other words, from the flow we get the adjoint map T ---, C(X; X). Dually, we can define the orbit map OF : X ---* C(T; X), thinking of the elements of C(T; X), the maps from T to X, as sequences in X.
OF(X)(t) = Ft(x).
(4)
Thus, oF associates to x in X its orbit sequence
{x, F(x), F2(x),...}. There is a natural metric on C(T; X) yielding the topology of uniform convergence on T: D(a,/3) = sup{d(a(t), fl(t)) : t E T}
(5)
for all a, fl E C(T; X). We denote by C(T; X ) the set C(T; X) equipped with the metric D. Regarded as a map from X to C(T; X) the orbit map OF is usually not continuous. In fact, OF is continuous at x E X if and only if the sequence of iterates {F t : t E T} is equicontinuous at x, because
D(OF(X), oF(y)) < ~ r d(Ft(x), Ft(y)) <_~ f o r a l l t E T. 46
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
(6)
(8)
(9)
because 0 E T and F ~ is the identity map by convention. Furthermore, inequality (7) is obvious. Usually, however, dF yields a strictly finer topology (more open sets) than does d. In fact, the two topologies agree exactly when OF is continuous at every point of X. The stronger condition of uniform continuity for OF holds exactly when for all r > 0, there exists 6 > 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) <_ e r d(a(t), fl(t)) < e for all t E T such that p(t) < e -1
(12)
Using (! 2) and the monotonicity of p, it is easy to check that D is a metric. We will denote by C(T; X) the set C(T; X) equipped with the metric /). Continuity of OF:X --* C(T; X) requires merely continuity of F. We will prove this result by expressing it in terms of the pull-back metric
dF(x, y) - b(oF(x), oF(y)) = sup{min[d(Ft(x), Ft(y)), 1/p(t)]: t e T}.
(13)
Clearly, we have, extending (9), (14)
On the other hand, given e > 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)) <_ -~
QED
f o r a l l s _< 2K"
(17)
Mathematics Department The City College New York, NY 10031 USA THE MATHEMATICAL [NTELLIGENCER VOL. 16, NO. 4,1994
47
Jeremy J. Gray*
Eduard ( ech Eduard Cech (1893-1960), the internationally famous topologist, was the leading Czech mathematician of his generation. He was born in Stracov in northeastern Bohemia, and went to the Charles University in Prague to study mathematics in 1912. But his university studies were interrupted by the First World War, and he only graduated in 1920. By then he had become interested in projective differential geometry, which studies those features of the geometry of embedded curves, surfaces,
Cech's work emphasised results about tangency, correspondences between manifolds, and.., the systematic theory of duality in projective spaces. and higher-dimensional spaces that are projectively invariant. ~ech s work emphasised results about tangency, correspondences between manifolds, and (notable for his later work in topology) the systematic theory of duality in projective spaces. His developing interests in this relatively new field soon earned him a scholarship to study with Fubini, its acknowledged leader, in Turin in 192122. The visit was fruitful, and they went on to write two books on the subject, in 1927 and 1931. ~ech returned to his native country and, after his habilitation (the degree that qualified him to be a university teacher), took up an extraordinary professorship at the Masaryk University in Brno. There he lectured on analysis and algebra for 12 years; this may well have deepened his interest in topology. In 1928 he became a full professor, and, inspired by the papers in the Polish journal FundamentaMathematica, * Column Editor's address: Faculty of Mathematics, T h e O p e n University, Milton Keynes, MK7 6AA, England.
he turned increasingly to topology: all his papers after 1931 were on that subject. The subject of topology was already dividing into two branches: point-set topology, which studies topological spaces directly, and algebraic topology, which studies spaces by passing to algebraic objects that may be attached to them. One might also distinguish between the study of t h e topological properties of most spaces and the study of important special cases, or, if you prefer, the properties of spaces in general and the special properties of perverse spaces. Most topological spaces which arise naturally in other branches of mathematics have simple properties. The difficulty arises in giving suitable, intrinsic foundations for the subject of topology, and the role of the otherwise strange, artificial spaces studied by point-set topologists is to test the utility of definitions and the generality of proofs. The Polish school excelled in work of this kind. Algebraic topology, on the other hand, aims to distinguish between, even to characterise topological spaces by computing algebraic objects, such as groups or rings, that can be attached to them. It is not restricted to the study of well-behaved spaces, but such are the computational problems which arise that in practice most interest attaches to the study of tractable examples. Happily, these have tended to be the spaces that arise naturally in other contexts. However, the tools of algebraic topology are often insensitive. For example, there are many contractible spaces that are not homeomorphic, but all necessarily have the same homology groups. Throughout the 1930s there was considerable interest in extending the tools of algebraic topology to deal with such problems. Cech's first contributions in topology were characteristically broad and aimed at keeping the subjects of algebraic and point-set topology together. In his first two papers he developed a homology theory for arbitrary spaces, and established general duality theorems
48 THE MATHEMATICALINTELLIGENCERVOL,16, NO. 4 (~)1994Springer-VerlagNew York
for manifolds, generalising the classical duality of subspaces of projective spaces. Cech's approach to homology theory was deliberately intended to be very general, as the title of his paper in Fundamenta Mathematica for 1932 makes clear ("General homology theory in an arbitrary space"). It is based on the idea of studying all finite
Cech's originality lay in introducing the concept of an inverse limit to obtain homology groupsindependent of the choice of covering. open coverings of a given space. The paper introducing this idea has become one of his best known, although many of the ideas in it can be traced back to earlier work of Vietoris and Alexandrov. Cech's originality lay in introducing the concept of an inverse limit to obtain homology groups independent of the choice of covering. The approach turns out to work very well for compact spaces, and this paper is the origin of what is now called the Cech homology theory. The corresponding cohomology theory worked less well for noncompact spaces, giving unexpected results even for the first cohomology group of the real line. A better w a y forward was found by C.H. Dowker, who replaced Cech's finite coverings with arbitrary coverings. With this modification, ~ech cohomology was shown by Eilenberg and Steenrod to satisfy all the axioms they postulated for a cohomology theory (unlike the Cech homology theory with integer coefficients, which fails the excision axiom). In another paper, presented to the International Congress of Mathematicians in Ziirich in 1932, Cech presented his ideas on the definition of the higher homotopy groups of a space. Unhappily, the report on his talk was brief and obscure, and Cech never returned to the subject himself, thus leaving it open for the decisive contributions of Hurewicz. Hurewicz later wrote that his definition agreed with Cech's, and Alexandrov was to single out ~ech's contribution in 1961, when commemorating Cech's life and work, and to lament that it had been misunderstood. In 1934 Cech extended his approach to homology theory to obtain a theory of local homology, appropriate to the study of neighbourhoods of a topological space. His reputation grew, and when he reported on his results at an international conference on topology held in Moscow in 1935 Lefschetz invited him to visit the newly-founded Institute for Advanced Study i n Princeton. Upon his return, Cech founded a topology seminar at Brno which applied itself to the work of Alexandrov and Urysohn. In three years the seminar published 26 papers, including Cech's "On Bicompact spaces" which appeared in Annals of Mathematics. In this paper he introduced the idea Of the (Stone)-Cech compactification of a regular topological space. The seminar continued
Eduard Cech
until 1939, when the Germans invaded Czechoslovakia and closed all the universities. Thereafter it continued in the flat of Cech's student B. Pospi~il, until 1941, when the Gestapo arrested Pospi~il. The seminar had a lasting influence on the development of mathematics in Czechoslovakia, because it marked the introduction of the method of working on mathematical problems in groups. After the War, ~ech moved to the Charles University in Prague. N o w in his fifties, he began an intensive administrative career. In 1947 he was appointed Director of the Mathematical Research Institute of the Czech Academy of Sciences and Arts and in 1950 a Director of the Central Mathematical Institute. In 1952 the Central Institute was incorporated into the Mathematical Institute of the Czechoslovak Academy of Sciences, with Cech as its first Director. That year he returned to the Charles University to head its newly-founded Mathematical Institute. But he also found time to redirect his mathematical interests; in the 1950s he wrote 17 papers on differential geometry. He also deepened his interests in the teaching of mathematics. He wrote seven textbooks for secondary schools and held seminars on elementary mathematics at both Brno and Prague. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
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Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous ini-
tials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels ? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.
Mathematics in G6ttingen (1737-1866) Riidiger Thiele Whatever our search after bygone times, the temporal gap will remain unbridgeable. Spatially it is another matter: We can revisit the scenes of historical events. G i i t t i n g e n a n d Its U n i v e r s i t y
George II, from 1727 King of Great Britain and Ireland and Elector of Hanover, founded a university in G6ttingen, which in 1737 was formally inaugurated as the Georgia Augusta and would soon be one of Europe's foremost universities. Lichtenberg called it the "queen" of universities, and from far-off K6nigsberg, Immanuel Kant (1724-1804) counseled students to study in G6ttingen. Lectures began in October 1737 with a lecture on physics delivered in a granary by S. C. Hollmann, but it was the Faculty of Law that lent the University its pre-eminence in the time of the revival of learning. There were significant mathematicians, astronomers, and physicists in G6ttingen before Gauss made its mathematics famous throughout the w o r l d - - such as A. Segner, A. K/istner, G. C. Lichtenberg, and T. Mayer. The princeps mathematicorum was loath to reveal all the fruits of his intellect, so Bessel, the astronomer (1784-1846), reproached him in 1837 in the following terms: "Where would the mathematical sciences be today, not merely in your house but in the whole of Europe, had you but said all that you could say! " *Column Editor's address: Mathematics Institute, University of Warwick,Coventry,CV4 7AL England. 50
Figure 1. Turmstrasse viewed from the intersection with Nikolaistrasse. The gable wall visible at the fax end of the street is that of the house in Kurze Strasse occupied by Gauss
from 1807 to 1816.
