The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Sheldon Axler.
The Banach-Tarski Paradox The Banach-Tarski paradox is a fascinating topic, and any attempt to bring it to a new audience is to be welcomed. The cover article in your Volume 10, Number 4 issue by Robert M. French was clearly designed to do this concisely and efficiently, for no references were given. Of course the classic original papers, at least, are easy to find. H o w e v e r , I do think that Stan Wagon's superb book, published three years ago and suggestively titled The Banach-Tarski Paradox, deserved a mention. To begin with, it has a comprehensive list of references to the extensive literature. Also, its first few chapters have m u c h the same aim as Robert French's article; they form an entertaining and lucid introduction to the "paradox" which is perfectly accessible to any graduate student. Furthermore, the book as a whole is an in-depth guide to the subject and will probably be the standard text for many years. R. J. Gardner Department of Mathematics King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia
THE MAI]-tEMATICAL INTELLIGENCER VOL. I1, NO. 2 9 1989 Springer-Verlag New York 3
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Sheldon Axler.
The Banach-Tarski Paradox The Banach-Tarski paradox is a fascinating topic, and any attempt to bring it to a new audience is to be welcomed. The cover article in your Volume 10, Number 4 issue by Robert M. French was clearly designed to do this concisely and efficiently, for no references were given. Of course the classic original papers, at least, are easy to find. H o w e v e r , I do think that Stan Wagon's superb book, published three years ago and suggestively titled The Banach-Tarski Paradox, deserved a mention. To begin with, it has a comprehensive list of references to the extensive literature. Also, its first few chapters have m u c h the same aim as Robert French's article; they form an entertaining and lucid introduction to the "paradox" which is perfectly accessible to any graduate student. Furthermore, the book as a whole is an in-depth guide to the subject and will probably be the standard text for many years. R. J. Gardner Department of Mathematics King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia
THE MAI]-tEMATICAL INTELLIGENCER VOL. I1, NO. 2 9 1989 Springer-Verlag New York 3
Allen Shields*
Egorov and Luzin: Part 2 This column for vol. 9, no. 4 (1987) dealt with the mathematicians N. N. Luzin and D. F. Egorov. Since then additional material has come to m y attention, some of it while I was visiting the Soviet Union for six weeks in May and June 19889 We begin with Egorov. If one looks at vol. 35, no. 1 (1928) of the journal Matemati~eski~ Sbornik, published by the Moscow Mathematical Society, one finds a list of all the members of the society at that time, with the date they joined the society. The president is Egorov (1893), and the vice president is Luzin (1910). (Egorov was born in 1869 and Luzin in 1883.) Curiously, Kolmogorov is not listed, although he had already published a number of important papers (he was born in 1903). Inside the front cover there is a note in French and in Russian to the effect that the journal will publish papers in Russian, French, German, English, and Italian. Papers in Russian must have a r~sum~ in one of the other languages, and papers not in Russian must have a Russian r~sum(~ (which will be supplied by the editors u p o n request). The instructions about figures include the restriction that there be at most one figure for each four pages of text. Authors received 50 free reprints. Finally, all correspondence is to be sent to: Prof. D. F. Egorov, Moscow 69, Borisoglebskff per., No. 8, Apartment 5. (Inside the back cover the addresses of the officers of the society are given; Luzin's address is: Moscow 2, Arbat, No. 25, Apartment 11, telephone 3-35-16.) Three years later, inside the front cover of vol. 38, no. 1-2 (1931) of the journal one finds a different message from the editors: Starting with the next number the journal Matemati~eski~ Sbornik will appear with the title Sovetski~ Matemati~eski~ Sbornik. Until recently the Mathematical Society had retained its cliquish academic character. Prof. Egorov, a reactionary and a churchman, headed the society. He opposed the
* Column editor's address: Department of Mathematics, University of Michigan, Ann Arbor, M148109-1003USA
policies of Soviet p o w e r . . , under the guise of defending "academic traditions" and "pure apolitical science." The growth of the proletarian strata among the graduate students . . . and their struggle against the reactionary wing of the professors decisively turned the basic cadres of Soviet mathematicians.., to active participation in social construction . . . . In the Mathematical Society an initiative group, including many prominent mathematicians, was formed to reorganize the society9 The declaration of the initiative group was accepted by the Mathematical Society (it was published in the journal Nau~ni~Rabotnik no. 1, 1931). The Society expelled Egorov and other reactionaries, and replaced them mainly with graduate students . . . . 9. . The editors of Sovetskff Matemati&skir Sbornik call upon Soviet mathematicians to close ranks around the journal and help to turn it into a fighting organ of Soviet mathematics. The editors invite the collaboration of foreign scholars sympathetic to the Soviet Union. Inside the back cover is a notice that material for the journal should be sent to editor in chief Lazar Aronovi~ Lyusternik, or to the Secretary: Aleksandr Iosifovi~ Gel'fond. [Note: they seem to mean Aleksandr Osipovi~ Gel'fond; no A. I. Gel'fond is listed in Mat 40 [1959].] The name "Sovetskii Matem. Sbornik" was not in fact ever used by the journal. As regards the initiative group [Lev], Note 12 gives the following reference: L. A. Lyusternik, L. G. ~nirel'man, and others, "'Declaration of the Initiative Group for the Reorganization of the Mathematical Society," Nau~nif Rabotnik 1930, no. 11/12, pp. 67-71. [The title is a translation of the Russian original. I have not seen this article, and if anyone could send me a copy I would be very grateful.] In no. 3 - 4 of the same vol. 38 of the Sbornik one finds an appeal to Soviet mathematicians to support the journal, and to break with the old tradition of publishing their best works abroad 9 Because they want to have an international journal, articles in Russian will continue to have foreign r~sum~s, and articles will be accepted in the four other languages indicated above. The editors in chief n o w are P. S. Aleksandrov and A. F. Bermant. Manuscripts are to be sent to Bermant. Authors now receive 100 free reprints. In the same issue there is a report on the first all-
THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2 9 19~9 Springer-Verlag New York
Ernest
Kol'man
Russian conference on planning in mathematics, which met June 5-9, 1931. The report stresses the need for a close connection between theory and the practice of socialist construction, as called for by the Central C o m m i t t e e of the Party and by comrade Stalin. The necessary planning is said to be impossible in bourgeois countries. In bourgeois mathematics one observes the isolation of theory from practice. This tendency in the present period of the decay of capitalism is expressed in mathematics in formalism, which treats all mathematics as a chess game with meaningless symbols, and by subjective intuitionism. There follows a long section presenting materials from the conference. First is a resolution based on a speech by E. Kol'man at the final session. It states that the attempt to fit the developing content of mathematics into an idealist formal scheme has led to the crisis in the foundations (Brouwer, Hilbert, and the French school of the theory of functions; Sierpiriski and Luzin are mentioned as adherents of the French school). Further, although Soviet mathematics has attained one of the top places in world mathematics, it still does not have a form that expresses the socialist character of our Revolution. Until recently a basic obstacle had been the attitude of certain reactionary professors (Gyunter, Egorov, and others). Other resolutions deal with the need for more institutes, for financial support, and for autonomy. Then there are reports on different areas of mathematics and their connection to applications. Another reference, given to us by Prof. ~afarevi~ (Shafarevich), is to a journal called Mathematical Sciences for Proletarian Cadres, vol. 1, 1931. The article, entitled "Reorganization of the Moscow Mathematical Society," states that the Society expelled the reactionaries Egorov, Finikov, and Appel'rot. In addition, the pedagogical units were completely reorganized and have become a section in the Society, a tool in the struggle for Marxist pedagogy in mathematics. The Society elected a n e w presidium, including: Kol'man, Golubev, Gel'fond, Orlov, Burstin, Khin~in, Khotinski, Raikov, Frankl', Yanovskaya, Lyusternik, Lavrent'ev, and others9 6
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Soon after being expelled from the Mathematical Society, Egorov lost his position at the University. He was arrested and intensively interrogated, apparently falling sick in prison, and then exiled to Kazan. Elsewhere in this issue of the Intelligencer there is a very interesting interview with Shafarevich by Smilka Zdravkovska. We quote: "As Chebotarev told me, Egorov died in Chebotarev's home, in his arms." (This was in September 1931.) Chebotarev was born in 1894, he was at Kazan University from 1928 until his death in 1947. There is more about Egorov and Luzin in the Shafarevich interview later in this issue. Luzin's troubles in 1936 began innocently enough with a short article of his in the newspaper Izvestiya (June 27, p. 4) entitled "A pleasant disappointment." I have not seen this, but [Lev] gives an account, and a recent article in Sovetskaya Kultura has several quotes from it. Luzin describes attending the final exams in a secondary school in Moscow. He concluded by expressing complete satisfaction with the results of the exams, writing: "I could find no weak students, the students differed from one another only in that some answered quickly and some slowly, but all performed very well. On this occasion I found precisely that deep understanding of the laws of mathematics, of whose lack I have so often complained." He especially acknowledged the school director's personal contribution. On July 3 Pravda, p. 3, carried an article by the school director entitled "An answer to Academician N. Luzin." He states that the students' performance was not outstanding, for example on the trigonometry exam out of 33 students 11 received the highest grade of 5, 12 received a grade of 4, and 10 received 3. The results for algebra and for geometry were similar. These results were normal, and did not at all justify Luzin's statement that he could not find any weak students. Luzin is severely criticized for overpraising the school, instead of providing comradely criticism. The very next day Pravda, on p. 2, has an unsigned article entitled "'On enemies in Soviet masks." It indudes the following: "A careful examination of the activity of this academician in recent years shows that the deliberate overpraising of these students is far from accidental. It is but one link in a long chain, and is very instructive in showing the methods by which enemies mask themselves. We well know that N. Luzin is an anti-Soviet p e r s o n . . . . Academician Luzin could have become an honest Soviet scholar, as did m a n y from the older generation. But he didn't want this; he, Luzin, remained an enemy, counting on 9 . . the impenetrability of his mask . . . . It w o n ' t work, Gospodin Luzin!" ["Gospodin" means gentleman, or mister. After the Revolution it was replaced by "grazhdanin," which means citizen.] Further articles in Pravda followed, on July 9, 10, 12 (p. 3 in each case). The article of July 9 was entitled
"Traditions of servility." It complains a b o u t the tradition of p u b l i s h i n g abroad, of j u d g i n g as valuable only those w o r k s that receive foreign approval: To take only mathematics, the majority of Soviet mathematicians (Aleksandrov, Kolmogorov, Khin~in, Bern~tein, and others) publish their works abroad and not here, in the USSR, in the Russian language . . . . It is known (see the articles in Pravda for July 2 and 3) that Academician Luzin sent his best works abroad, with the definite political intention of publishing in the USSR, as he himself cynically expressed it, "any rubbish . . . . " One must not suffer such a situation any longer. The Soviet Union is not Mexico, not some sort of Uruguay, but is a great socialist power. And the Russian language is the language of a mighty people, spoken by at least 150 million people . . . . The conditions have been created in our country for such a great flowering of science, as has not been known, and is not known, in any capitalist country. As regards the reference to U r u g u a y , [Lev], N o t e 26, states that that c o u n t r y h a d recently b r o k e n diplomatic relations w i t h the Soviet Union. O n e of those w h o r e a d the articles in Pravda on July 2 a n d 3 w a s the physicist Pyotr Leonidovi~ Kapitsa. H e w a s o u t r a g e d , a n d w r o t e a letter, d a t e d July 6, 1936, to the C h a i r m a n of the Council of Ministers of the USSR, V. M. Molotov. Last year this letter w a s rep r i n t e d in the n e w s p a p e r Sovetskaya Kultura, M a y 21, 1988, p. 6. We q u o t e s o m e excerpts: The article in Pravda on Luzin puzzled, astounded, and angered me and, as a Soviet scholar, I feel that I must tell you what I think about this. Luzin is accused of much. I don't know if these charges are correct but even if they are completely justified this does not change my negative attitude to the article. First let us begin with some accusations of a petty character. He did not publish his best works in the Soviet Union. Many of our scientists do this, and primarily for two reasons: 1) printing and paper are of poor quality here; 2) by international custom, priority is only given for works published in French, German, or English. And if Luzin published bad works in the Soviet Union, then this is the fault of the editors of the journals that accepted them.