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 ~)1994 Springer-Verlag New York
Figure 2. Residence of Lichtenberg at I Gotmarstrasse (at the corner of Prinzenstrasse). Here also were the offices, the press, the bookshop, and the storerooms of the publisher Dieterich. By the end of Gauss's life, Berlin had become another important German mathematical center, and with the annexation of Hanover to Prussia in 1866 the University of G6ttingen was largely dominated by Berlin, despite acquiring Dirichlet as Gauss's successor and Riemann to succeed him. We now go "mathematical sightseeing" through the first six-score years of the university (more exactly, 17371866). Marble plaques, commemorating the most important places of mathematical interest, were first put up in 1874. Their coordinates are easily given, because the early town very nearly forms a circle delimited by the city walls. Weender Strasse and Geismar Strasse form the north-south axis and Barftisserstrasse and Groner Tor Strasse the east-west axis; they meet almost at the Market Place. As of 1854 the Kassel-Hanover rail route touched the city walls in the west. Gauss was very interested in railroad construction, his eldest son being employed by the railroad administration, Hanover. He was present at the opening ceremony on July 31,1854. He had nearly come to grief a few weeks earlier while observing the progress of the work, when his carriage horses shied at a locomotive, upsetting his carriage. Along the Weender Landstrasse, an old strategic highway, which enters the town ~n the north and continues as Weender Strasse, came the student Carl Friedrich Gauss on October 11, 1795, possessor of a grant from the Duke of Brunswick worth 158 thalers and free board. (For comparison, lodgings would cost 25-35 thalers a year, supper 2-289 thalers a month, lecture fees 5 thalers per semester.) Almost exactly 12 years later the stagecoach
carried the famed astronomer Dr. Gauss, with his wife and child, along the same route to his place of w o r k The University Observatory. In between, his masters G. C. Lichtenberg (in 1799) and A. K~istner (in 1800) had been borne along this ~r route to their last resting place in the Cemetery of St. Bartholomew's Without the Walls (Bartholom/ius Friedhof), formerly churchyard of St. John's. Later Dirichlet (in 1859) and a son-in-law of Gauss, the noted orientalist Heinrich Ewald (in 1875), would follow. The cemetery is about 20 minutes on foot from both the station and the Market Place; it can be reached in about 5 minutes from the Weender Tor (fig. 19). Alongside the Weender Tor, the Auditorium Maximum was built between 1862 and 1865, and inaugurated by blind King George V of Hanover. The Faculty of Arts (including Mathematics) was quartered in the North Wing, facing away from the town. One of the figures at the portal, the one on the right, represents Gottfried Wilhelm Leibniz (1646_,.1716), who was in the service of the Duke of Hanover from 1676. The Grand Hall of the University, Wilhelmsplatz, built in classical style, can be reached from the Market Place by proceeding east along Barfiisserstrasse. On the occasion of the 100th anniver-
Figure 3. Wing of the building at 1 Gotmarstrasse, where Lichtenberg conducted his famous practical lectures. THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
51
Figure 4. Lichtenberg Memorial in the Quadrangle of the Kollegiengeb~ude (old University Library). Across the way from Lichtenberg is a seat for the visitor, again with a book lying on it. The book is opened at Lichtenberg's aphorism: D a s viele Lesen hat uns eine gelehrte Barbarei zugezogen (much reading has brought upon us a breed of learned barbarians).
books daily. The Library originally housed some 12,000 volumes. It had 160,000 when Gauss was a student, and 250,000 during his time as professor. With almost 4 million volumes today, it is one of the most important repositories of learning in Germany. In the University's early years, rooms in the Library served as lecture rooms. The Department of Manuscripts and Rare Books collects papers, such as unpublished works of Cantor, Dedekind, Gauss, Hilbert, Klein, and Riemann; manuscripts (about 10,000); and autograph letters (in excess of 100,000), currently totaling more than 1700 meters of shelving. The Physical Sciences Collection in the so-called Museum Wing of the Library had Wilhelm Weber as Curator from 1821 to 1837. He is buried in the Municipal Cemetery (Old Main Walk Plot 7). The wing housing this collection was pulled down in 1877. It stood roughly at the corner formed by Prinzenstrasse and Papendiek, across the road from a hostel for students of noble birth which had the name of Londonsch/inke. It was here in the Physical Sciences Collection that one of the terminal points of the electric telegraph system was set by Gauss and Weber on Easter 1833 (fig. 17). It remained in use until Weber's departure in 1837, and it was destroyed in a thunderstorm in 1854. The other terminal was at the Observatory (11 Geismarlandstrasse), about I kilometer away. Between these two points the line was attached for support to the Accouchierhaus, Germany's first gynecological clinic, built 1785-1791 at the Geismar-Tor, and to the shorter of the two towers of the Johanniskirche (fig. 15). Weber's reconstruction of the telegraph system for the Vienna World Fair in 1873 can be seen at the First Institute of Physical Sciences, 9 Bunsenstrasse, next door to the Mathematical Institute. S e g n e r , M a y e r S e n i o r , a n d K~istner
sary, it was donated to the University by William IV, the last Hanoverian King of Great Britain and Ireland, whose statue stands in the square. If you are lucky, in the office, ground floor right, you may be entrusted with the key of the Hall and the Detention Room, where recalcitrant students languished as recently as the start of the Second World War, and whose walls are adorned with their graffiti. In front of the Grand Hall (Aula) can be seen plaster casts of busts of Hilbert and Gauss, that of Gauss originating as a medallion created by H. Hesemann only a few weeks before Gauss's death. A copy of the medallion is to be found on Gauss's gravestone. The building also housed the Akademie der Wissenschaften zu G6ttingen, founded in 1770 by A. von Hailer, which today is just behind the Grand Hall in the Theaterstrasse. The old building of the Nieders/ichsische Staatsund Universit/itsbibliothek is a few minutes' walk northwest of the Market Place and, together with the 14th-century Paulinerkirche, can be approached from Papendiek (pape, "monk," diek, "embankment"). Here began the modern academic research library. In G6ttingen it was possible to use the Reading Room and take out 52
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
Segner (1704-1777) was the first Professor of Mathematics in G6ttingen, his professorship dating from 1735. In 1743, the decision was taken to set up a University Observatory in an old tower on the city walls, which until that time had served as a firehouse. Together with T. Mayer (1723-1762), who was called to G6ttingen in 1750, Segner was put in charge of construction, which was completed in 1751. Altercations led to Segner leaving G6ttingen in 1755 with Euler's assistance, to become successor to the famed philosopher C. Wolff in Halle. Shortly before his departure from G6ttingen, Segner made the important discovery that every solid body has three axes of inertia. Segner made practical use of Daniel Bernoulli's theoretically postulated "reaction effect" (Hydrodynamica, 1730) to drive a horizontal waterwheel, a principle today used in lawn sprinklers. In turn, Segner's simplereaction waterwheel gave Euler the impulse to work on turbines, which led in early 19th-century France to practical grinding mills, as approvingly mentioned by Carnot, Perier, and Prony. In N6rten there was a cornmill driven by Segner's simple-reaction waterwheel as early
as 1750.1 N6rten is approximately 12 kilometers north of G6ttingen, easily reached along the A7 autobahn by turning off at N6rten-Hardenberg, or from Federal Highw a y 3, leaving G6ttingen in the direction of Northeim, and in N6rten following tl~e sign Burgruine Hardenberg. The beautiful mill house can still be seen today. The Burgrave Friedrich Karl von Hardenberg not only showed an interest in Segner's work; as a diplomat he also influenced the construction of the first Observatory. A noted member of this family is Friedrich, Freiherr von Hardenberg (1772-1801), known as Novalis, poet of the romantic movement, who was deeply interested in mathematics: "The life of the gods is mathematics; all god-sent envoys must be mathematicians." Mayer is famed for his astronomical catalogue and lunar charts. The charts posthumously earned him [in the company of Harrison (1693-1776) and Euler (17071783) a British government award. The mural quadrant (Bird, 1756) used by Mayer hangs in the lecture hall of the new Observatory. After Mayer's death (grave unknown, somewhere in the Albani-Friedhof), the post of Director of the Observatory remained vacant for some years, until occupied by A. K/istner (1719-1800) from 1775 to 1789. K/istner came from Leipzig to succeed Segner. He was the author of numerous textbooks, including the first history of mathematics to be written in German (1796 -1800). Montucla's Histoire des mathdmatiques had appeared 40 years before, in 1758. K/istner also wrote poetry. Gauss is supposed to have said that he was the best poet among the mathematicians and the best mathematician among the poets. But no less a literary figure than Gotthold Ephraim Lessing held him in high esteem, and Lichtenberg described him as a dictionnaire encyclopddique. From 1766, K/istner lived at 26 Nikolaistrasse (directly next door to the old Observatory, fig. 1). He was buried in the Bartholom~us Friedhof along the Weender Landstrasse (in close proximity to the G6ttingen publisher Ruprecht, fig. 19). The Observatory had been in a poor state of repair since 1759, and in 1791 the proposal was made to build a new one. Construction started in 1803, modeled on Gotha and Oxford. Gauss was Director of the Observatory from 1807. Because of the Napoleonic occupation of 1806-1813, construction was not completed until 1816 (French troops for a time used the old Observatory as a powder magazine). In 1822 the old Observatory, in 1897 the Tower, and finally in 1981 the relevant section of the city walls were demolished: There is nothing in Turmstrasse today to call the Observatory to mind.
from the Paulinerkirche (at the rear of the old University Library). Appointed Professor of Pure and Applied Mathematics in 1770, he took up his teaching office in his return from England in 1776 with lectures on mathematics and astronomy. Experimental physics, a subject that was to spread his fame beyond the bounds of G6ttingen, followed from 1778 until his death. Lichtenberg used his own apparatus. He sold his collection to the University in 1789, and today by previous arrangement these historic devices may be seen, together with others in the First Institute of Physical Sciences, 9 Bunsenstrasse. From 1778 Lichtenberg lived in premises owned by his friend and publisher Dieterich in Gotmarstrasse (No. 1, at the intersection with Prinzenstrasse and somewhat to the north of the Johanniskirche, fig. 2). This was one of the largest houses in G6ttingen, accommodating up to 60 tenants. In the wing was Lichtenberg's lecture room, accommodating 100 students or more, eager to witness his fascinating experiments, which his assistants repeated on Sundays for genera~ interest (fig. 3). Of ~he students, Lichtenberg remarked, "They cut lectures already, till we get to the thunder and lightning, "2 by which he was referring to the appeal of the artificial lightning, the detonations, and the hovering bubbles during his lectures. He further remarked, "It is incredible how ignorant students are when they come to the University. I have only to spend ten minutes mathematicizing and a quarter of them are sure to doze off. "3 In 1787 Lichtenberg rented a cottage at 37 Weender Landstrasse where in 1794 he installed a lightning conductor. In 1907 the house was pulled down and rebuilt in slightly enlarged form at 34 Brauweg (not far west of the Mathematical Institute). His house in Hospitalstrasse (roughly where No. 3 now stands) was also equipped with a lightning conductor (fig. 5). 2 "Sie s c h w ~ n t z e n aber jetzt schon, bis es blitzt u n d donnert." 3 "Es ist u n g l a u b l i c h , wie u n w i s s e n d die s t u d i r e n d e J u g e n d auf Univers[it~ten] k o m m t , werm ich n u r 10 M i n u t e n rechne oder geometrisiere, so schl~ft 88derselben sanfft ein."