If he envied his students and therefore sometimes had inequitable relations with them, this phenomenon, unfortunately, occurs even among the strongest scholars. Thus there remains one, very serious, charge against Luzin, that he hid his anti-Soviet feelings behind flattery, though somehow this doesn't seem such a great offence. Here we h a v e . . , a very important question of principle: how should one treat a scholar whose ethics do not correspond to the spiritual needs of the epoch? Newton, who gave mankind the law of gravity, was a religious maniac. Cardano, who found the formula for solving cubic equations and made a number of important discoveries in mechanics, was a wastrel and libertine. What would you have done with them if they had lived here in the Soviet Union? Or suppose that someone close to you fell ill. Would you call a brilliant doctor if his ethical and political convictions were contrary to your o w n ? . . . I don't want to defend Luzin's moral qualities, but there is no doubt that he is a very strong mathematician, one of our four best mathematicians. Besides, he did more than
Nikolai Luzin
anyone else to gather and to educate the pleiad of young Soviet mathematicians we now have. I consider that a country having strong scholars such as Luzin must first of all do everything so that their talents are used as fully as possible for mankind. People such as Luzin, who are ideologically not suitable to us, must, first, be given conditions in which they can continue their scientific work without having wide social influence. Second, everything possible must be done to reeducate them in the spirit of the epoch to make good Soviet citizens of them . . . . Was everything possible done to reeducate Luzin, and people like him in the Academy of Sciences? And can methods such as the article in Pravda attain this g o a l ? . . . In general how did you go about reconstructing the Academy? You began by selecting party comrades for the Academy. This would be the best method if among the party members there were strong scholars. Leaving aside the social sciences, our party academicians are much weaker than the academicians of the older generations, and their influence is therefore small . . . . Our principal scientific capital continues to be the older generation . . . . Having in hand the economic achievements and political gains of our country, I don't understand why one cannot reeducate any academician, no matter whom, if only one proceeds in a considerate and individual manner. Recall, for example, the case of Pavlov . . . . [Editor's note: see below.] From these considerations... I see in the Pravda article only a hostile step towards our science and our Academy, since it does not reeducate our scholars and does not raise their prestige in the country. And Luzin's name is sufficiently well known in the West that such an article will not pass unnoticed. The article is weak and unconvincing, and can give rise to . . . ridicule. Seeing the harm that all this does to science in the Soviet Union, I felt that I had to write you about this. A d a y or t w o later a m e s s e n g e r arrived at Kapitsa's institute w i t h a p a c k a g e f r o m Molotov for Kapitsa. Kapitsa w a s at his dacha, in the village of Z h u k o v k a , a n d the messenger went there. Kapitsa opened the p a c k a g e a n d f o u n d inside his o w n letter. In the u p p e r left corner of the first page, in a precise h a n d w r i t i n g , w e r e the words: " N o t w a n t e d , r e t u r n to citizen Kapitsa. V. M o l o t o v . " K a p i t s a w a s v e r y a n g r y at Molotov's a n s w e r , a n d in his archives there is the draft of a letter to Stalin a b o u t it. The letter w a s n e v e r sent, however. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 7
After the Pravda articles meetings were held in universifies and scientific institutes to discuss the articles. A commission was set up in the Academy to study the case. The commission's report is given in Pravda, Aug. 6, 1936, p. 3. Among other things Luzin is accused of publishing, without credit, results due to various students of his, including Suslin, Aleksandrov, Kolmogorov, Lavrent'ev, and Novikov (see point 5 in the report). Also, there is reference to Luzin's refusal in 1930
Luzin lost his position in the university but was not expelled from the Academy of Sciences. to sign a letter of protest to French scientists for interfering in Soviet affairs. The French, and others in western Europe, were protesting the trial of the socalled Industrial Party. (See point 4 in the report, and see [Lev].) Luzin lost his position in the university but was not expelled from the Academy of Sciences. Incidentally, the 1988 issue of Sovetskaya Kultura referred to above has two other letters by Kapitsa. One was written in 1937 to protest the arrest of the physicist Fok, and the third was written in 1980, to Andropov (then head of the KGB) to protest the treatment of the physicists Sakharov and Orlov. Kapitsa was in an unusual position when he wrote to Molotov about Luzin. For years Kapitsa had been at Cambridge University, working in the Cavendish Laboratory with Rutherford. The Soviet Union was interested in getring him back. Kol'man describes an occasion in the s u m m e r of 1931 w h e n he and Bukharin were attending a conference in London. One day they went to Cambridge to see Kapitsa, who invited them home to dinner. After the meal Bukharin tried to convince him to return, and promised him "mountains of gold" if he would return (see [Kol], p. 176, in the chapter entitled "Return to science"). Kapitsa was hesitant, but his wife was against it. Kapitsa spent his summers at his dacha in the Soviet Union. He did this again in 1934, but w h e n he wanted to return to England the Soviet authorities refused permission. He was given his own institute in Moscow, and since he knew they needed him he had a certain independence. The 1980 letter of Kapitsa describes Lenin's relations with I. P. Pavlov. Pavlov was hostile to Soviet power after the Revolution. He took every opportunity to criticize the authorities, he wore his Tsarist medal, and he crossed himself w h e n e v e r he passed a church. Lenin simply paid no attention to all this. For Lenin, Pavlov was a great scholar and Lenin did everything possible to provide him with good working conditions. In the early 1920s the food situation was catastrophic in Petrograd, but Lenin ordered that Pavlov's 8
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dogs should receive their full food rations. Kapitsa remembers Academician A. N. Krylov telling him of how he bumped into Pavlov one day and said to him: "Ivan Petrovi~, may I ask a favor of you? . . . . Naturally," Pavlov replied. "Please take me as one of your dogs." Finally we say a few words about Kol'man, who was mentioned above, and also in this column, vol. 10, no. 3, for his defense of the plant breeder and "geneticist" Lysenko against Kolmogorov. Kol'man (1892-1979) was born in Prague in a middle-class German Jewish family. He received his doctorate at the Charles University in Prague, was mobilized into the Austro-Hungarian army, and sent to the Russian front, where he was captured in 1916. He supported the Bolshevik revolution, joined the Communist Party, and stayed in the Soviet Union. He was connected for a time with the Institute of Red Professors, and with the MarxEngels-Lenin Institute. After the Second World War he returned to Prague. In 1948, following the change in government in Czechoslovakia, he was arrested and deported to Moscow, where he spent three and a half years in Lubyanka Prison. After being released he took up science writing again. In 1953 Kol'man was apparently the first (in the Soviet Union) to defend cybernetics, which had been much disparaged (see [Gr]). He returned to Prague where he took a critical attitude to the entry of Soviet troops in 1968. Eventually he emigrated to Sweden, where he published in memoirs (see [Kol]), in German and Russian, under the title We shouM not have lived that way. The memoirs have a list of all persons cited - - t h e r e is no mention of Egorov or Luzin. There are three references to Kolmogorov--first that Kolmogorov and Aleksandrov read in manuscript a book of Kol'man in 1937, second that in 1974 they joined the chorus denouncing Solzhenitsyn w h e n he was expelled from the Soviet Union, and third that Kolmogorov had first (October 1956) been negative towards cybernetics, and then in April 1957 changed his mind and supported it. One must conclude that the title of his book does not refer to any change of heart toward his role in the attacks on Egorov and Luzin.
Bibliography [Kol] Arnogt (Ernes0 Kol'man, We should not have lived that way, New York: Chalidze Publications (1982). [Russian]. [Gr] L.R. Graham, Science and philosophy in the Soviet Union, New York: A. Knopf (1972). [Lev] Aleksey E. Levin, Anatomy of a public campaign: "Academician Luzin's case" in Soviet political history, preprint. Mat 40 [1959] Matematika v SSSR za sorok let, 1917-1957, vol. I, II, Gos. izdat, fiz.-mat, lit., MoscowLeningrad (1959). (Russian).
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Sheldon Axler.