Lichtenberg
Lichtenberg enrolled as a student of mathematics and physics on May 21, 1763. From 1764 to 1767 he lodged on the first floor of 3 Paulinerstrasse, just across the road 1 Annalen der Physik 43 (1813), 166.
Figure 5. Plan drawn by Lichtenberg's own hand for a lightning conductor (1794) for his house on Weender Landstrasse. To counter the awe of local inhabitants, Lichtenberg elucidated the construction and its purpose in the local paper. THE MATHEMATICALINTELLIGENCERVOL.16, NO. 4,1994
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Gauss
Figure 6. Gravestones of Lichtenberg and his wife (right) in the B a r t h o l o m ~ i u s F r i e d h o f , Weender Landstrasse.
Lichtenberg had read astronomy under T. Mayer and described it as "perhaps that science in which the least has been discovered by chance, where human intellect is seen in all its greatness, and in which man best grasps how insignificant he really is. "4 Lichtenberg lies buried with his wife in the Bartholom/ius Friedhof, Weender Landstrasse (figs. 6 and 19). Einstein (1879-1955) made the following estimate of Lichtenberg: "I know of no other who is so quick on the uptake. "5 Lichtenberg, a man of science, was also no mean man of letters. In his celebrated S u d e l b f i c h e r , a collection of notes and aphorisms, he says of infinitesimal calculus, "The grand stratagem of regarding minute deviations from the Truth as being the Truth itself, on which the whole edifice of Differential Calculus is erected, is at the same time the basis of our deliberations, whose whole frame so often would collapse, were we to regard the deviations with philosophical strictness" (A, fascicle 1). (fig. 4). 4 [Die Astronomie] "ist vielleicht diejenige Wissenschaft, w o r i n das w e n i g s t e d u r c h ZufaU e n t d e c k t w o r d e n ist, w o der m e n s c h l i c h e Vers t a n d in seiner g a n z e n GrOsse erscheint und w o der M e n s c h a m besten lernen kann, wie klein er ist." s "Ich kenne keinen, d e r m i t solcher Deutlichkeit d a s Gras w a c h s e n hOrt." 54
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Gauss enrolled as a student at the Georgia Augusta on October 15, 1795. He attended lectures by K~stner, Lichtenberg, and the classicist C. G. Heyne. From 1795-1796, he lodged in a tailor's house in 11 Gotmarstrasse, where today is a store, and from 1796 to 1798 he lodged at 16 Geismarstrasse, the house of a w i d o w (fig. 7). When he was changing lodgings and on vacation back in Brunswick, Gauss came to a momentous decision. On March 30, 1796, before getting up in the morning, he solved the problem of how to construct a regular heptadecagon, just as later, on January 23,1835, while still in bed, he was to formulate the law of induction. Thereupon he commenced a journal in which the first entry reads: "Principia quibus innitur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim partes etc. Mart. 30 Brunsv." (Basis of the division of the circle, geometrically divided in 17 parts, March 30, 1796, Brunswick) (fig. 16). Thus was the balance tipped in the direction of mathematics instead of classics. As a student, Gauss was a closer intimate of Lichtenberg than is commonly supposed. When Gauss received an inoculation against smallpox, in those days a procedure not without its risks, Lichtenberg put at his disposal his cottage at 37 Weender Landstrasse (pulled down in 1907) for his quarantine. After becoming Director of the Observatory in 1807, Gauss lived at 15 Kurze Strasse (a continuation of Weender Strasse), near the old Observatory (fig. 8). His wife was not satisfied with the accommodation. A letter of hers speaks of "shabby, filthy rooms, a smoky and draughty kitchen, ancient and phlegmatic landlord and landlady." But it was here that in 1809 Gauss wrote his immortal T h e o r i a m o t u s c o r p o r u m c o e l e s t i u m . . . ("Theory of the motion of the Heavenly Bodies revolving round the Sun in Conic Sections"). His wife did not survive long in the unattractive quarters, dying in 1809. To the same address Gauss brought a second wife in 1810, but she too soon became sickly, and died in 1831. The newly appointed physicist W. Weber (fig. 11) was at hand to pilot Gauss through this bleak passage by his scientific commitment. We might call the 1830s a "period of physical sciences" in Gauss's life. The two men organized magnetic measurement worldwide, and in 1833 in the precincts of the new Observatory, directly at the point formed by the way to the Observatory and Geismar Landstrasse, they erected a "nonferric" observatory known as the "Magnetic House." On its inauguration in 1834 Gauss used the following terms in explanation: "Without exception, everything in the building that in the normal w a y would be of iron, locks, door hinges, window fittings, nails, etc., is made of copper. Provision has been made as far as possible to exclude any draughts." After Gauss's death, the house was enlarged by Weber. In 1902 it was dismantled and re-erected in the grounds of the Geophysical Institute at 180 Herzberger Landstrasse,
Figure 7. Gauss's sketch-map in a letter to his friend Wolfgang (Farkas) Bolyai (1775-1856) showing the way from the Weender Tor via the Market and Barfiisserstrasse to Gauss's address, 16 Kurze Geismarstrasse (Cod. Ms. Gauss Briefe B: Bolyai) (by kind permission of the Handschriftenabteilung der Nieders~ichsischen Staats- und Universit~tsbibliothek G/~ttin~en).
Figure 8. Where Gauss lived in Kurze Strasse from 1807 until his move to the new Observatory in 1816. On the right can be seen Turmstrasse, where the old Observatory was situated. Nearly opposite Gauss's house is the spot where once stood the house where W. Bolyai (1775-1856) lodged as a student from 1796 to 1798, marked by a memorial plaque on the existing building, 2 Kurze Strasse. Figure 9. The #Magnetic House" on the grounds of the Geophysical Institute. In it, students now do practical work. The condition of the building is no longer completely iron-free. at H a i n b e r g (fig. 9). H e r e are p r e s e r v e d various geodetic i n s t r u m e n t s used b y G a u s s When land surveying, a m o n g t h e m a Gaussian heliotrope. You w o u l d do well to m a k e an a p p o i n t m e n t before visiting. The H e r z b e r g e r Landstrasse begins at the city ring road, initially leading w e s t out of G6ttingen. In the d a y s of Klein and Hilbert the m a t h e m a t i c i a n s o f G 6 t t i n g e n took a w a l k to H a i n b e r g regularly on T h u r s d a y afternoon, 3 o'clock sharp. THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 4,1994 5 5
Figure 11. Portrait of W. Weber, drawn by the topologist B. Listing. (By kind permission of the Handschriftenabteilung der Nieders~ichsischen Staats- und Universit~itsbibliothek G6ttingen, Cod. Ms. Gauss Posthum. 26.) Figure 10. Model of the Gauss-Weber Memorial, clearly showing the occasion of its erection: the electro-magnetic telegraph. In the Memorial as constructed, the communicator can only be seen from the rear. Neither model nor actual monument suggests the discrepancy in age between Gauss and Weber, except in that Gauss, the eider partner, is offered a seat. (By kind permission of the Handschriftenabteilung der Nieders~chsischen Staats- und Universit~itsbibliothek G6ttingen, Cod. Ms. Gauss Posthum. 28.)
Going t h r o u g h Kurze Strasse in the direction of the city walls, past the W6hler M o n u m e n t (1890, for E W6hler, a chemist, b u r i e d in the municipal cemetery) opposite his chemical laboratory, one reaches the w e l l - k n o w n G a u s s - W e b e r M e m o r i a l (1899). Both m o n u m e n t s are the w o r k of H a r t z e r (fig. 10). It w a s in a Festschrift for the unveiling of the G a u s s - W e b e r M e m o r i a l in 1899 that Hilbert's pioneering Grundlagen der Geometrie first appeared.
Figure 12. The new Observatory at 11 Geismar Landstrasse, which in Gauss's time was beyond the city limits. On the left, the west wing leads to the rear, where Gauss lived from 1816 onward. On the upper floor left is the Observatory's little Museum; in the right wing is a lecture room containing T. Mayer's mural quadrant. The Rotunda houses Gauss's heliometer (1814), seen in the well-known lithograph
by Rittmiiller, which shows Gauss standing on the terrace of the Observatory, before the Rotunda. On the left-hand side of the building is affixed a plaque commemorating the electromagnetic telegraph of 1833; a plaque is affixed to the right-hand side in memory of the founder of astrophysics, K. Schwarzschild, Director of the Observatory from 1901 to 1909.
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In 1816, on completion of the new Observatory b e y o n d the Old Town, in Geismar Landstrasse (No. 11, some 10 minutes walk east of the Gauss-Weber Memorial). Gauss took u p official quarters in the west wing (fig. 12). At an elevation of 161 meters, the Observatory served not o n l y for watching the skies but also for more m u n d a n e matters: From this point, Gauss took note of the n u m b e r of paces to various points in G6ttingen that were of importance to him. He died, in his living quarters in the west wing, on February 23, 1855. The m u s e u m of the Observ a t o r y displays the c o p p e r memorial plaque w h i c h King George VJof H a n o v e r presented for Gauss's deathbed. Gauss is buried in the Albani Friedhof (figs. 18 and 19). Gauss's first lecture, in the s u m m e r semester of 1808, was on astronomy. Later, in the new Observatory, Dedekind 6 in 1850 describes the reluctant lecturer Gauss7: "Across the r o o m from the door,.., s o m e w h a t r e m o v e d from the t a b l e . . , sat G a u s s . . . wearing a light, black skull cap, a m e d i u m long b r o w n frock coat, gray breeches; usually sitting easily, bending slightly forward and gazing d o w n w a r d s before him, hands clasped across his body. H e spoke without notes, very clearly, and with simplicity."
Figure 13. The velvet cap worn by Gauss, his quill pen, and his telescope, with a focal length of 0.43 meters, in the Museum of the Observatory (Photograph by Professor Voigt).
6 Dedekind read mathematics in G6ttingen from 1850, obtained his doctorate under Gauss in 1852, was habilitated in 1854,and remained as Privatdozent in G6ttingen until 1858. From 1864 he was a professor in Brunswick. 7Gauss spoke of the "thankless task" on which his "precious time" was dissipated in a letter dated October 26, 1802to Olbers (1758-1840).
His skull cap, together with other belongings, and historical astronomical instruments, can be soen in the Observatory's little m u s e u m during office hours, preferably by a p p o i n t m e n t (figs. 13 and 14). O t h e r implements used by Gauss are located in the First Institute of Physical Sciences, Bunsenstrasse; and the municipal museum, on Ritterplan, also possesses some historical exhibits of this nature. Gauss f o u n d e d a school of a s t r o n o m y but not a school of mathematics, though here too he had his pupils,
Figure 14. Old instruments on view in the Observatory museum. The Museum also has a "piece of electric wire used
in the first telegraph," vouched for by Professors Heyne and Weber. (Photograph by Professor Voigt). THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994 5 7
Figure 15. The towers of St. Johannis, originally a romanesque church. The present construction is that of a 14th-century "hall church," a church with nave and aisles built to the same height. The left-hand tower served as a support for Gauss's telegraph wires. Today the tower houses students, who will permit visitors to view it on Saturdays.
Figure 16. First page of Gauss's journal, with the entry on the construction of a regular heptadecagon. The journal was discovered only in 1898 by P. St~ickel (18621919). (By kind permission of the Handschriftenabteilung der Nieders~chsischen Staats- und Universit~itsbibliothek Giittingen, Cod. Ms. Gauss Math. 48 Cim.).
Figure 17. The telegraphic code used by Gauss and Weber, with the encoded text of what was presumably the first message transmitted, W i s s e n v o r M e i n e n , Sein v o r Scheinen (Knowing and not deeming, being and not seeming), and the time taken to transmit it: 489minutes for the 30 characters. In a present-day repeat, it proved impossible to transmit the message on the reconstructed telegraph in less than 6 minutes, in spite of choosing a night-time transmission, when there was not much interference. 58
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among them J. B. Listing who obtained a doctorate under Gauss in 1834. In 1849 Listing was back in G6ttingen as Professor of Physics. The mathematical term "topology" derives from his Vorstudien zur Topologie ("Preliminaries of topology," 1847). He also coined the term "geoid" for the figure of the earth everywhere normal to the direction of gravity. The M6bius strip, a surface with only one side and only one edge, named for August Ferdinand M6bius (1790-1868), another of Gauss's students, was discovered independently by Listing in 1858. Of the various places where Listing resided in G6ttingen, including the Observatory, only 21 Prinzenstrasse (at the corner of Am Leinekanal) exhibits a memorial plaque (inner courtyard of the Michaelishaus, once the Londonsch/inke). Anyone who happens to have a 10-Deutschmark note can find on the obverse sid.e Gauss's portrait and some views of G6ttingen - - to wit, level with Gauss's eyes, the new Observatory; beyond, the two towers of the Johanniskirche (fig. 15) and the tower of the Jacobikirche; level with his mouth, the Grand Hall of the University. On the reverse side of the 10-mark note is Gauss's geodetic heliotrope, constructed from an English sextant (the vice-heliotrope can be seen in the First Institute of Physical Sciences). In the bottom right-hand corner is a section of the Basic System (Hauptsystem) for the Measurement of Arc as used in the Kingdom of Hanover (1820-1848). Gauss himself surveyed the Basic System between 1821 and 1827, including the famous triangulation Brocken-Inselsberg-Hoher Hagen, with sides of a maximum length of 106 kilometers.
Figure 18. The grave of C. F. Gauss (30. 4. 1777-23. 2. 1855) in the Albani Friedhof. In the background can be seen the grave of the chemist J. F. G m e l i n (1748-1804), one o f the earliest authors of a history of chemistry (1797).
The Hoher Hagen, 508 meters in height, is in the vicinity of Dransfeld, some 15 kilometers west of G6ttingen, reached along Federal Highway No. 3, in the direction of Hannoversch-M(inden, passing on the way the municipal cemetery of G6ttingen with the graves of F. Klein, D. Hilbert, J. B. Listing, C. L. Siegel, W. Schmeidler, W. Weber, M. Planck, M. von Laue, M. Born, W. Nernst, O. Hahn, and E. Wiechert. About 1 kilometer before reaching Dransfeld, turn south toward JiJhnda, bearing west some 3 kilometers along the way, at H/igernhof (always following the sign Gaussturm), you will find a Gaussturm, a post tower, with an excellent view of the town, and refreshment rooms in which mementos of Gauss (pictures, a bust, geodetic heliotrope, etc.) are on display. Dirichlet
En route from Bonn to Breslau, Dirichlet chose to travel via G6ttingen so as to meet Gauss. Gauss, who described Dirichlet's papers as "jewels," would have dearly loved to have their author on his staff, but it was not until Gauss's death that Dirichlet came to G6ttingen, as his successor. H. Minkowski writes, "In the fall of 1855, Dirichlet took up residence in G6ttingen. Here he set up house in Miihlenstrasse No. 1, a pleasantly situated house with a garden, furnishing it according to his taste, and the peace of a small town, something he had not enjoyed since his youth, was sufficient recompense for the superficial convenience of living in a large town like Berlin. True, he had fewer students than in Berlin... [but he] attracted many young mathematicians to G6ttingen."
Figure 19. Location of graves in the cemeteries Albani Friedhof and Bartholom~ius Friedhof.
Before taking the house in M6hlenstrasse, Dirichlet had lived at 1 Gotmarstrasse. The house and garden in M/ihlenstrasse, described in his diaries by Varnhagen von Ense, is not far away. It was acquired by Dirichlet in 1856, and here he lived until his death in 1859. He lies buried in the Bartholom/ius Friedhof, together with his wife, the younger sister of the composer Felix Mendelssohn (figs. 19 and 20). THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994 5 9
Figure 20. Grave of P. G. Lejeune Dirichlet (13. 2. 1805-5. 5. 1895) and his wife (right) in the Bartholomtius Friedhof, Weender Landstrasse.
It was Dirichlet's supreme honor to succeed his revered princeps mathematicorum Gauss in G6ttingen. Gauss's biographer, Sartorius von Waltershausen, writes that Dirichlet always carried Gauss's Disquisitiones arithmeticae and expounded on it, just as a clergyman always carries the Bible. Riemann
Riemann enrolled in G6ttingen on April 25, 1846, to read theology and philosophy, switching to Berlin in the summer semester of 1847, where he also attended lectures by Dirichlet. In the summer semester 1849 he returned to G6ttingen to read mathematics--living, as he had done before, at 27 Rote Strasse, a street leading eastward from the Market Place, where after him the future Imperial Chancellor Bismarck lodged as a student. From 1849 to 1851, he lived not far from the Jacobikirche, 18 J~denstrasse, after obtaining his doctorate s under Gauss (1851), at 4 Weender Strasse; and finally, from 1854 to 1857, at 18 Barf6sserstrasse (corner of J/idenstrasse, only a few yards east of the Market Place), where a memorial plaque was set up in 1890 on the proposal of Felix Klein. Riemann completed his habilitation in 1854 with a paper "On the representability of a function by a trigonometric series"; the proceedings included the famous lecture "On the hypotheses which lie at the foundations of geometry." Philosophically, he was influenced by Herbart (1776-1841), who, coming from Kant's Chair in K6nigsberg, taught in G6ttingen from 1831. Herbart lies buried in the Albani Friedhof (close to the Schildweg boundary), as do his successor Lotze (1817-1881, near the Schwanenteich), and Gauss (fig. 19). Riemann's precarious financial situation did not improve until he succeeded Dirichlet in 1859. Ill health obliged him to undergo treatment in Italy at various times, and it was in Italy that he died in 1866.9 8 See R. R e m r a e r t , the R i e m a n n - f i l e , Mathematical no. 3, 44 ~8, for a n a c c o u n t of t h i s event. 9 See E. B e l t r a m i , R i e m a n n ' s I t a l i a n t o m b , (1987), no. 3, 54-55.
Intelligencer, 15 (1993),
Mathematical Intelligencer, 9
60 THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 4, 1994
Acknowledgments
I am indebted to Professors Beuermann, Siebert, and Voigt for their information and helpful suggestions, and also to Professor Voigt for permission to use two photographs. For his assistance and for permission to use illustrations, my thanks go to Dr. H. Rohlfing of the Nieders~chsische Staats- und Universit~tsbibliothek G6ttingen. I thank G. L. Baurley for his careful translation. Bibliography
Biermann, K.-R., C. F. Gauss. Der "Ffdrst der Mathematiker" in Briefen Gesprdchen, Urania, Leipzig, and C. H. Beck, M6nchen, 1990. Dedekind, R., Gesammelte Werke, vol. 2, E Vieweg, Braunschweig 1931, p. 294. Georg Christoph Lichtenberg, 1742-1799. Wagnis der Aufkla'rung (Catalogue to an exhibition). C. Hanser, M/,/nchen, 1992. Minkowski, H., P. G. Lejeune Dirichlet und seine Bedeutung, Jahresbericht des Deutschen Mathematikervereins XIV (1905), 152. Neuenschwander, E., and Burmanns, H.-W., Die Entwicklung der Mathematik an der Universit~it GOttingen, Georgia Augusta 47 (1987), 17-28. Nissen, W., Gb'ttinger Gedenktafeln, Vandenhoeck & Ruprecht, G6ttingen, 1962; supplement, 1975. Voigt H.-H., Geschichte der GOttingerSternwarte, Georgia Augusta 56 (1992), 27-38.