Paradise Lost: The Troubled Government Support of Science Saunders Mac Lane 1. It Was Paradise Throughout the Western world, for about 40 years, the progress of mathematical research has been vitally advanced by the actions of governments in providing funds, equipment, and thus encouragement to those engaged in the otherwise lonely search after new mathematical results. In France, the CNRS (Centre National de Recherche Scientifique) supported large numbers of young scientists; in Great Britain, the UGC (the University Grants Council) distributed funds to scientists at universities; in the Federal Republic of Germany, there are Max Planck Institutes and the Deutsche Forschungsgemeinschaft, while the Alexander von Humboldt Stiftung has brought both young and mature scientists as visitors to Germany--and so it went in many countries, to the mutual benefit of the country and science. The experience of World War II had shown that mathematicians of all types could make vital contributions on topics new to them in a time of need. Science, as Vannevar Bush put it, is an endless frontier. Hence in the USA, after WW II, the ONR (the Office of Naval Research) set the tone for the coming dispensation by providing contracts to support research on a wide variety of mathematical problems, not just those of interest to the Navy. When the NSF (National Science Foundation) was established the first NSF director, several of its program directors, and much of the style 9 of its operation came from the model provided by the ONR. Gradually, there was much wider government support for mathematical research. In the relatively rare cases when the NSF did not support some field of 10
activity, the defense agencies (for example, the OOR, the Office of Ordnance Research, and the AFOSR, the Air Force Office of Scientific Research) often could provide research contracts for that field. This was the arrangement which provided science with "Multiple Sources of Support." This support soon created in the USA a sort of paradise for science and thus in particular for mathematical research. Unfortunately, in the last few years, most of the best aspects of this paradise have been lost, and painfully so. After a brief discussion of the prior Garden of Eden, this essay will discuss the loss: Why
THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2 9 1989 Springer-Verlag New York
it happened (especially in the USA) and what might now be done as a restorative. Ample government support meant that most mathematicians who could do research had financial encouragement to do so. The government grants provided for students and for visitors and sometimes for sabbatical leave or for portions of the academic year salary. Moreover, a system of "summer salaries" for the principal investigator provided by grants came into effect in all the sciences. (In the USA, government support is usually managed uniformly across the sciences.) These government-funded summer salaries at first provided one-third of the "academic year" salary (i.e., of the total university salary); this was intended to cover research work during the three
It may be that the availability of summer salaries meant that the universities did not sufficiently increase the level of academic year salaries. s u m m e r months. Subsequently, at the NSF the amount was usually 2/9ths of the academic year salary, on the grounds that everyone should take a one-month vacation after nine months of teaching. These summer salaries were often justified by observing that research mathematicians had previously been forced to earn needed extra money by teaching in summer school, to the detriment of their possible research. At that time, there were a few people skeptical of the summer salary policy, but its provision did clearly promote more research (not always better research), even on the beaches of Copacabana, as the saying then went. This policy (and related moves) did increase the general level of prosperity of mathematicians in the USA--for example, the quality of mathematicians' housing visibly improved in the period 1958-1970, and the signs of such well being may have encouraged more able young people to take up mathematical research. (There was a big increase in the number of Ph.D.'s granted.) On the other hand, it may be that the availability of summer salaries meant that the universities did not sufficiently increase the level of academic year salaries. Inevitably, people became more dependent upon the government and less on their university, and they became accustomed to this added 2/9ths salary. (Currently, this can cause considerable pain when a grant with its attendant summer salary is not renewed.) Moreover, the need for frequent grant applications may encourage people to stick to standard problems in a familiar field so as to produce a longer list of published papers. One may think also of mathematics in Canada, where the government does provide grants to support research, but
with nothing in the way of a summer salary. In both countries, teaching loads have gone down and research has advanced. Beginning in the 1950s, mathematical research flowered in the USA. There came to be many specialized conferences, often supported in part by government grants. They made it possible for people to learn more promptly of the latest developments in their field, while conference proceedings (sometimes not very carefully refereed) spread the good news to those who had not had a subsidy to attend. These conferences and the like may have encouraged overspecialization, but there were many good results: famous problems were settled and many new ones proposed. It was a heady period, which embodied a sort of paradise for mathematics--and for other sciences. This generous support came from the taxpayer's money. Why was this possible? Partly because of national pride--for example, mathematics in the United States rose to a position of undoubted world leadership. But the support was probably chiefly the result of the experience in World War II, when US scientists with all sorts of prior training pitched in to identify and then to solve the immediate scientific problems which arose from the war effort. It was not by any means the case that only applied mathematicians were effective, because many problems did not belong to any previously recognized part of science. The war experience led to the development in the USA of new centers of applied mathematics, and this development
The ~lan of mathematical research is spread over many different institutions (many more than can have centers) and depends upon new thrusts introduced by individuals or by collaborations instituted by individuals and not by artificial "centers." was timely and effective. But the promoters of applied mathematics, such as Richard Courant and his intellectual descendants, who tend to cIaim that applied mathematics (espedally PDE) is the field of central national mathematical interest, are simply wrong. For the actual real world, the whole of mathematics matters. The role that pure mathematicians can play in attacking practical problems is well illustrated in a recent article by Peter Hilton [3] about the British team led by Alan Turing at Bletchley Park, where the codes of the German Enigma machine were broken. There were similar triumphs of new mathematics in the USA, including the development of operations research and the work of "pure" mathematicians such as Hassler Whitney, who shifted for the time his interests from THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989 1 1
topology to airborne fire control. In brief, government was led to support a wide range of science because a large cadre of scientists is a national treasure, ready to h a n d at times of great need. A n d it seems that treasures abound in paradise. 2. T r o u b l e in a n English Eden In Great Britain, much of the support of research at the various universities comes from the government. This support has been good, though perhaps not quite on the scale customary in the United States. But in 1988 the British government announced that no new tenure would be granted for university staff. This announcement appears to mean that those lecturers, readers, and professors who now have tenure (and such tenure is often granted quite early in a British academic career) will continue to hold that tenure, but that all new appointments and all promotions will not have tenure. (In the extreme case, this appears to mean that a lecturer now having tenure could be promoted to be a reader or a professor and then, in the next year, be fired.) It is of course explained that such dismissals will occur only w h e n the individual in question is no longer productive. It is not explained how this productivity is to be measured.
The British government announced that n o new tenure would be granted for university staff. This move, cancelling future tenure, is an incredible assault upon the whole British academic community. It is not just that it gravely endangers freedom of speech and other academic freedoms. Even these freedoms may not have been absolute beforehand (Bertrand Russell, a pacifist in World War I, at that time lost his fellowship at Trinity College, Cambridge). It is also not just the economic injustice involved--previously it was generally thought that the income of academics was well behind that of equally capable denizens of the City. This is a real point, since academic salaries in Britain are noticeably lower than comparable ones in the USA, and it seems very unlikely that any government decree adjusting these salaries will be issued. The real point about tenure is much deeper: Tenure for university faculty makes it possible for such faculty members to have long-range plans to tackle major intellectual problems, including the elusive ones, where there is no assurance of final success. The inevitable fact that some t e n u r e d faculty m a y not have the courage to make such long-term commitments and that others may try but fail is irrelevant. What is relevant is that tenure does make it possible for some people to try those basic problems, without any con12
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cern about this year's scientific fashion, next year's grant, or next year's bottom line. Thus there are mathematicians w h o try but fail to solve o u t s t a n d i n g problems such as the Poincar6 conjecture or Fermat's last theorem. Academic freedom helps to make such bold t r i e s m a n d more modest ones--possible, and indeed may also cover recent successes such as that with the Bieberbach conjecture on univalent functions. 3. Is There Protection from "Defense"? In the United States, the current troubles are not with t e n u r e but with misdirected support. As already noted, the government support for mathematics has come both through the National Science Foundation and the various "defense" agencies. There are difficulties w i t h both channels. The old-line d e f e n s e agencies (ONR, OOR, AFOSR) initially had supported mathematics across the board, but now their interests are much narrower. This came about first through the imposition by the Congress of the so-called Mansfield Amendment, which instructed the agencies to support only that science which is relevant to the agencies" mission. This amendment is no longer in legal force, but the intent is still powerful. Both the older defense agencies and the newer, more specialized ones now focus research support narrowly on issues of immediate interest to them. There certainly are such issues, and they do need attention. However, this attention is sometimes based on such ample f u n d i n g that the whole seriously threatens the balance of mathematical research in the USA. Thus, if professors in (say) cryptology get much more liberal grants, with ample support for students, the result may be to attract floods of students to this field, beyond the real need. This now tends to be the case in a number of such applied fields. Of the government support of academic mathematics in the USA, at least 35% and perhaps as much as 50% now is channeled through defense agencies. (Exact figures are hard to come by, in part because the definition of mathematics is ambiguous.) The result is a considerable distortion of the choice of a field of research by new mathematicians. Clearly they should not be concentrated chiefly in the fields of current defense interest. (Even these can change!) This is of special concern today, when the number of new graduate students is much lower than 12 years ago, and at a time when the extreme commercialization of computers has made computer science a financially very attractive competitor. The imbalance toward defense has been accentuated in the USA by the Strategic Defense Initiative (SDI = star wars). Funding of this type of research has increased by massive a m o u n t s - - a n d this without adequate prior scientific evaluation of the whole project. It
Worldwide, the support of scientific research is tied too closely to military concerns. has brought forth a sharp response from a large segment of the members of the American Mathematical Society (see the Referendum report in the Jan. 1988 Notices of the AMS [5]). In o t h e r c o u n t r i e s , the problems of mathematical research are different, but they are still substantial. For instance, Soviet mathematicians meet great obstacles in sending manuscripts to Western journals. The cause of this particular difficulty is apparently u n d u e concern about security--in other words, defense related. This illustrates my general thesis: Worldwide, the support of scientific research is tied too closely to military concerns. Mathematics matters in m a n y other aspects of human life. The military emphasis should be cut back or balanced by more civilian-oriented research support.
4. What Ails the NSF? In the United States, the National Science Foundation also faces real troubles. Some years ago, in 1983, the eloquent David report [2] on "Renewing US mathematics" was issued. It observed, on the basis of considerable evidence, that mathematical research was badly underfunded. Since that time the NSF budget for mathematics has been substantially increased and effectively "protected" by NSF officials, but some of the resulting funds have been misdistributed, so that the results have been gravely disappointing. Thus the number of mathematicians supported by individual grants, not counting postdoctorals, is about 1400, and has been at about the figure 1300-1400 for some years, without real increase. With the considerable number of n e w young mathematicians, this has meant that many active older mathematicians with prior grants have been d e n i e d renewals, sometimes for good reasons and sometimes not: specific examples of real injustice are well known in nearly every good department of mathematics in the country. Misjudgments in individual cases aside, one must ask w h y this has happened. Here are some possible reasons: A. Typical budgets for grants have substantially increased, because of considerable salary inflation and the inclusion in grants of more funds for graduate students and undergraduate research. This means fewer grants. B. The NSF now p~oudly gives extra large grants to certain "Presidential Young Investigators.'" Moreover, when such a PYI gets additional money from some industrial source, the NSF matches that money (up to a specified maximum) by NSF money. In plain lan-
guage: W h e n a PYI gets industrial support, some other m a t h e m a t i c i a n m u s t get cut out of a m o r e modest grant. And this can happen without any real comparison of merit. (The peer-review process for prospective PYI's is much less rigorous.) I simply do not believe that any elite group, such as that of the PYI's, can be chosen with any such accuracy. Indeed, one may suspect that the choice is often made in part because of a fashionable field of s t u d y - - i n imitation of physics, mathematics has of late become too subservient to fashion. C. That same emphasis on fashion has meant that departments of mathematics pursue for appointment just a few people who are generally regarded as stars. This tends to bid a few selected dollar salaries up into six figures; then the resulting added 2/9ths for the summer cuts out most of a grant to some lesser (really lesser?) star. Currently, this effect may be somewhat mitigated, because of the recent announcement of a salary cutoff at $95,000, beyond which there will not be federal support. D. It seems to me vital (despite the occasional public statements to the contrary) that the NSF grants continue to be based upon peer review. Such reviews, though never perfect or replicable, are far superior to the alternative, since that alternative is essentially decisions by regional considerations or by pork-barrel pressures in the Congress. Unfortunately, there is some evidence that there has been a loss of quality and care in the peer reviews sent to the NSF. Thus I happen to have seen two-line reviews which were essentially vacuous, and other reviews which did little more than express one-sided prejudice. There was even one review which made the irrelevant complaint that the university in question had abolished the reviewer's favored department 15 years ago. My observations about the quality of peer reviews are not based upon extensive data, but I may still suspect that m a n y fields today are so specialized that there is not an extensive choice of qualified possible
In imitation of physics, mathematics has of late become too subservient to fashion. r e v i e w e r s - - a n d today, nearly all potential reviewers are overwhelmed by those many requests for letters of reference in connection with potential university appointments. There is a deeper reason for the troubles with peer review: About 1974 the National Science Board (over m y protest) decided to make the text of peer reviews, with the reviewer's name deleted, available to the principal investigators. This happened because of a sort of populist revolt against the supposed power of the "Old Boys' Network." The effect has been the expected one: The reviewer knows that the THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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review, signature deleted, will go to the principal investigator, and so writes guardedly and cryptically, to avoid possible identification and the c o n s e q u e n t trouble. The review is then less useful to the program officer. This loss of effectiveness is, alas, due to the decision of the National Science Board.