Karl-Sudhoff-Institut fiir Geschichte der Medizin und der Naturwissenschaften Universitfit Leipzig D-04109 Leipzig, Germany
A Closer Look at the Cast Ironwork of Australia Heather McLeay In the spring t e r m of 1993, I h a d the good fortune to be a visitor to the University of S y d n e y and, travelling a r o u n d the city, I h a d the o p p o r t u n i t y to a d m i r e a n d s t u d y the great variety in the cast iron "lace" which can be seen on balconies, railings, d o o r w a y s , a n d civic architecture a r o u n d e v e r y corner in the city's suburbs. Australia has a rich heritage of decorative cast ironw o r k . M u c h of the cast iron a p p e a r e d d u r i n g the 1870s a n d i880s, a n d even c o m m o n p l a c e objects such as pillar b o x e s and drinking fountains were v e r y ornate. The m o s t o b v i o u s examples of this decorative i r o n w o r k occur in railings and balustrades for v e r a n d a h s and usually h a v e a repeating pattern. I a l w a y s appreciate a g o o d pattern. Where there is pattern, m a t h e m a t i c s is a l w a y s present. For instance, a strip p a t t e r n or "frieze" pattern u s e d to decorate a p o t or building can be one of just s e v e n distinct types. D u r i n g m y visit, I decided to h a v e a look at the different frieze types occurring in the a b u n d a n t cast iron of Australia.
tion, reflection a n d rotation there are just seven w a y s in which this can happen. 1 Figure 1 s h o w s h o w all s e v e n are constructed.
First Type m Simple Repetition (p111) The simplest w a y to create a frieze pattern is to design a basic m o tif a n d to k e e p repeating it, that is to translate it, as in Figure 2. t There is an internationally recognised notation for each of the seven frieze types and, in this article, this is given in brackets after the descriptive name. It consists of the letter p followed by three symbols; for example, pmm2. The first symbol after the p is used to indicate vertical mirror lines, the second to indicate horizontal mirror lines and the third to indicate 180 degree rotations (haIf turns). Thus, if vertical lines of reflection exist the first symbol is m; otherwise it is 1. If horizontal mirror lines exist, the second~ymbol is m, whilst for 8 glide reflection (see fourth type plal) it is a and for no horizontal mirror line the symbol is 1. The last symbol is 2 if there are centres of 180-degreerotations; otherwise it is 1.
T h e S e v e n Patterns A frieze pattern has the characteristic that it repeats e v e n l y or regularly in a precise way. By using transla-
Figure 2.
Figure 1. THE MATHEMATICAL INTELLIGENCERVOL. 16, NO. 4 (~)1994 Springer-Verlag New York 6 1
Figure 3.
Figure 4.
Figure 5. With regard to cast iron lacework, this type occurs more frequently in the "fringe" or "apron" above a verandah. The traditional "rinceau" design 2 is shown in Figure 3.
Second Type m Vertical Reflection (pm11) In Figure 1, the team pulling to the left in the tug-of-war is the mirror image of the team pulling to the right. The vertical mirror lines can be thought of as fold lines along which the design can be folded over onto itself.
2 Rinceau f o r m s were illustrated in A New Booke of Drawings b y Jean Tijou in 1693. Tijou w a s a French s m i t h w h o worked in E n g l a n d for Sir C h r i s t o p h e r Wren.
62 ~ E MATHEMATICALINTELLIGENCERVOL. 16, NO. 4, 1994
This type of pattern is by far the most common in the cast iron all over Australia. An ornate example from the Paddington district of Sydney is shown in Figure 4.
Third Type--Vertical and Horizontal Reflection (pmm2) The third type of frieze pattern is obtained by simply reflecting the whole of the second frieze in a horizontal mirror line so as to turn it upside down (see row 3 of Figure 1). There is now a horizontal fold line along which the design can be folded onto itself as well as the vertical fold lines. This structure is also quite common throughout Australia. Interestingly, it seems to be more common in Brisbane than in Sydney. Figure 5 is just one example from Norman Parade, Eagle Junction, Brisbane.
Figure 6.
Figure 7.
Figure 8. Fourth T y p e - - G l i d e Reflection (plal) The next design which I noticed as I travelled around is the same in structure as the pattern made by footprints in the s a n d - the motif is first reflected and then slid along a specific distance each time. This type almost invariably occurs in the fringe above the verandah. Figure 6 is another "rinceau" design from Fortescue Street, Brisbane.
very elusive. The fifth is exemplified very well in the pattern of the Art Nouveau bracelet shown in Figure 1. It has vertical lines of symmetry as well as horizontal glide reflections. An example in cast iron is shown in Robertson [1] on a house in Mangalore, Tasmania. I also found an example in wrought iron on the Polo Club in Brisbane (Figure 7).
Fifth Type--Vertical Reflection and Glide Reflection
Sixth Type D H a l f Turn (p112) The sixth pattern has the same structure as the well-known Greek key design shown in Figure 8. This has rotational symmetry. The
(pma2)
The three remaining design structures were
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
63
basic motif is rotated through half a turn and then back again to give the continuous design. There is a cast iron version of the Greek key design on a house in Park Street, South Yarra, Melbourne. Figure 9 is a different example of the sixth frieze type, from a house in Clayfield, Brisbane, where the design is placed at the base of a simple railing pattern in order to give additional interest.
Seventh Type--Horizontal Reflection ( p l m l )
The last and most elusive design in the cast iron work of Australia has a horizontal line of symmetry running through it. I found this type only as a vertical support to canopies.
Figure 9.
Figure 10.
Figure 11. 64
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
This frieze type really gives a sense of "going somewhere," just like the ladybirds in Figure 1, and does not lend itself to a static balcony situation. Well, that is my explanation as to why this is so unpopular! See what you think. Figure 10 is an example placed horizontally; Figure 11 shows where it occurs in the verandah itself.
Complex Designs The artisans creating these panels to decorate verandahs had the opportunity to use one of these seven structures. Sometimes they would use their skill to tantalise us. An impression is given of one structure while another is imposed. For example, the design
Figure 12.
Figure 13. in Figure 12, from the Palace Hotel in Perth, is deceiving. It looks as if it is based on vertical reflection (second type, pm11) but is actually a simple repetition (first type, p111). Take a closer look at the spiral in the lower part of the design. This motif has no symmetry in contrast to the central and the upper motifs which have vertical lines of symmetry. The design in Figure 13 is composed of a motif which, were it placed true to the vertical, would be a seventh type, p l m l . There would be a horizontal line of symmetry running through the whole pattern. But, because the motif is leaning to one side, this fold line for the design is destroyed and the frieze becomes the first type (p111) with only translational symmetry. Figure 14 is another design which I found in Sutherland Avenue, Ascot, Brisbane. At first glance, it appears to have the fifth structure type, but it doesn't quite make i t - - have a closer look at the extent of the spirals and the "fleur de lys" parts of the pattern. And how frequently should those circles in the centre occur? So which type do you think it is?
Our aesthetic appreciation of design relies,on an innate subconscious ability to recognise pattern, and our personal preferences for certain designs are deeply rooted in our culture. Washburn and Crowe [2]'explain that within a given culture there will be preferred symmetry or symmetries in the decoration of objects. Anthropologists have noted that the pattern types found in the designs used by different cultures can be used to authenticate pottery and other objects and also to date the developments and changes within those cultures. For example, archaeologists have found that in the pottery of some North American Indians, only one structural pattern was present. Although all seven designs have been found to occur in the clay pipes of Ghana, 72% are type pmm2. In the case of the pottery of the Incas, all seven types exist, but the predominant structure is again pmm2 (40%) with pro11 coming second (20%) and pma2 third (11%). In the cast ironwork of Australia, by far the most common structure is pm11, and in second place pmm2. This observation is drawn from my own experience and also from the illustrations in books on the subject. However, in Brisbane there is a noticable difference from Sydney in that there are many more examples of pmm2. The exact proportions I must leave for another study. Finally, on returning to Britain after my stay in Australia, it seems that a great deal of work has been going on replacing much of our own cast i r o n w o r k - - or was it there all the time and I just hadn't noticed before? References
1. E. G. Robertson, Decorative Cast Iron in Australia, Viking O'Neil, Penguin Books Australia, Sydney, 1990. 2. D. K. Washburn and D. W. Crowe, Symmetries of Culture, University of Washington Press, Seattle and London, 1988. Figure 14.
University of Wales Bangor, Wales THE MATHEMATICAL INTELLIGENCER V O t 16, NO. 4, 1994 6 5
Jet Wimp*
Bayes or Bust?: A Critical Examination of Bayesian Confirmation Theory by John Earman Cambridge, Massachusetts: The MIT Press, 1992. US $35.00, xiv + 272 pp. ISBN 0-262-05046-3
Reviewed by Evelyn Mitchell The Reverend Thomas Bayes died in 1761. He left a considerable estate and a jumble of personal papers. Among these were two essays, one dealing with asymptotic series, and another which was subsequently published in the Philosophical Transactions under the title, "An Essay Towards Solving a Problem in the Doctrine of Chances." Bayes's essay was mainly ignored, even by the early giants of probability theory, until it was reprinted in facsimile in 1940. Bayes's ideas are now so important in statistical inference theory that most people working in that area are known either as "Bayesians" or "non-Bayesians." "Bayesianism" is probably the predominant view held by those in the philosophy of science concerned with statistical inference, or, more generally, with the confirmation of scientific hypotheses. As might be expected, emotions run high among proponents and non-proponents of Bayesian doctrines, mainly because these ideas impinge on how one sees the world in general. Bayes or Bust? is a significant contribution to the discussion concerning the strengths and weaknesses of Bayesian Confirmation Theory. John Earman is lucid throughout, whether discussing the historic roots of probability theory as it applies to the problem of induction, or proposing an alternative to a Bayesian model. This is not a complete tour of the territory claimed by Bayesians. Earman is primarily concerned with the questions raised in scientific confirmation theory that lend themselves to a Bayesian discussion. Still, he does not neglect interesting side roads like those leading to Formal Learning Theory and Kuhn. Roughly speaking, Bayesians hold that it is possible to estimate the parameters of a probability density for * Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA. 66
a phenomenon before making observations. They argue that it is always possible to determine a distribution in a coherent way by making a subjective judgment of the relative possibilities of the values of the parameters. Non-Bayesians insist that inference should not involve subjective judgments, that it should be based entirely on objective considerations, i.e., the relative frequencies measured in an actual population. Its proponents believe that Bayes's theorem on conditionalizing provides a mechanism by means of which statements can be driven to conform more closely to reality. Statements conform to reality when the subjective probability with which a rational agent holds to a belief is close to the objective probability of the event in the world. An agent who consistently applies conditionalization over prior probability assignments will never suffer the penalty that irrational agents would of losing an infinite series of bets, for instance, to an imaginary Dutch bookie in the notorious Dutch-book scheme. That classical scheme, thoroughly explained in this book, attempts to convince us that the degree of belief (of, say, a player in a game) can provide a means of justifying the probability axioms of the game. This is opposed to the idea of assuming that the degree of belief of the player is controlled by the principles of probability. The former is the approach Bayes himself took in exploring the connection between belief and betting behavior. This is not to say that any agent in consistently applying Bayesian mechanics always would reach a state of belief which closely conforms to reality, only that the player would eventually hold a belief as closely resembling reality as that held by any other rational agent. Earman begins with a presentation of Bayes's original proof of the application of probability to the problem of induction. He believes that many of the pitfalls encountered by contemporary Bayesians could be avoided by understanding Bayes's original presentation of the probabilistic approach to induction. Bayes's intent was to solve the problem of scientific inference in the context of a set of similar trial runs. He was not attempting to formulate a theory of subjective knowledge. This distinction is one that Earman uses to good advantage in his discussionof the challenges that Bayesianism presents to one presently interested in the problem of scientific induction and empirical knowledge.