5. The National Science Board The policies for the National Science Foundation are set by the National Science Board (NSB), which consists of the Director of the NSF and 24 people, each appointed for a six-year term by the President, with the advice and consent of the Senate. This board meets monthly and, with input from the director and other officials of the Foundation, considers problem cases and sets policy. Initially and for a considerable period, the NSB had a large proportion of disting u i s h e d scientists as m e m b e r s , t o g e t h e r with a number of university presidents and other administrators. Thus, in mathematics Marston Morse, E. J. McShane, Mina Rees, R. H. Bing, Saunders Mac Lane,
The promoters of applied mathematics, such as Richard Courant and his intellectual descendants, who tend to claim applied mathematics (especially PDE) is t h e field of central national mathematical interest, are simply wrong. and Peter Lax served in succession as board members. At the present time, there is no mathematician on the NSB; this is typical, since there are now few working scientists on the NSB. The effect upon NSF policy is clear and devastating. For example, the director, Erich Bloch, has enthusiastically pushed [7] for a doubling of the NSF budget, but the principal vehicle for this doubling is to be the creation of a number of new "Science and Technology Centers." They are intended to strengthen the nation's "economic competitiveness"; note also that because of long-continued propaganda by engineering deans, "technology" gets equal billing with science. At the moment, because of budgetary constraints, only eleven of these planned centers are to be funded in fiscal year 1989. Whenever they appear, it is inevitable that they will emphasize group research and tend to channel that research toward industrial applications. This emphasis bodes ill for mathematics, since the 61an of mathematical research is spread over many different institutions ( m a n y more t h a n can have centers) and depends upon new thrusts introduced by individuals or by collaborations instituted by indi14
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viduals and not by artificial "centers." There is already too much emphasis upon the latest fashion in mathematics; packaging this research in proposals for a few centers will distort the situation still more. The crux of the trouble is the notion that scientific discovery should primarily serve what is called "economic competitiveness." Scientific discoveries do sometimes help old industries or create new technologies with the resulting improvements in the economy, but this should not be the main thrust of the search for new scientific knowledge. Yet the issue of competitiveness seems to be much on the minds of the leading members of the NSB. Thus Erich Bloch, the director, said in an article [1] that the "NSF is the principal instrument of the administration's competitiveness initiative." Just imagine--"the PRINCIPAL instrument!" And at a congressional hearing on competitiveness held on April 28, 1988, Lewis Branscomb (recently c h a i r m a n of the NSB) said, " T h e NSF's research centers have been very helpful." Inevitably, other NSF officials echo this slogan. This general attitude, sponsored by the present NSB, sacrifices the long-range interests of science and mathematics for an immediate economic objective. Now, economics is a dismal science, and I cannot here enter into all the complications, but I may summarize as follows: In recent years, American industry has lost some of its international dominance. Much of this loss is due to short-sighted policies--the failure to modernize plants (e.g., steel plants [8]) because the payoff is too distant. Thus the habit of many in business (few of w h o m understand science) is to look for the immediate bottom line, and not for the long run. More pungently, the captains of industry in the USA have been so b u s y b u y i n g up each o t h e r ' s c o m p a n i e s a n d floating bonds to make this possible that thay have paid little attention to what the companies now produce or could produce in the future. Thus a dreadful gap in competitiveness has arisen. The NSB proposes to throw the planning of science into this gap--giving up the long-range interests of progress and scientific discovery to take up the slack left by careless industrial management. This is the effect in the USA of an imbalanced constitution of the NSB. Similar effects may appear in other countries by other paths. One can only hope that this potentially disastrous policy will soon be reversed. The problem at issue is essentially the old one of patronage. The patron (here the government) has a definite interest in the use (or the misuses) of the funds it provides for the intellectuals. In the case of the NSF, the taxpayer's m o n e y cannot be spent without the oversight of the taypayer's representatives (the Congress and the NSB). But this is oversight of an essentially creative process; if carried out too narrowly, it disturbs and eventually corrupts that process, as may now be the case with summer salaries.
The oversight cannot successfully focus just on the practical results because such results often are elusive. This is the case for science, as in the historically famous instance of Maxwell's equations for the electromagnetic field, where the practical use was not at first apparent. It is even more the case in mathematics, because mathematics is ultimately the consideration of the patterns common to different branches of science. The Laplace equation is used not just in potential theory, and groups and their representations have m a n y different applications. PDE's and groups are patterns and the mathematician is there to study pattern and form, wherever it may lead. Who could have known in advance that the algebra of operators would be used in knot theory and that the results would be important to the study of DNA? (See Vaughan Jones [4]). In brief, science cannot prosper and make its expected contributions to a better life until it escapes the domination by defense agencies and industrial managers.
3.
4. 5. 6. 7. 8.
Peter Hilton, Reminiscences of Bletchley Park. 1942-1945, A Century of Mathematics in America, Part I (Peter Duren, ed.), Providence, R.I.: Amer. Math. Soc. 35 (1988), 291-302. V.F.R. Jones, A new Knot Polynomial and yon Neumann algebras. Notices Amer. Math. Soc. 33 (1986), 219-225. Referendum of January 1988. Notices Amer. Math. Soc. 35 (1988), 13-14. I.M. Singer, Letter to the Editor. Notices Amer. Math. Soc. 34 (1987), 503-504. J. Mervis, Erich Bloch's campaign to transform NSF. American Scientist 76 (Nov-Dec 1988), 557-561. Mark Reutter, Sparrows Point: Making Steel--the Rise and Decline of the American Industrial Might, New York: Summit Books (1988).
Department of Mathematics University of Chicago Chicago, IL 60637 USA
References 1.
Erich Bloch, Current Issues in NSF support for research and education. Notices Amer. Math. Soc. 33 (1986), 229. 2. Edward David, Renewing US Mathematics--Critical resources for the future. Report of the ad hoc committee on research in the mathematical sciences. Notices Amer. Math. Soc. 31 (1984), 435-466.
THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989 1~
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Listening to Igor Rostislavovich Shafarevich Smilka Zdravkovska
What follows is a collage from several conversations with Igor Rostislavovich, held around 3 June 1 9 8 8 his 65th birthday. Already his age sets him apart from most Soviet mathematicians: the Nazis invaded the Soviet Union w h e n bright youngsters from his generation were ready to begin university studies, and instead many were sent to the front; but by then Shafarevich had finished his undergraduate studies Oust before t u r n i n g s e v e n t e e n ) a n d was a l r e a d y doing mathematics. Here is a quote from Crelle's journal, in the volume dedicated to Shafarevich's sixtieth birthday: Shafarevich belongs to those mathematical personalities of this century to whom our science owes decisive advances. A great part of his life work is dedicated to the scientific school he founded (in number theory and algebraic geom-
etry). This school includes outstanding mathematicians and its scientific influence extends outside of Moscow and the Soviet Union over the whole mathematical world. A m o n g his fundamental contributions to number theory, algebra, and algebraic geometry, one should mention his work on the inverse Galois problem, on the generalized law of reciprodty, on the problem of towers in class field theory, and on moduli of K3 surfaces. Shafarevich has written several monographs and textbooks, notably Number Theory (with Z. I. Borevich) and Foundations of Algebraic Geometry, which have been hugely popular. Both have been translated into many languages. He has also written a number of nonmathematical books and articles, including the book Socialism and a report to the Committee on H u m a n Rights on the status of religion in the Soviet Union. He is an associate member of the Academy of Sciences of the USSR, and a member of the US National Academy of Sciences, the American A c a d e m y of Arts and Sciences, the London Royal Society, and the German Academy Leopoldina. He holds many awards and prizes, including the Lenin prize, the H e i n e m a n n prize of the GOttingen Academy of Sciences, and an honorary doctorate of the Universit~ de Paris. And it is a pleasure to listen to him. Zdravkovska: Please tell us about the history of Soviet mathematics as you have experienced it. Shafarevich: When I try to recollect, a very clear picture comes to me. Since I was 15 or 16 I have r u b b e d elbows w i t h m a t h e m a t i c i a n s . That was starting in 1938, in the last pre-war years. I have the impression that our mathematics was then a very interesting experiment. There was a large group of
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mathematicians, m a n y very good mathematicians; they had all the journals, practically all were accessible. And all contact stopped there. Here is some sort of sociological experiment: h o w will such a group develop? It is very hard to image the degree of our isolation. I remember very well one of the strongest impressions in my l i f e - - m y first conversation with a foreign mathematician. That was Loo-Keng Hua, w h o was from China (he died recently). He arrived here in the late 1940s, and the Scientific Council of the Steklov Institute called me. I was asked to give a lecture in English, so that he could understand. The second foreign mathematician w h o m I met was Erich K/ihler, who invented K~ihler geometry. He also came here. The third one was Leray. Then in the early 1950s many others followed. The older generation, Pontryagin for example, rem e m b e r e d t h e [ t o p o l o g y ] c o n f e r e n c e [1935, in Moscow] which Cartan had attended and where he posed the problem of computing the Betti numbers of the simple Lie groups, which Pontryagin solved. He met Lefschetz. For them these were individuals, and, vice versa, they themselves were individuals for the others. But we developed in total isolation. Here I was, a r o u n d thirty years old, meeting m y first foreign mathematician. At the same time, there were m a n y v e r y good mathematicians from here. That's w h y mathematics developed, even though somewhat one-sidedly. Topology was well developed here, thanks to Pontryagin (until Pontryagin quit working in topology, and the students of Caftan began: Serre, Thom . . . . ; for some time that was a breaking point), then functional analysis, and analysis. But some things were absolutely unknown. It was very typical, and very interesting as far as the mathematics were concerned. That is w h y it was extremely appealing to try to penetrate there. For example, algebraic geometry. I remember h o w Chebotarev told me that he had tried many times to understand the Italian algebraic geometers, and had come to the conclusion that it was perfectly hopeless. Then he gave it up. He assured me that the same opinion was shared by the West, that in his time he had talked to van der Waerden and someone else, and that the consensus was that it was a hopeless undertaking. I think that it wasn't quite thus, that he simplified a little, because of course there were people in the West who still had contacts with the Italian algebraic geometers and w h o understood them. Also, algebraic number theory was completely unknown. Many questions of higher-dimensional analysis, say, Hodge theory, harmonic integrals--all these were completely mysterious things. I came across them in the following way. During the very end of the war and in the first post-war years I lived very far from the Mathematics Institute and I had to commute for an
hour each way. I would board the trolleybus and take along the journals that I wanted to return to the library. I would leaf through them (I didn't look at the papers that I had read). It was interesting to find out what they contained. All of a sudden I would come across totally shattering discoveries. So I found a survey by de Rham and learned de Rham and Hodge theories. Another time I e n c o u n t e r e d Nevanlinna theory. It created a feeling of something interesting, because right nearby there was some undiscovered, mysterious domain. I think that the following generation already had a completely different attitude toward mathematics. You could learn everything through conversations. As for now, I have the feeling that many mathematicians simply are born knowing everything and that there is nothing to teach them. But I remember h o w I learned that way: I w o u l d take a journal with some interesting mathematical article and I would see that a little is understandable, and then something totally unintelligible begins. There are references. Some of these would be even more unintelligible. That was intriguing. On the one hand it increased my interest in mathematics and created a special relationship toward it. It created the expectation of something unknown, and mysteries at each step. On the other hand, it stimulated very much the creation of students. In any branch that I would start working on--first algebraic number theory, then algebraic geometry--there was almost no one to talk with. And in order to create a circle of people with w h o m I could communicate, it was simply necessary to seek out y o u n g people and give them the taste for it. I would like to add that the situation that I describe reflects in some sense two states: on the one hand the youth of Soviet mathematics, and on the other, my own youth. In my recollections it is hard to tell them apart. I r e m e m b e r that mood. It is beautifully described by Goethe in Faust. He says: ~ a gtebel mir bie ~eIt tJerbillIten ~ie ~no~l~e ~3unber noc~ Der~l~racl~. He exactly describes the psychology of youth. Indeed Soviet mathematics was young: It had started with Luzin's school. Or I was young. It's hard to tell the difference. Probably, both played a role. You asked me about Soviet mathematics. It seems to me that n o w it differs in no w a y from world mathematics. The isolation is completely over. It may exist on a personal level; some people have fewer contacts, some have more. But on the whole, it is nonexistent. And the danger in mathematics is the same as in our civilization in general. It consists in that mathematics can acquire the character of a mechanized, automatized, artificial, controlled activity. In mathematics this is connected, of course, with the very large growth of the number of publications and the number of matheTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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maticians. I think Andr~ Weft said that earlier the trouble was that there were too many bad mathematicians; but that is only a half trouble, n o w there are too many good mathematicians. That is much more complicated. And indeed, a completely n e w method of communication has developed: there are year-long conferences--a year of algebraic geometry, a year of number theory; the technique of preprints, continuous trips here and there, invitations. Here also I could quote Andr~ Weft. It seems that he was here for the first time in 1935; there was this famous topology conference I already mentioned. Then the next time he was here was about 15 years ago. And he told me that when he had first been here he had discovered a cer-
Andr~ Weil said t h a t earlier the trouble w a s t h a t there were too m a n y bad m a t h e m a t i cians; b u t t h a t is o n l y a h a l f trouble, n o w there are too m a n y good mathematicians.
tain new social phenomenon. He understood immediately that it was very important and predicted a great future for it in the West. In Russian it is called "koman-di-rov-ka" [business trip]. Simply put it is a trip to various places at someone else's expense. And indeed that was an element of contacts, a mixing of information, that he took as something new, surprising. N o w it is ubiquitous. The impression is that all mathematicians form a unique nervous system. And thanks to that, the property of individualism of the mathematician is weakened. For example, Poincar~ invented topology by himself. N o w that is already impossible. Such is the situation. I do not think it is just my wish that it were not so. It is also impossible because when graphs are drawn in science statistics, they all turn out to be exponential. The budget for science will very s o o n - - a r o u n d the year 2000--overcome the global human production. So that this cannot go on. And the situation is already changing. Earlier, all competent mathematicians w h o finished Meh-Mat [Depart~Lent of Mechanics-Mathematics at Moscow State University] were immediately and eagerly hired there. They became docents, then professors. But n o w a mathematician is far from finding work so easily. I hear that the West has the same problems. But that is natural. Of course, theoretically, there could be a society consisting 3/4 of mathematicians, provided the physicists, biologists, medical doctors, engineers, etc. gave up the same claim. But they cannot all have their way. All our life has the character of a race, acceleration, condensation, or as they say, revolution. Napoleon has said that bayonets m a y be a good thing, but sitting on them is uncomfortable. The same goes for revolution. I don't want to discuss whether it is a good or bad 18
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thing, but at least it is also uncomfortable to always exist in a state of revolution. They lead either to a crash or, as they say here, to a perestroika. Perestroika should happen in mathematics too, in some sense. Because the current direction cannot continue infinitely. Zdravkovska: You complained the other day about the solution of the 4-color problem. Shafarevich: T h o s e are of c o u r s e d a n g e r o u s symptoms, when proofs appear that are not only impossible to understand, but can only be run by buying time on a powerful machine. Another example is provided by the finite simple groups. I was struck when, recently, the Canadian mathematician Coleman told us h o w he had studied the old work of Killing (it's either 100 years old, or will be next year). He had discovered that it contained the whole theory of simple Lie groups, i.e., what is n o w called Coxeter groups, Weyl groups, Dynkin diagrams. Slowly they were extracted from there. But is there a single person w h o understands the work on finite simple groups? At any rate, there is no single exposition. There are two volumes giving a description of only the apparatus necessary for understanding that work. Zdravkovska: What do you think of Meh-Mat and Petrovskii"s role in it? Shafarevich: I would like to answer more generally about Moscow mathematics. I have the impression that Moscow mathematics has played an exceptionally great role in the USSR. We are closer to France than to the States or Germany. In France everything is concentrated in Paris. We also have Leningrad, and a few more schools in some other places. But there is a great concentration of mathematicians in Moscow (espedaily in earlier times). And Moscow consisted of two components: Meh-Mat and the Steklov Institute. Of course, each strong mathematician had an input in the development of both centres, via work and influence on students. But if one speaks of an administrative, social input, then Petrovskii was the one who had the greatest input at the University, and Vinogradov at the Steklov Institute (for the period that I have been able to observe). Petrovskii played a very great role both at the whole University and at Meh-Mat. I think that in mathematics and in m a n y other sciences, such as physics, the end of the 1940s and early 1950s saw a certain slowing down of development. M a y b e the isolation increased or interest waned. The number of y o u n g mathematicians decreased. Maybe it was caused by the war, the death of very many talented mathematicians in the war. In the University it was also connected with the persecution. The opinion was that if a man works too much in mathematics he becomes antisocial, that he loses
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touch with the collective, sets himself off against the collective, b e c o m e s an egotist and individualist. There, they say, a professor invites a second-year student, offers him tea. It's clear that the student will become after that an individualist. I remember it because I was that professor. There were years when only one or two, and sometimes no, students w h o were finishing Meh-Mat would be left to do graduate work. It is strange that it was then done openly. In a big auditorium (that was in the old building) a list was put up of all the students w h o m the professors proposed for graduate work. [There are two ways of becoming a graduate student in mathematics: as an undergraduate, a recommendation of your professor is needed; as an employed mathematician, the recommendation of your workplace is required.] Then, one after the other, they were crossed out as unsuitable. The list consisted of 30-40 people.
cancer. The goal in his life, that by which he judged himself, was his personal mathematical work and the Steklov Institute. That is why, his prejudices, sympathies, and antipathies notwithstanding, his goal was nevertheless for the Institute to be good. His first reaction was always negative. In psychology it is called negativism. You would propose that he should hire s o m e o n e w h o m he likes in every respect, b u t he w o u l d say: "'No, there are too many difficulties. I don't know, I don't know." That was always the first reaction. But masses were hired on his initiative. All our section [of algebra] was hired under Vinogradov. And many others. I would come here to meet with Delone w h o would invite me to talk with him. And I would meet almost all the mathematicians whose names I knew: Keldysh, Gelfand, Shmidt and many others would be here. [We were at the Steklov Institute w h e n this conversation took place.]
Zdravkovska: Who would cross them out? Shafarevich: Some kind of committee consisting of representatives of the "dekanat" [chairman's office] and the local party organization. This was an especially dark period in the physics department. There, it was simply considered that quantum mechanics and relativity theory are the result of rotting bourgeois science, which believes in mysticism. A n d Petrovskii managed to overcome it all, when he became rector. That was probably 1952 (I do not remember exactly) just before the death of Stalin. It also had a propitious influence on me, as I was not allowed to teach at the University from about 1948 or 1949, but Petrovskii invited me back. I was a witness w h e n , in the late 1940s, during a meeting of the Meh-Mat Scientific Council, Lev Abramovich Tumarkin said: "What next? A secondyear undergraduate publishes a notice in the Doklady: W h a t kind of a f a s h i o n is that? We create n e w Newtons in our deparh~-tent." Boris Nikolaevich Delone [pronounced as the name of his French relatives Delaunay] was very apologetic: "He is m y student, I assure you he is quite innocent, it is I w h o dragged him into number theory." This student, w h o after that meeting began working in applied mathematics under Petrovskii, later became an associate member of the academy in Novosibirsk. So, about Petrovskii. He pulled the university out from a depression, as I see it. On the other hand, the person of Vinogradov was quite unique. He created the Institute and directed it from 1930 until his death. It is hard for me to name a single well-known Moscow mathematician w h o has not worked at the Steklov Institute for a quite long period. That was connected with various peculiarities of Vinogradov. He was a somewhat troubled person, terribly alone. He never had any family, or close friends. His one sister died of 20
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Zdravkovska: Couldn't one meet them at the University? Shafarevich: Yes, the majority would also be there. Vinogradov himself could never teach classes, he tried but was unsuccessful. Then he never did it again. Though in some sense he was a great pedagogue, as he has written a beautiful book on number theory where he gave his view of number theory: number theory is not a field that can be developed gradually by axioms, it is a certain art, artistry. The book consists of m a n y e l e m e n t a r y and boring s t a t e m e n t s from number theory. But 3/4, or even Yl0 of the book is occupied by problems. The idea was that number theory can be understood through problems--just as a handicraft, one has to use one's hands. But Vinogradov could not teach orally. Zdravkovska: But Vinogradov also had his bad sides. Shafarevich: You know, Vinogradov was a victim, or we were victims of the fact that the present remarkable rule (about retiring at age 70) did not exist then. N o w it is a perfectly ironclad rule. Zdravkovska: When was it put into effect? Shafarevich: This year. In our Institute five people are n o w retiring from being heads of sections, including the Director of the Institute. It is a colossal change of guard, and n o w it will continue. Of course, when a person grows old, he becomes subject to many influences, he loses some firmness, some strength of character. If luck would have it, he could stay at the level of his position, but that is often not the case. Of course, until he turned 70, he was the ideal of an Institute director. Of course, there is always personal fric-
tion in such a background, but it has the character of a second order improvement to the asymptotic term, so to speak. To finish m y reminiscences about Vinogradov and Petrovskii I am glad to say that, unlike some other outstanding mathematicians, neither of them signed statements condemning Sakharov or Solzhenitsyn. Zdravkovska: When did you start teaching? Shafarevich: Late in the war, in 1943 or 1944. Zdravkovska: You were 20 then. When did you enter the University? Shafarevich: I never entered it. I was a bold kid. While a schoolboy, I came to the dean [chairman] and told him that I had been reading textbooks and that I did not know whether I understood them correctly, whether I really understood them. Could I try to take an exam? He said I would be made an "external" student. The first one he sent me to, to take an exam from, was Delone (analytic geometry), the second was Kurosh (algebra), and the third was Gelfand (analysis). So I met all three. They were very cordial and good towards me, they worked a great deal with me, i.e., gave me a lot of literature to read on top of what was required. That was a period w h e n school was changing: sometimes it would consist of ten grades, sometimes of nine. By the time I finished ninth grade, I had passed almost all the exams for the university. I was allowed to enter as a final year student so that I have no high school diploma. I turned 17 as soon as I finished the university. I consider Delone and Kurosh my teachers. They represented not just two different, but two diametricaUy opposite mathematical types. Delone was a classical representative of the Petersburg school. He was interested in Diophantine analysis of a very classical type. All more or less abstract directions in mathematics were completely foreign to him; for example, general topology and the direction in algebra that stemmed from Emmy Noether. But Kurosh was, on the contrary, a radical. He told me that mathematics can be divided into two parts: philosophy and accounting. All formulae, i n t e g r a l s - - t h a t was accounting. A n d everything that existed before the twentieth century had already lost its interest. That was a psychology analogous to the psychology of futurism, avant-gardism in art. This two-sided influence helped me in some sense to find a middle road. That is w h y when the fundamentally new directions started appearing from the West, such as homological algebra, I felt that that was some kind of synthesis of abstract notions with concrete problems of the theory of complex manifolds or algebraic geometry. I felt that it was something close to me.