THE MATHEMATICAL INTELLIGENCERVOL. 16, NO. 4 (~)1994 Springer-Verlag New York
After establishing a historical foundation, Earman presents a passionate look at the strengths and weaknesses of Bayesian Confirmation Theory. He is neither an uncritical defender nor a dogmatic proponent of Bayesianism. As Bayes or-Bust? shows, he is aware of both the strengths and the weaknesses of Bayesianism as it applies to problems in scientific reasoning. "I confess that I am a Bayesian" he writes, "---at least I am on Mondays, Wednesdays, and Fridays .... On Tuesdays, Thursdays, and Saturdays, however, I have my doubts not only about the imperialistic ambitions of Bayesianism but also about its viability as a basis for analyzing scientific inference." Unlike the case of a recent book which also discusses Bayesianism as it relates to scientific reasoning, Earman clearly states both the strengths and weaknesses of a probabilistic approach to reasoning. Further, his presentation of modern Bayesianism is rigorous and thorough,
... Are the beliefs of the bettor controlled by the laws of probability? and though the less mathematically fluent reader may find the going difficult, the effort is well spent, for the reward is a sound understanding of the assumptions and mechanisms of Bayesian probability. Earman's primary concern, however, is the application of Bayesianism to philosophical questions, particularly those raised within the context of scientific confirmation of theories. Philosophers of science see Bayesianism as a w a y of managing the thicket of scientific reasoning by appeal to a model which is simple enough for wide application to many different historical contexts and examples, yet is powerful enough to provide a normative standard. Bayesianism, therefore, would serve not only as a way of explaining scientific decision making, but also as a w a y of guiding scientists in maintaining valid reasoning when deductive logical methods are not available to them. Bayesianism, in some versions, has successfully met challenges raised against other theories of confirmation and belief change. These traditional challenges amenable to a Bayesian solution include the Raven's Paradox, as well as Quine and Duhem's Problem. In addition, problems which arise only within a probabilistic context, such as the problem of Zero Priors, Earman shows here to be solvable by a Bayesian approach. The heart of the book is in Earman's presentation of serious challenges to Bayesianism. The simplicity of a Bayesian approach to the problem of scientific induction grants it great explanatory power. However, the approach requires that an agent be able to calculate the change in probability entailed by the change in credence of a particular evidence statement. A problem arises
when the experience of the world remains the same, but the explanation an agent puts upon the experience requires a change in the probability attached to a series of belief statements. This problem, in which evidence which is already known with certainty seems to drive a change in probability allocations, is called the problem of Old Evidence. This is a severe challenge to Bayesians. It requires either that an agent reprocess a probability statement by passing it through the conditionalizing engine without justifying the reprocessing by reference to an experience of the world, or that the agent be mistaken in his or her previous allocation of probabilities when first confronted with a piece of evidence. This second problem raises bizarre situations when the decisive piece of evidence in support of the old theory has been known for a very long time, and so the mistake could not be in the agent's allocation of probabilities. Several other severe challenges arise out of the subjective nature of a Bayesian explanation of induction when it is confronted with the objective requirement of scientific knowledge. We tend to view scientific knowledge as a rational and objective collection of beliefs about the world. A Bayesian says that this objectivity rests ultimately upon the subjective state of a rational agent in its allocation of prior probabilities and the credence with which it holds hypotheses. This explanation is dissatisfying to those who believe that scientific knowledge imparts a special quality of certainty to knowledge gained through experience. Earman poses another challenge to Bayesians in his presentation of a possible adjunct to the theory. In one simple form of Bayesianism, an agent is required to hold to all possible hypotheses and all logical evidence statements. When confronted with a confirming instance of an evidence statement in experience, an agent is required to adjust his or her probability allocations in accordance with this experience. A problem arises, however, because even in the simplest belief system consisting of two hypotheses, an infinite number of statements is possible. It is over this infinite set of statements that the total quantity of credence must be allocated. In such a statement of Bayesianism, it is obvious that no amount of evidence from experience could drive the agent's probability allocations to such a degree that he or she would discard an hypothesis in favor of another. Earman proposes as a solution to this problem a form of eliminative induction in which groups of hypotheses can be discarded not on the basis of any one piece of evidence, but because of an understanding of the requirements of the form of a solution to the problem. The example he gives is that of the large number of competing theories of gravitation considered by physicists in this century. Some of the possible theories were discarded not because of any particular piece of evidence, but because they entailed certain statements about gravity which were known to be false. In weeding the garden of theories, fewer healthy theories can be watered with an agent's allocation of probabilities. THE MATHEMATICALINTELLIGENCERVOL. 16, NO. 4,1994 6 7
Following this, Earman discusses Bayesianism in relation to the questions raised by Kuhn regarding scientific revolutions, as well as the problem of the relation of observation to evidence. Earman finishes his tour of the landscape of Bayesian confirmation theory with a discussion of an argument made by Putnam against Carnap's version of Bayesian~ ism. John Earman has written a thoughtful and challenging book about an intriguing area of philosophical and mathematical argument. He has set out to examine the historical roots of Bayesianism, the mechanism of its application to scientific decision making, and the problems peculiar to a probabilistic formulation of the solution to the problem of induction. He has produced a work which never shirks from the double task of clarity and rigor while still managing to entertain and enlighten. The journey is not an easy one, but this is not the fault of Earman, but rather the difficulties of the geography of Bayesianism, which promises oases to the weary traveler that upon closer examination turn out to be mirages. The journey is not in vain, however, for the rewards of Bayesian vistas are more than worth the effort expended by the thoughtful reader.
Philosophy Department University of Saskatchewan Saskatoon, Saskatchewan Canada S7N OWO
Selected Problems in Real Analysis Volume 107, AMS Series of Translations o f Mathematical Monographs by B. M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov Providence, RI: American Mathematical Society, 1991, x + 370 pp. US $112, ISBN 0-8218-4559-4 Reviewed by Jet Wimp
This book is full of marvels. I feel obliged, though, to caution any potential readers with addictive personality traits: animal rescuers from multicat households, closet chocolate bingers, spenders whose spouses beg them to scissor their credit cards, and most certainly those who have waged midnight battles against the urge to solve just one more tiny problem in some problem book. There's some of the addict in all of us. I'm reminded of the vendor who made a living during the depression by standing on a street corner and handing out exactly one free roasted cashew to each passing pedestrian. The American Mathematical Society, that staid and venerable old institution, has missed a shot at a canny if unsavory marketing ploy. The Society should have purchased advertising slots in the usual journals, including 68 THE MATHEMATICALINTELLIGENCERVOL.16, NO. 4, 1994
this one, and into each dropped a single, or perhaps several, roasted cashews such as I
Let an be a positive sequence with S = ~ a,~ convergent. Show that S* = ~ a~-1/~ is convergent. Yes, indeed - - scores of usually sedate mathematicians clamoring to claim their full bags of roasted nuts, screaming their credit card numbers over the telephone to the bemused office workers in Providence. However, the AMS seems to have no idea of the treasures in this book and has devoted no special effort to advertising it. To the Society, it is simply Volume 107 in a dogged series of translations of highly specialized Russian mathematical monographs. For the teacher, for the researcher, this book will prove a priceless resource: an aid in the preparation of course problems or doctoral exams, an energizer of one's research efforts, an avenue to new and often surprising ideas. The only book I can think of which compares with it is similarly excellent and a much worn inhabitant of my library, Problems in the Theory of Functions of a Complex Variable, by L. Volkovysky, G. Lunts, and I. Aramanovich (MIR, Moscow, 1977). Many are the Drexel graduate students who, unknowingly, have weathered its assaults. This and the book under review are perfect desert island books. I selected the above problem for two reasons: It is characteristically lucid and simple, just right for my intermediate analysis class (though requiring, perhaps, some sort of hint), and the solution provided in the book, unexpected and ingenious, not only demonstrates the convergence of the problem series but can be nudged to reveal that S* G (1 + v/S) 2 [11. I was charmed. Show that the series Y~ I sin n21/n is divergent. Because of our experience in calculating various lim infs and lim sups of various sines and cosines, the problem may assume for us a number-theoretical flavor. Dirichlet's theorem, maybe? Kronecker's theorem? Some slick application of the approximation of 7r by rationals, we imagine, and we can effect an easy slide to home. Alas, it's not to be. Number-theoretical considerations are irrelevant. Only after we decide that the convergence of this series is equivalent to that of
E
[ sin(n - 1)21 + [ sinn2[ + [ sin(n + 1)21, n
since the given series differs from one-third of this by a convergent series, can we mount a successful attack [2]. The above two problems have something in common, and that something is shared by many other problems in the book: they are truly baffling. There doesn't seem 1
Here and in w h a t follows, I will italicize p r o b l e m s selected from the book.
to be a way to get a handle on t h e m - - n o facile slathering on of some hoary theorem will provoke the dutiful appearance of a solution. Often the solutions turn out to involve a trick, although I hesitate to use that disparaging term. We've all heard a ~trickdenounced as something that works only once, in contrast to the more respectable method, something that works lots of times. Nevertheless, the tricks required here can frequently be used to solve more ambitious problems, problems that transcend the merely problematical to become full-fledged theorems. The same method used to solve the first problem can be used to show: Let an, Cn be positive sequences with y~ an convergent, Cn = O(1/Inn). Then y~ a~-r converges.