I. R. Shafarevich
I would like to add that Delone gave me two particularly good pieces of advice: one was to read Hilbert's Zahlbericht, and the second to read Gauss. As a result of that lecture I chose the field in which I worked for about 20 years. Zdravkovska: How old were you when you read Hilbert's Zahlbericht? S h a f a r e v i c h : Fifteen. At such an age it produces an especially strong impression. At fourteen Delone reco m m e n d e d that I read Galois theory. But that, I completely did not understand. More so as the exposition was very heavy: I read Chebotarev's book. That is a very heavily written book, confused. Then I figured it out from van der Waerden's book. The following year I read Hilbert, when I was 15. I already knew a little algebra to be able to understand it. It does not use any particular mathematical apparatus. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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Zdravkovska: You told us a little about Delone and Kurosh. What about Gelfand?
won't be captivated by them, but Galois theory, that's something else."
Shafarevich: I r e m e m b e r that Gelfand made me read many books saying that he wouldn't give me the exam if I didn't. But these were really very good, very useful books. As I was then interested in algebra, teaching me analysis was very useful. This was the first special topics course that I attended. And I suspect it was the first that Gelfand taught, but I am not certain. Springer now publishes my collected works and I added notes to m y papers: I tried to remember how I had written them. Here, I recall my first published paper: it was rather abstract, reflecting the atmosphere of Meh-Mat of the time. It was very hard to learn something nonabstract. This paper is concerned with abstract topological fields and the existence of a metric on them. There is a notion that, as Gelfand told me, one encounters in the theory of linear s p a c e s q t h e notion of b o u n d e d set. There I applied it to algebraic fields. I talked a lot with Gelfand. For a while he was considered my official advisor (in 1940, when I was a postgraduate). I think before the war I had two supervisors simultaneously: Kurosh and Gelfand. Then, when the war started, Gelfand went to one place, the university to another, so Kurosh remained my sole supervisor.
Zdravkovska: When did you have your first students?
Zdravkovska: How do you pick your students? Shafarevich: I think I have a fairly standard approach toward m y students. Basically, I would acquire them teaching first-year undergraduates. The atmosphere at Meh-Mat consisted in hunting for the talented students. By the third year they were all taken. Very rarely would a student change directionmas a rule they would do that after becoming independent mathematicians. Within the limits of the university, it would be very rare for a student to change supervisors. Therefore the majority of my students grew from the first or second year compulsory algebra course. Then I would spend a lot of time with them. I would talk with them, tell them things. I would prepare each student for the exam. I w o u l d think out what problems should be given him ahead of time, what he should do before coming to take the exam. The students were strictly trained. At least that was true for the average that I saw at Meh-Mat. But Delone did it completely differently. His influence consisted of two things. First of all he could create a colossal enthusiasm, a feeling that it was all extremely interesting. Second, he would give advice. Some strange advice occasionally; he would advise a boy who knew little to read Galois theory. Though I do not know whether I knew what a determinant was. But he w o u l d say "Determinants are boring, you 99
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Shafarevich: It is hard to remember. Whom to consider a student? There were people with whom I interacted, but their field was still little known to me or to them. It was hard to start. From the moment that students began appearing regularly, the first one was A. I. Kostrikin. He studied in a provincial university, and a special group of students was selected to continue in some particular field. They had to write their diploma theses and I was asked to give them subjects. I already had the habit of giving weaker students diploma works of the following sort: learn some difficult chapter from an old book, say Jordan's, and rewrite it more or less in a modern and understandable way. That was useful for others as well, as they could later read that work. So I gave a few such subjects, and then, to m y surprise, I got carried away and said: "'Well, in this area (group theory was close to them and I had been asked to give them diploma works related to it) there is a so-called Burnside problem and some strange things about it have started to happen in the last few years. Almost nothing has been done, but clearly it is moving. If anyone would like to, he could read such and such papers." Then all of a sudden a young man came to me and said "I would like to." In the beginning I became very scared, so I said: "First start by reading a part of van cler Waerden and solving the problems in it." In a few days he had solved them all. Soon I found out that this was a very talented man. He solved the restricted Burnside problem in a very complicated way, which is probably still not completely untangled. This method is now called "Kostrikin sandwiches." Then we worked together. Very pleasant memories are left over from then. Piatetski-Shapiro is one of my older students. He worked with many: He started as a student of Nina Bari (he has a paper on trigonometric series). Then he wrote a paper under Gelfand's influence. But I think his most interesting papers are in several complex variables. Even though he wrote them completely independently, and even chose the problem himself, during that period he interacted with me; we had a seminar on the theory of modular functions of several variables. The second generation of my students arose when Petrovskii called me back to the university. That was when the new building was opened [1953]. There, for the first time, I taught first year algebra. There I met Manin. That was a very strong year. It was the beginning of a very strong wave of mathematicians. That year Manin, Anosov, Golod were freshmen, then the next year Arnold and Kirillov, but them I knew from before: I had given them both prizes in the high school
mathematics olympiads. Then came Novikov, and my student Tyurina. I would like to mention two of my students who no longer work in mathematics. One is Tyurina; she had an outstanding gift for mathematics; and she died young; her work is still well known in singularity theory. The other is Arakelov who had a serious disease and quit working in mathematics. But he is remarkable in that he invented an apparatus for solving problems coming from number theory, which later turned out to be useful in many questions. Arakelov metrics can be found in dozens of papers, even in physics (the theory of bosonic strings). I know many papers entitled " O n the Arakelov metrics . . . . " Moishezon was approximately the same age as Manin, Arnold, and Novikov, but he appeared later-he graduated from Stalinabad (now Dushanbe), in Tadzhikistan, not from Moscow University. The very last of my brilliant students was Kolyvagin. He solved a problem that had bothered number theorists for a long time. It is connected with the theory of elliptic curves. Their arithmetic is determined by a certain group [called the Shafarevich group] about which it was not clear whether it was finite or infinite. No one was able to find it for any example. He found a new method connected with a whole series of other questions. That is one of the sensations in the last year in this field.
Zdravkovska: At some point you stopped having students. Shafarevich: Yes, Kolyvagin was really one of the last. He started working with me as an undergraduate, precisely w h e n I stopped teaching at the university. I left him to Manin in inheritance. By then, Kolyvagin had already written his diploma work, and he became a graduate student of Manin. So that he is our joint student. This was more than ten years ago. Recently I had to make a list of all the postgraduate students whose official supervisor I had been. I had to remember them all. I counted 33. These were candidate's degrees [=American Ph.D.], and then 10 doctorates. One has to take into account that it all ended about ten years ago. I now live on old resources. Zdravkovska: Looking at this list of your students, I find that I know the names of about a third of t h e m . . . Shafarevich: Of course, that is natural. Mathematics is a pyramid. When you teach first-year undergraduates, you meet masses of talented people. Toward the end much fewer go on to do graduate work. An even smaller number go on to do research work after the candidate's degree, and often some quit doing research after the doctor's degree. Moreover, my observation is that this is not by any means a selection of
the most talented, but according to some additional criteria. I remember exceptionally brilliant first-year students w h o stood out among all their fellow students w h o themselves became more famous mathematicians. They stood out by their ingenuity, the und e r s t a n d i n g of the mathematical ideas b e h i n d a problem rather than solving it via a complicated calculation. Then, toward the end of their studies at MehMat, they simply lost interest. Smilka, please forgive me, b u t it occurred most often with women. I remember h o w I was always struck by the number of undoubtedly outstandingly gifted first-year students. But their interest simply went into some other direction, maybe a much more important one; for example, bringing up children.
I d o n o t u n d e r s t a n d w h y I w a s n o t f i r e d much earlier. Zdravkovska: Did it bother you that you were not allowed to teach? Shafarevich: Yes, the pain has gradually subsided. But for a long time I would simply see in my dreams that I lecture. It was a constantly recurring dream: I feel it is the beginning of a special topics course. I like the w a y I have thought out the plan for the whole year. In front of me there is a large auditorium and I have an uplifting feeIing. That was a constantly recurring dream. It is a very lingering thing. You only feel when it's over how captivating an occupation teaching is. Zdravkovska: Would you lecture now? Shafarevich: I don't know. It needs a great deal of effort. I don't know. Zdravkovska: Why were you fired twice from Meh-Mat? Shafarevich: It is quite clear w h y the second time. I do not understand w h y I was not fired much earlier. Each society has to be judged by the norms existing in it, and not by some abstract scheme. So, by the existing norms, m y crime deserved much more than being fired from the university. I published a book on socialism and a series of [nonmathematical] articles. I collaborated with Solzhenitsyn, with Sakharov. And me, a university professor, teaching students! It was of course a strange phenomenon. Zdravkovska: Why were you allowed to transgress so much? Shafarevich: The last years it was purely due to Ivan THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989 2 3
Georgievich [Petrovskii]. He would say to me: "You know they ask me for your blood.'" And I would answer: "What's to be done. They a s k - - s o spill it." "No, no, no. You know, I use ruses. I tell them that the students like you, w h o knows what would happen [if you were fired], it isn't convenient to do it in the middle of the academic year . . . . By the end of the year, they have forgotten." That was my last conversation with Petrovskii. It is remarkable that he was a member of the Presidium of the Supreme Soviet, i.e., according to the formal constitution he belonged to the collective that is our head of state. So he managed to delay it to his death. Then his successor, Khokhlov, also managed to delay it for some time. Basically, the initiative came from Meh-Mat. The first time, in 1949, many were fired; it was a dark atmosphere at Meh-Mat. It seems that if a teacher had many students, that was considered bad. The desire was for a more bureaucratic way. All should have the same number of students. Many that had parttime positions were fired. It is not as though we were stripped of our jobs. Just as the second time, I continued to work in the Steklov Institute. Foreign mathematicians phoned me asking whether I couldn't feed my family. That was never the case. In 1949 there was no rule that you couldn't teach somewhere else. But many of those w h o worked somewhere else and also taught at Meh-Mat were fired. Luzin, for example, and many others. Shmidt was fired also, and he was an ex-member of the government under Lenin. Zdravkovska: Could you tell us about Egorov who died in Chebotarev's home in Kazan? Shafarevich: That was all long before I came to Meh-Mat. So it's all what Americans call "hearsay." I heard it from tales of the elders. In an American court it would not be taken as evidence. I have the impression that an approximately equally important role in the creation of the Moscow school of mathematics was played by two mathematicians: Luzin and Egorov. But they were completely different individuals. Luzin was very extroverted. He was a beautiful lecturer. Some even said that they didn't like his theatricality during lectures. He made a colossal impression on students, especially the younger ones. They became around him a small sect of adoring followers. But it seems that Egorov was a much more reserved and closed man. However, from stories I have the impression that his influence on the creation of the Moscow school was no smaller. Both the theory of functions of a real variable and the first impulses toward the creation here of a school in functional analysis stemmed from Egorov. He was the president of the M o s c o w Mathematical Society a r o u n d 1930. I could show you a copy of the report on the situation in the Moscow Mathematical Society from the journal 24
THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989
Matematicheskie znaniya v massy published by Kol'man. He describes how the reactionary Egorov was fired. Three reactionaries are mentioned: Egorov, Finnikov, and Appelrot. I have never heard of Appelrot. About Finnikov I know very well. He suffered too. Nothing catastrophic happened, but it was hard for him. He was probably fired. But Egorov was arrested. For some time he was under inquest, and then he was exiled in Kazan. He seems to have fallen sick in prison. As Chebotarev told me, Egorov died in Chebotarev's home, in his arms. Chebotarev had been living in Kazan for a long time, I think right after the civil war. He had graduated with Delone. He came from D. A. Grave's school. Grave himself didn't become particularly k n o w n as a mathematician, but he had a brilliant school in Kiev from which at least three students became known: Delone, Chebotarev, and Shmidt. Zdravkovska: How do you choose what to work on? You have always worked on what was not fashionable around you. Shafarevich: Each mathematician tries to find something that he deems interesting and unknown. You have to find an interesting problem, i.e., something people didn't pay attention to before. It is a question of luck. In my life I've had half-successes, successes, and failures. I remember, for example, h o w Kostrikin and I went out of Moscow for a walk, and returning on the electric train, started discussing. At that time I was interested in the work of Elie Cartan on "infinite" Lie groups or, as they are now called, "pseudogroups.'" It is partly a geometric, partly an analytic theory. Kostrikin was studying finite-dimensional Lie algebras over a field of finite characteristic--a purely algebraic question. So, when he started comparing the classification given by Cartan with the known examples of Lie algebras, it turned out that they coincided. We conjectured that, though these are completely different fields, there is complete parallelism, and that thanks to it, one could transfer Cartan's techniques there. We made the first steps toward its proof. So it was a half-success. But not a complete success. I regret very much that my school didn't bring this to an end; clearly there were people that could have done it. It was solved a few years ago by two Americans, R. Block and R. Wilson. Very good and classical algebraic questions were solved completely. There were cases of complete failure that have remained unknown to all. For example, the following p r o b l e m has always b o t h e r e d me. Take all lines passing through the origin. They form a projective space; we have here a fiber space over a projective space. Now, what does the set of trajectories of an ordinary differential equation form? Completely unclear. If a solution winds on a limit cycle, these two points
should be considered as infinitely close, inseparable in the topology. On the other hand, locally, there is a uniqueness theorem: each solution intersects a transversal plane only once. So it occurred to me that Grothendieck had created an apparatus in algebraic geometry for studying precisely that kind of phenomenon (which can be well described locally, but nothing comes out globally), I mean the Grothendieck topology. I got m y student Parshin interested in it. We worked extremely hard on this problem for several m o n t h s completely unsuccessfully, i.e., nothing worked. It seemed that we constructed some topological invariants, such as the Euler characteristic, Betfi numbers . . . . We hoped that in the end these would be invariants of the differential equation that would enable us to estimate the number of limit cycles or something of the sort. The reason for the failure is now more or less clear. It seems that Alain Connes from Paris has been able to do something in that direction. But he had an additional idea that didn't even enter our heads: the rings of functions on these spaces could be noncommutative. In connection with this I remember that Menshov (he is now about a hundred years old [Menshov died in December 1988]) a long time ago told me that when Hadamard came here, there was a dinner in his honor, and Menshov asked him how he chooses subjects on which to work. Hadamard answered with an Eastern tale. Some sultan became bored with the w o m e n in his harem, and he called his chief eunuch and ordered him to find a n e w concubine who should please the sultan. He gave him two days. Should he not carry out the order, his head would be cut off. So the eunuch went to t h e bazaar to look, and he saw that the bazaar was full of women. But they were all the same as far as he could tell. All of a sudden he saw an old friend of his, a merchant, w h o had just come back from a trip, looking young and sun-tanned. The merchant asked the eunuch the reason for his sadness. The latter described his problem. The merchant was surprised: "What kind of a problem is that? Just take this one here." The eunuch decided to risk it. The following morning the sultan called him and said: "Go to my treasure-house and take as many riches as you can carry." The eunuch was happy. But then he decided that such a situation could occur again. So he ran to his friend the merchant: "Listen, tell me the secret. H o w did you choose her?"--"You know, one has to have eyes." So there, said Hadamard, in order to be able to find problems in mathematics, one should have eyes. Zdravkovska: What are your current interests? Shafarevich: They are connected with memories from youth. A memory from youth: it is a general phenomenon, when you first come across something, it
I. R. Shafarevich and friend
all raises questions, seems nonobvious. I remember that two things surprised me very much. That in algebra, especially in Galois theory, in algebraic number theory, in class field theory, in the theory of algebraic functions, there is a beautiful theory connected with various coverings, groups, that are abelian. In case the corresponding Galois group or fundamental group is nonabelian, all this.theory ceases to exist. I had the idea that, at present, abelian mathematics has been created, but that the mathematics of the future will be nonabelian mathematics. Then I discovered a year later, to my greatest surprise, the work of Andr6 Weil in which he expresses the same idea. On top of that, he had done something very interesting in this nonabelian mathematics. N e v e r t h e l e s s , I c o n t i n u e d working on it, and for a long time it was the theme of m y studies. Even recently there was the following case: I was invited to give a lecture at a meeting of the Moscow Mathematical Society for a general audience of students, orienting them to mathematics. I thought that this was suitable for them because, though something has been done, the situation has remained about the same--everything that is connected with nonabelian groups remains largely a mystery. Unfortunately, a week before the lecture I was called and told that "for some reason" the lecture had been cancelled. I think that no one has ever summed up the present situation. This remarkable work of Andr~ Weil is of course famous, but no one has given a general survey of what has followed from it. I shall write an article on this for the Mathematical Intelligencer. The second impression that has stayed with me since my youth: I was very surprised that in the theory of algebras (and there are various algebras--associaTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 2 ~
tive and Lie algebras, they all have manifold applications) everything ends with the case of semisimple algebras. The algebraists with w h o m I have talked always said for some reason: "These algebras are nilpotent and there is nothing interesting there." The only person from w h o m I have found support was the same Andr~ Weft and in the same paper. He encounters there at some moment nilpotent algebras and writes that on sait qu'on ne sait rien about them. And later at some point I correspond with him about it. He said that he still thinks that these algebras are some sort of mystery, and not something meaningless. People work on them but this is not one of the prestigious directions of mathematics, even though there are clear, hard problems which seemingly many have tried to solve, but were unsuccessful. They have applications, for example, in the classification of singularities of mappings. In their book, Arnold, Varchenko, and Gusein-Zade state vividly that a certain object becomes infinitesimal, not perceivable with the eyes; that is w h y we have to describe this object via the ring of functions on it, and it turns out to be a nilpotent algebra. As I am at an age w h e n no one expects me to do anything, I am not afraid to work on it. If nothing comes out, it won't be a scandal9 I hope to at least draw attention to it. Clearly, there are these connections with various fields, e.g., vector bundles. I think that it is an interesting field that does not enjoy due respect from the mathematicians. Zdravkovska: Who is close to you mathematically? Shafarevich: That has changed in the course of my life. For a long time the closest person for me was D. K. Faddeev because we were both in the same situation in algebraic number theory: we started working on it when there was no such tradition here. There was Chebotarev, but Chebotarev was a classic. First of all it was hard to understand him, he has a very peculiar mind. So one had to learn it all from the literature. It was very interesting. Aside from that, Faddeev is a very rare man w h o relates in a well-disposed way to other people's thoughts. He derives greater pleasure from listening to w h a t he is being told than from trying to tell something. So he was for a long time the closest friend I had. Then I started having students: Manin, Tyurina, 9 . . This is the generation of students with w h o m we lived in the atmosphere that nearby there was something unknown. We have a very interesting seminar. We studied the classical Italian algebraic geometry, surface theory. It was very difficult. For example, you have to talk in the following week's seminar; you read Enriques's book and it is totally unclear whether you will be able to extract something from it to give your talk. But it was very interesting. We wrote the book Algebraic Surfaces from it. 26
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Another period of my life arose when contacts with foreign mathematicians became possible. A m o n g them the people who were closest to me are Tate and Serre. Contacts with foreign mathematicians were very useful to me. I was very excited when I met them for the first time (I knew them all very well, knew their work, but had not met them); and a great number of them simultaneously, at that, during the Edinburgh congress. I met these people in the flesh9 Zdravkovska: Do you remember this picture [from Halmos's I Have a Photographic Memory]? Shafarevich: Yes, I also remember the boat. That was in Edinburgh, yes? It was very pleasant. These p e o p l e w e r e all y o u n g , and t h e y p l a y e d a game switching badges. Somebody pinned Serre's badge on me, and mine on Serre. Then Cartan saw me and said: "Bonjour, Monsieur Serre!" I saw his badge saying "Cartan" but I thought that maybe he is the same kind of Cartan as I am of Serre. Zdravkovska: Why have you stopped going to meetings? Shafarevich: You know, it seems to me rather that it was an exception when I did go to meetings. Earlier, it was impossible to travel abroad, and when it became possible, it was hard to refuse. But I have the feeling that life is stripped of silence and peace. And under these conditions to fling from one end of the world to the o t h e r . . . I had the following happen to me: I lectured in India, returned to Moscow, then I had to travel to Paris. There were two strange years like that. It seems to me that everybody lives that way in the West. Already then it seemed unnatural. But it all got regulated irrespective of my wishes. Zdravkovska: Would you travel now? Shafarevich: No, though I have many invitations. For example, I've now been invited to G6ttingen, but it was not clear whether it wouldn't be connected with some unpleasant conversations [to obtain an exit visa]. At any rate, I refused without regrets. But maybe, at some point I will travel some more. Zdravkovska: Please tell us what interested you as a child. Shafarevich: My first sensation of a perfectly new phenomenon, which I do not even know h o w to formulate, culture maybe, was connected with history. I remember h o w I read a book, some German text on the history of Rome (later I wondered that it had produced such an impression--it is a very boring book). But it opened my eyes to the fact that the world does not finish with my personal sensations, and extends in
all directions to infinity. I threw myself into reading history books and for some time I was convinced that, of course, I shall be a professional historian--I'll work all my life on history. I tried to remember h o w I turned to mathematics and could not recall the logic or reason for it, though I think that to work on history then would really have been hard. It was considered an ideological discipline, and the possibility of choice of various points of view was very limited. I could hardly have understood that as a 10-12-year-old boy, but maybe I felt it subconsciously.
books that could, as a model, shed light on the current situation: ecological, concentration of people in cities. A sharp change from the civilization in which 9/10 of the people lived in nature, in villages, to the urban civilization. Such a phenomenon occurred also in antiquity. Very similar. I n o w read books about it.