We have to ask ourselves if tricks are always getting such bad press, and the objects on which we use them we belittle with the label "problems," why is the mathematics created with them often so beautiful? I was thinking of just this the other day as I was attending a lecture given for mathematics students by a mathematical hobbyist, by trade a physician. He was discussing topics in elementary number theory, material with which most of us are familiar. It's all too easy for us to dismiss such people as dilettantes as we pursue our frenzied quest, say, to lower McThistlebottom's estimate of e = 3.84 to the earth-shaking ~ = 3.835. I found the talk refreshing. 2 What was refreshing about it was the lecturer's unadorned affection for the beautiful result, the lucent, elegant proof. Number theory especially is full of such satisfactions. He devoted several minutes to Dirichlet's proof concerning the approximation of irrationals by rationals: If c~ is an irrational number, there are an infinite number of integers q such that for some integer p, Ic~ - P/ql < 1/q 2.
The proof, which proceeds via the pigeonhole principle, is very simple. The pigeonhole principle has proved to be uncommonly powerful in the rapidly expanding field of combinatorics, and articles examining its philosophy and methodology have recently appeared. I want to discuss the principle a little here, as it seems to me this technique shuffles uncomfortably back and forth between the domains of methods (used to prove theorems) and of tricks (used to solve problems). In this book it gets a thorough workout, particularly in the solutions of the problems on inequalities (Section 1.2). I think of the principle as a trick--not because of its conceptual transparency, but because those situations for which it is appropriate do not announce themselves. In analysis, it is often clear just @hen one should reach for a Helly convergence theorem or for the Fourier inversion 2 Paul Roberts, in a recent sci. math posting, addressed this issue: "[I] wonder whether in other fields the interested amateur is held in such apparent contempt by the professionals, or is this true only in mathematics?"
formula. On the other hand, though many problems in combinatorics seem intuitively amenable to the pigeonhole principle, vulnerable problems in other fields, such as number theory, may appear remote from it. Doesn't its frequent use qualify it as a bona fide method? Perhaps, but prepping any given problem so that the principle can be used may involve imaginative trickery. Some writers try to bestow profundity on the principle by gussying up its statement. The following quotes are from two recent books on discrete mathematics: FIRST AUTHOR: The pigeonhole principle. If n pigeons fly into k pigeonholes and k < n, some pigeonhole contains at least two pigeons.
(There follows an illustration of boxes containing pigeons milling about.) SECOND AUTHOR: The pigeonhole principle. For positive integers ra and n, let A be an m-element set, B be an n-element set, and f be a function from A to B. If m > 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<,~ -n
The first sum does not exceed )~~ an and the second does not exceed ~ )~I-'L Minimizing the upper bound so obtained, s - 1) + )~~ an, over all A > 1, yields the inequality cited. This kind of argument is very familiar to those working in the trenches of hard analysis, for instance, the theory of the Riemann zeta function. 2. We show that the numerator in the sum above is bounded from below by some small positive number for all n. If not, then for some n we have
n2=Trk+e,(n-1)2=lrk'+c ',
( n + l ) 2 =Trk"+~",
k, k', k" are integers, and }el,le'l, ]e"I < 1/4. Taking differences gives 2 = (n - 1 ) 2 - 2 n 2 q- ( n q- 1 ) 2 ~r(k' + k" - 2k) + e' + ~" - 2G which is impossible. This result has obvious generalizations. 3. No, because ~--~(
=
1 )~--~2 a,~+n-~-~a~ _> -=C~.n
John von Neumann and the Origins of Modem Computing by William Aspray Cambridge, MA: MIT Press, 1990. xvii + 376 pp. US $43.00 hardbound, ISBN 0-262-01121-2
Reviewed by Peter C. Patton The author of this book is Director of the Center for the History of Electrical Engineering at the Institute of Electrical and Electronics Engineers. He has furnished us with a well researched history of the computing contributions of one of the twentieth century's greatest mathematicians. The text narrative is 270 pages, followed by 72 THEMATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
82 pages of end notes and two bibliographies totalling 30 pages. In May 1958, the Bulletin of the American Mathematical Society published a retrospective of von Neumann's mathematical contributions (vol. 63, no. 3, part 2). In June 1961, Pergamon Press published a six-volume set of his collected works edited by A. H. Taub, of which Volume 5 treats computer design, automata theory, and numerical analysis. Later, in 1972, Herman Goldstine, one of von Neumann's closest collaborators in computing, published The Computer from Pascal to yon Neumann, Princeton University Press. This book discusses von Neumann's contributions to the ENIAC, the EDVAC, and the IAS (Institute for Advanced Studies) computer in a factual historical narrative fashion. Aspray's book goes much deeper to discover von Neumann's motivation for setting aside doubly brilliant careers in mathematics and mathematical physics to take up numerical computation, a subject not highly regarded at the time. Aspray also explores von Neumann's unique contributions to the development of computing machinery. His contributions are so fundamental and pervasive that today both computer scientists and computational scientists take them for granted. Aspray does an excellent job of unrolling history to present it as it happened, in context rather than just looking down the time tunnel toward the past. Today's computer architects sometimes dismiss von Neumann as the father of sequential stored-program computers without giving him credit for his vision. In fact, von Neumann anticipated parallel processing, which the development of new logical devices would one day make feasible. It was a study of neurophysiology that led von Neumann to some of his early computing notions. His first logic diagrams employed McCulloch-Pitts "neurons" as an abstraction of vacuum-tube-based "and," "or," and "not-and" circuits. Aspray documents how von Neumann deliberately chose a studied sequential approach to computer logic design, logic function, and programming because technology and human experience were simply not up to doing numerical computation in a parallel fashion. We need to give him credit for his vision (parallel processing) as well as his achievement (the sequential stored-program computer). Aspray points out that most of the leaders in digital computing following World War II, such as J. Presper Eckert and Jay Forrester, were young men who had not yet established their scientific reputations. On the other hand, most senior members of the American scientific community who had an interest in computing, such as Vannevar Bush and Samuel Caldwell, were wedded to analog technology. Von Neumann was unusual among the scientific elite for his knowledge of and dedication to digital computing technology. His stature with the Atomic Energy Commission (precursor to the Department of Energy), the U.S. Air Force, the U.S. Navy, and IBM was essential to the early development of the
computer industry. No lesser mathematician or scientist could have convinced his colleagues at the Institute for Advanced Study to undertake the construction of a computing machine, nor managed to secure the funding and support to complete the task. Aspray documents the national, academic and institutional politics of the IAS computer project in an interesting and entertaining manner. This book also documents another aspect of von Neumann's vision, namely, his anticipation of computational science as a third branch of discovery alongside theoretical and experimental science. Von Neumann saw the computing machine as a mathematical laboratory for research on methodology, not just as a device to solve physical problems. Other computing pioneers, such as Charles Babbage and even Howard Aitken, saw the computer primarily as an interactive tool for generating mathematical tables for use in problem solving rather than as the machine that itself produced the solution. To fulfill his vision, von Neumann spent most of his later years developing and popularizing numerical analysis. His legacy today is not only the von Neumann architecture, but the prescience that anticipated parallel processing and the personal leadership that revitalized numerical analysis. Today we can demonstrate performance improvement in scientific computation of 10s or more since the first computer; of this, half comes from improvements in the computing hardware and the other half from improvements in mathematical modeling and numerical methods. We owe much for both developments to one of the greatest mathematicians of our century. I recommend this book to mathematicians, applied mathematicians, computer scientists, and computer architects. Office for the Vice Provost for Information Systems and Computing University of Pennsylvania Philadelphia, PA 19104 USA
A H i s t o r y of Mathematics: A n I n t r o d u c t i o n by Victor J. Katz N e w York, Harper-Collins, 1993. xiv + 786 pp. US $62.50 hardbound, ISBN 0-673-38039-4 R e v i e w e d by Judith Victor Grabiner
You can't tell a book by its cover, but we know where we are with the full-color cover of this one. Astronomical instruments and timepieces rest on a tablecloth woven with geometrical patterns, all in perspective. The sixteenth-century picture from which this is a detail, The Ambassadors by Hans Holbein the Younger, is reproduced as the frontispiece to Katz's book. Thus we are prepared for an attractive, well-illustrated book that respects orig-
D
A
E
H
C Figure 1. Angle trisection. (All figures are from Katz, (~) 1993, reprinted by permission of HarperCollins College Publishers.) inal sources and presents material from them, and that stresses the cultural setting of mathematics. The history of mathematics from ancient times to the present is an oft-told story. Katz's table of contents provides few surprises: one chapter on ancient mathematics, notably Egyptian, Babylonian, Indian, and Chinese; four chapters on Greek mathematics; one on medieval China and India; one on the mathematics of Islam; one on mathematics in medieval Europe; two chapters on pure and applied mathematics in the Renaissance; one on seventeenth-century mathematics; one on the invention of calculus; two chapters on eighteenth-ceritury mathematics, including analysis, probability, algebra, and geometry; three chapters on nineteenth-century mathematics with one on algebra, one on analysis, and one on geometry; and finally a lone chapter on aspects of twentieth-century mathematics. I can assure those mathematicians wanting a history of mathematics, or contemplating teaching the Department history course, that Katz covers the standard topics. However, these same readers will want to know how this book differs from others. Katz's book has two outstanding features. One is that his expositions of important mathematics closelyfollow original sources. Though Katz often modernizes the notation, he reproduces original diagrams and explains the thinking of the mathematician in a w a y recognizable to anyone familiar with the original. The other is his excellent treatment of matherfiatics outside the European tradition, especially in China, India, and the Islamic world. Among the many examples from Greek sources, my students liked the deployment of elementary geometry in Apollonius's trisection of an angle using a hyperbola. (See Figure 1.) Given angle ABC, we seek the line BE such that angle EBC is 1/3 of angle ABC. Construct A C perpendicular to BC, then complete rectangle ADBC. We choose point E (if we can) so that, if BE meets A C in F, FE = 2AB. Why does BE trisect the angle ABC? Bisect EF at G. Then FG = GE = A G = AB, so the original angle A B G = AGB. Angle AGB is the exterior angle to the isosceles triangle AGE, so AGB = 2AEG. Since AEG and FBC are alternate interior angles, AEG = FBC, so angle ABG = AGB = 2FBC. Thus FBC trisects the angle ABG. But how could we arrange that FE = 2AB? Draw CH parallel to FE and EH parallel to AC. The point H lies on the circle with center C and THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994
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radius FE (=2AB). By similar triangles D E / D B = BC/CF, or D E / A C -= D A / E H , yielding D A 9 A C = D E 9 EH, so H arctan t ---- lim E also lies on the hyperbola whose asymptotes are D B and n----+OO D E and which passes through C. So, to find E, we need that hyperbola and that circle; we find their intersection point H, and we then drop a perpendicular from H to A D Expanding the denominator and grouping the terms extended. The foot of the perpendicular, E, lets us trisect yields the angle. arctan t The more modern examples include many steps to[ ( t3 ) ward the invention of the calculus in the seventeenth = im t (12+23+---+(n-1) 2) century. Among these are a short and understandable account of Fermat's method of finding tangents; a description of Descartes's method of normals; and Fermat's and Roberval's method of finding the area under curves of the form y = p x k by inscribed and circumscribed rect- Using recursion (much as ibn al-Haytham had done), angles. While describing how Fermat and Roberval used Jyesthadeva derived formulas for the sums of integral the formulas for sums of powers of integers, Katz points powers that appear in the above formula and concluded out that ibn al-Haytham knew of such results 600 years that earlier. A problem at the end of the chapter on matharctan t = t - t3/3 + t 5 / 5 - t 7 / 7 . . . . ematics in the Islamic world asks students to use alThe Indian mathematicians cared about series like this Haytham's method to derive the formula for ~ = 1 kS. one, Katz tells us, because they were calculating the Katz describes such well-known achievements as TorriceUi's finding the volume of the solid formed by rotat- lengths of circular arcs for astronomical purposes. I chose this example because it may be new to many ing a rectangular hyperbola about the axis, and Wallis's readers, but I don't want to give the impression that this finding the area under curves with equation p = x p/q. book is a collection of nonstandard topics. Katz emphaHe also includes some lesser-known but equally imporsizes mathematics that has contributed to modern mathtant derivations. For instance, he describes Hendrick ematics. In the history of calculus, for instance, among van Heuraet's method, equivalent to adding up the sum of infinitesimal tangents to a curve, for finding the many other examples, Katz covers Newton's discovery arc lengths of various curves such as y2 = x 3, y4 = x ~, of the binomial theorem and the invention of the calculus. He gives the clearest exposition I have seen in English of y6 = X7, etc. how Leibniz discovered his version of the calculus. He elIn 1637, Descartes stated (in his Geometry) that the egantly presents Johann Bernoulli's solution of the probproblem of arc lengths couldn't be solved. In a splenlem of the curve of quickest descent and d'Alembert's did irony, van Heuraet's treatise appeared as a supplework on the wave equation. Among many nineteenthment to Frans van Schooten's Latin edition of Descartes's century examples, Katz features the first delta-epsilon Geometry in 1659. Katz observes that van Heuraet's use of the differential triangle for this purpose helped call proof, Cauchy's demonstration of this theorem: Let f (x) attention to the relationship between tangents and the be continuous on Ix0, X] and f'(x) exist there and be bounded. If A is the minimum, B the maximum, of f ~ ( x ) summing-up processes of integration. on the interval, then the ratio [f(X) - f ( x o ) ] / ( X - xo) lies After mentioning James Gregory's seventeenthcentury discovery of the arctangent series, Katz tells us that the Indian mathematician Madhava (1340-1425) had also discovered this series, and even presents the result as translated from sixteenth-century Sanskrit verses. He then gives the proof by Jyesthadeva (whose dates are claimed to be 1500-1610!) which rests on the following lemma: D c( D1 Let BC be a small arc of a unit circle with center at O. Draw the tangent to any point A of the circle. Let OB, O C meet the tangent at the points B1, C1. Then the arc BC is B1 given approximately by B1C1/(1 + A B 2 ) . Jyesthadeva's proof proceeds by using similar triangles. Once we have the lemma, designate AC1, the tangent to the arc A C , by t. Then, divide the tangent AC1 into n equal parts, apply the lemma to each part in turn, and 0 finally let n become indefinitely large. The lemma then implies (in modern notation): Figure 2. Jyesthadeva's derivation of the arctangent series. 74 THEMATHEMATICAL INTELLIGENCERVOL.16,NO.4,1994
between A and B. Then, assuming f'(x) is continuous on the interval, Cauchy concludes the mean-value theorem for derivatives. There are many other wonderful things in this book. Look at the "pop-up" solid-geometry illustration from Henry Billingsley's Euclid (Figure 3), and the hundreds of end-of-chapter problems from diverse sources, including Galileo's Two New Sciences, Benjamin Banneker's notebooks, and Gauss's Differential Geometry. Devotees of "Stamp Corner" in The Mathematical Intelligencer will enjoy the dozens of portraits of mathematicians on stamps. For instance, there are Ai-Biruni with a globe on a Syrian stamp; Kepler with his model of the universe on a Hungarian stamp; Euler with an armillary sphere on a Swiss stamp; Benjamin Banneker with his surveying instruments on a U. S. stamp; Sofia Kovalevskaya (with no prop at all) on a Russian stamp; Henri Poincar6 with a book on a French stamp. A dramatic Egyptian stamp shows a jet flying over the pyramids at Gizeh. Since this is a textbook, the reader may wonder how it works in the classroom. I used it this fall and was pleased with the outcome. For the first 12 of the 18 chapters, up through Newton and Leibniz, the prerequisite is a year of calculus. That's a semester course. In his preface, Katz gives suggestions for creating a different course by choosing specific topics (for instance, equation solving, probability and statistics, number theory, modern algebra). The clear labelling of the sections makes doing this easier. The index helps the reader find all materials on various themes: for instance, there is an entry on women in mathematics. The index also gives phonetic pronunciations for non-English names, a feature my students found surprisingly empowering. The book also gives due attention to cultural exchange. For instance, in the section on the fourteenth century, Katz includes Jewish, Islamic, and Christian mathematicians, and describes the way their mathematical traditions interacted. Katz discusses mathematics in China and India and the influence of that work wherever mathematically relevant, rather than relegating the material to a "Mathematics in the East" chapter. There is a brief section on the mathematics of Native Americans, the peoples of Africa, and the peoples of the Pacific. The study questions and problems after each chapter enriched the course. I found that requiring students to choose a couple of problems and one discussion question from each chapter worked well. It encouraged students to find a problem just within their competence. Some discussion questions ask students to prepare a lesson to teach some mathematical topic using a particular historical approach. These questions are not just for prospective teachers; other students benefit from the chance to reflect on the w a y we teach them mathematics. For example, consider "Outline a lesson teaching the quadratic formula using geometric arguments in the style of al-Khwarizmi." Even some mathematics majors
Figure 3. Page from Billingsley's translation of Euclid's Ele-
ments, containing a pop-up diagram. have never seen completing the square done geometrically. Or, "Compare Levi ben Gerson's use of 'induction' to that of al-Karaji. Should the methods of either be considered proof by induction? Discuss." In about the year 1000, the Muslim mathematician al-Karaji proved that 13 -{-23 -F"" q- 103 = (i q- 2 + ' " + 10) 2, by showing that the result for 10 follows from that for 9, that for 9 from 8, etc., down to 1. In 1321 Rabbi Levi ben Gerson proved formulas for permutations and combinations by reducing the case for n to the case for n - 1, and then showing the formula to be true for n = 1. Students who work this all out have to think carefully about both these specific proofs and the general nature of mathematical induction. Among the many examples from later time periods, four problems take students through Euler's discovery that
~ k=l
1 k2
_71-2 6 '
and students are asked to outline a lesson on manipulating matrices by following Cayley's 1858 treatment of the subject. The last problem in the book asks students to construct a circuit representing the addition of binary numbers, as outlined in the text's discussion of computers. I highly recommend this book as a text in a history of mathematics course. It's also good reading for anyone willing to work through the examples. Of course, every reader will think there is too much of some things and THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4, 1994
75
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
Robin Wilson* Swiss Mathematics L e o n h a r d Euler (1707-1783) was probably the most prolific mathematician of all time. He was born in Basel, Switzerland, where he studied with Johann Bernoulli. In 1727 he went to St. Petersburg on the invitation of Catherine I, where he became Professor of Physics and later (following Daniel Bernoulli) Professor of Mathematics. After remaining there for fourteen years, he went to Berlin to head the Prussian Academy, staying until 1766, when he returned to St. Petersburg for the rest of his life. Euler reformulated the calculus in terms of functions and created an algebraic language for discussing them. He contributed to number theory, differential equations, differential geometry, the calculus of variations, and the theory of equations. He conventionalized several of our standard notations, such as Y~, e, i and f(x). He solved the K6nigsberg bridges problem, discussed the motion of the moon and the shape of the earth, related the exponential and trigonometrical functions by means of the equation ei~ = cos ~ + i sin ~,
Jakob Bernoulli (1654-1705) was Professor of Mathematics at the University of Basel from 1687 until his death. With his brother Johann, he was among the first to develop Leibniz's calculus and apply it in a variety of areas. These included the study of curves such as the catenary and isochrone, the use of polar coordinates, the development of Bernoulli numbers and polynomials, and the study of mathematical probability and statistics. This stamp features a portrait of Jakob Bernoulli, painted by his brother Nicholas, together with his law of large numbers. It has been issued to commemorate the International Congress of Mathematicians in Zurich this summer.
as shown on the 1957 "Pro Juventute" charity stamp, and stated his polyhedron formula (vertices) - (edges) + (faces) = 2,
Jakob Bernoulli
illustrated on an East German anniversary stamp of 1983. The Swiss logarithmic spiral stamp was issued in 1987 to commemorate the 150th anniversary of the Swiss Engineers' and Architects' Association.
The Logarithmic Spiral
Leonhard Euler
* Column editor's address: Facultyof Mathematics,The Open University,MiltonKeynes,MK76AA,England. 78
THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 @1994 Springer-Verlag New York