Zdravkovska: What do you think of the current situation? Some youngsters are pessimistic. Are you an optimist?
Shafarevich: U n f o r t u n a t e l y , since I s t o p p e d teaching in the university my contacts with youth Zdravkovska: You told us once that, before the war, the have ceased. I am an optimist. I look at things irrationpsychology of people deciding to work on mathematics re- ally optimistically. From reading various historical sembled the psychology of people deciding to enter a monas- books, I have the impression that humanity finds its ways and makes its decisions as a result of a very tery. complicated process, more complicated than the creation of a single individual man. Therefore, w h e n Shafarevich: Yes, in some sense the attractiveness such a decision is made, it is incomprehensible to an of Meh-Mat before the war w a s c o n n e c t e d with individual man. If a historical situation arises, which leaving behind the hardships of life. Mathematics was, seems to be a crisis to an individual man, and he 9simply logically sees that there is no exit, that does not mean that an exit indeed does not exist, because an U n t i l the w a r , s c i e n t i s t s w e r e p o o r l y paid, exit can be found using the more powerful conscioush a d no p a r t i c u l a r prestige. The s t a n d a r d o f ness of humanity, not that of an individual. That is living o f scientists w a s extremely low. During why, on the one hand, it is probably an objective fact the w a r t h a t changed sharply. that this is a period of crisis. I remember continuously after the war that the one thing I would hear would be that we are at the brink of a catastrophe. (It is a situaon the one hand, a nontechnical science--the connec- tion reminiscent of the one in medieval times, when tion with painful practical problems did not arise. On people expected the end of the world.) First it was said the other hand, it was not the humanities; you didn't that the atom bomb had been invented, and now we feel in it the elements of ideological interference. I had are surely going to exterminate each other; then that the feeling that Meh-Mat was attractive for many as a the demographic explosion will kill humankind; n o w place where one can do science in its most concen- the ozone layer will be destroyed, and there will not trated form, without any outside influence. During the be enough clean water, or the earth will be poisoned war, the relation t o w a r d science c h a n g e d v e r y by chemical fertilizers. The invariant is that it seems strongly. Until the war, scientists were poorly paid, there is a feeling that the naturality of man's contact had no particular prestige. The standard of living of with nature has been broken. The time is probably scientists was extremely low. During the war that critical. But it does not follow that there will be no exit c h a n g e d sharply. The salaries increased all of a from that crisis. Probably there will be. Of course, as sudden by a factor of 2 or 3. The prestige changed very with any crisis, this is connected with shocks, but will much. Scientists began to be written about. Most pres- they be large or small, that, of course, is unclear. I think that man is a very strong creature. We have seen tigious, of course, was to be a physicist. The next enough h o w harm can be inflicted on man and nature, place, probably, was held by mathematics. One should say that this did not lead to an imme- but we do not realize h o w much strength remains. diate increase [of the number of gifted students] at Take now, for example: What people write in the M e h - M a t in t h e 1940s. T h e n s u d d e n l y [in t h e newspapers, h o w many thoughts have accumulated mid-1950s] for a reason that I cannot understand, it cut in people's heads! It seems the reaction should have through. But I have to say that during the period that I been that everyone would be indifferent and no one have observed, this was the only t i m e - - a n explosion should be interested in anything. On the contrary. All which literally in the course of some four years gave of a sudden there is a terrible explosion of activity, as such a wave of remarkable mathematicians. soon as such a possibility has occurred. H o w many societies are formed[ The societies Doverie, Pamyat, are Zdravkovska: What do you read now? different, opposite. It is clear that these are all young people w h o are really not at all indifferent to what Shafarevich: Nonmathematical? I read historical h a p p e n s in their country. They may completely diTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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verge in their view of what should be done, but not in that it concerns them, and not in that they themselves should be doing it. And they were brought up on the very opposite principle.
Zdravkovska: Things do change now in the Soviet Union... Shafarevich: Yes, it is something completely unpredictable. I think that in the end it will lead to something that no one is expecting, neither those who are for, nor those w h o are against perestroika. It seems to me that a process that no one can control has started. Of course, it could end with something sad, an explosion, for example. Zdravkovska: Your article Russophobia has now become known in the USA. It is widely discussed among mathematicians, and occasionally it provokes sharp disagreements. Some consider it unfair, and even accuse you of antiSemitism. It seems this article was written a long time ago. Has your attitude toward the questions discussed in it changed since then? Shafarevich: Since this article was written, the situation in our country has changed considerably: the word glasnost has even entered the English vocabulary! Unfortunately, this has only confirmed the point of view of my article. The counter-Russian ideology which is discussed in that work has n o w manifested itself much more vividly. Many very sharp judgments have appeared putting down Russian history, culture, and the Russian national character: on the immemorial slavery of the Russian soul, the shallowness of Russian literature, etc. N o w these are p u b l i s h e d in journals with wide distribution. The authors are partly those mentioned in my article, but many new ones have joined them. I am afraid that, should this ideology be accepted without reserve, it will have a destructive influence on the Russian national conscience. It would be hard to expect that it will not provoke objections. In my paper I tried to formulate and discuss those objections, keeping at the level of logic and facts. I allow that some places may seem offensive to Jewish national feelings: I know myself that in these cases feelings always speak before logic. But in that paper, I definitely do not say anything similar to the unfounded offensive Russophobic judgments which are extensively quoted therein. The Russians and Jews will have to live together for a long time to come and must learn to listen and discuss each other's opinions, even if they seem offensive. The most difficult questions are better discussed openly, and not s u r r o u n d e d by prohibitions and taboos. Zdravkovska: You also talked last time about music. 28
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Shafarevich: Other than mathematics, I am most interested in history (the applied science, which gives the possibility to understand what is going on now), and then music. I really used to go to the Conservatory very often. And even though with age it becomes harder to do it, I still go there regularly. But at that time, before Stalin's death, music had a special place in culture because it was in some sense uncontrollable. Not everybody could understand it. Last year was the 80th anniversary of Shostakovich's birth. On this occasion I listened to him very much, listened all over again. It was surprising. Of course, it is clear that the composer writes about his time, and what does one see there? Simply, an apocalypse. H o w could it have happened? It was for the initiated. There was a language with which one could communicate. There was no Solzhenitsyn, nothing. Music had a great weight in culture in general. Aside from mathematics and music, I also like hiking . . . . Of course, I still go hiking with my students. I stopped hiking in the mountains because that has become difficult for me. We have no mountain huts as in the West in which you can spend the night, and where you can get some food. You have to carry everything with you. And w h e n you have to carry around 30 kilograms in your backpack, it is hard for me. But we go outside of Moscow, always with students. My love for hiking was Delone's influence. He was a well-known lover of mountain hiking. His feeling for natural beauty was surprisingly strongly developed. If you w a n t e d to travel in the mountains where it is beautiful, the best way was to ask Delone. You could rely on him 100% there. He would always recommend a route, a pretty pass. He would say: "Everyone goes that way, but you go this way, it is more beautiful." [This is a fitting sentence to end an article on Igor Rostislavovich.] Acknowledgment I would like to express my gratitude to Askold Khovanskii and Allen S h i e l d s - - w i t h o u t their presence, help, and support this article would not have been written; and to I. Dolgachev, Yu. Manin, and A. Rud a k o v - - f o r enlightening conversations about Shafarevich. Smilka Zdravkovska Mathematical Reviews 416 Fourth Street P.O. Box 8604 Ann Arbor, MI 48107 USA Igor Rostislavovich Shafarevich Steklov Institute of Mathematics ul. Vavilova, GSP-1 SU-117333 Moscow, USSR
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 caf~ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the European Editor, Ian Stewart.
Sir William Rowan Hamilton Radoslav Dimitri4 and Brendan Goldsmith William Rowan Hamilton, undoubtedly the greatest Irish scientist, was born in Dublin on Dominick Street, around midnight between 3 and 4 August 1805. He was the fourth of nine children in the family of Sarah Hutton and Archibald Rowan Hamilton. It was to young William's advantage that both of his parents were intellectually oriented, but the greatest influence on the young man's education was his uncle James. James Hamilton was a curate in the village of Trim, 20 miles from Dublin, and a talented linguist: besides " o r d i n a r y " European languages, he was good at Greek, Latin, Hebrew, Sanskrit, Chaldee, Pall, etc., and it is no surprise that the child prodigy William was fluent in English, Latin, Greek, Italian, French, Arabic, Sanskrit, Persian and learning Chaldee, Syriac, Hindustani, Malay, Bengali, Chinese, Moabifish. To contradict once more the prejudice that talents for fine arts and sciences are incompatible, let us add that Hamilton was also able to write verse. Influenced by Zerah Colburn (1804-1839), an American calculating boy who happened to be at Westminster School in London at the time, Hamilton learned a good deal of mathematics and astronomy, so that by the age of 19 he was able to predict eclipses. Although he never attended formal schools, his self-education and the education given to him by his uncle enabled young Hamilton to be the first among one hundred candidates entering Trinity College, and later to win all possible language and mathematics awards. He finished the first draft of his distinguished treatise on systems of light rays, then fell in love With a lady who let him down by marrying a down-to-earth soldier (Hamilton could offer "'only" some poems to
her). However, because William Wordsworth did not think highly of his poems, Hamilton realized that science was his best bet, although he persisted in writing poems all his life. Hamilton's student days were finished in a grandiose m a n n e r w h e n he was offered a chair of astronomy at Trinity College and was appointed Astronomer Royal at the Dunsink Observatory at the age of 22, although he did not even apply for the position advertised. His great classical work, Theory of Systems of Rays, was published w h e n he was 23; it had the same effect on optics as Lagrange's Mdcanique analytique had had for mechanics. The ideas here involved the application of algebra to optics. Hamilton's discoveries in optics mathematically predicted an unexpected real phenomenon connected with the conical refraction in biaxial crystals. According to some this was the peak of Hamilton's career; according to Hamilton the discovery of quaternions was his greatest a c h i e v e m e n t . The f o r m e r opinion was supported by the fact that after the discoveries in optics, Hamilton was destroyed by his third marriage and alcohol. Hamilton also made significant contributions to dynamics with his most important paper On a General Method in Dynamics (1834). Both discoveries in optics and dynamics depended on the single idea of a characteristic function. The background for Hamilton's discovery of quaternions was his representation of complex numbers a + bi as ordered pairs (a,b) with suitable algebraic operations.
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Brougham Bridge, birthplace of quaternions.
The plaque on Brougham Bridge commemorating the discovery of quaternions. Hamilton wanted to do similar things for rotations in three-dimensional space and the great troubles he had led to the even greater discovery of quaternions. The discovery involved denying the commutative law for multiplication. It was a great surprise at the time, comparable to the discovery of non-Euclidean geometries. The quaternion algebra over a field F (as it is known today) is a set H=
{or + f3i + "yj + 8k ]ot, f~,'y, S E F}
with operations of addition and multiplication defined according to the basic rules for "orthonormal vectors": z~ = ]~ = k2 = - 1 , ij = k, jk = i, ki = j, ji = - k , kj = - i , ik = - j .
Hamilton was so impressed by the new idea that suddenly flashed through his mind (after 15 years of persistent work) while walking with his wife from Dunsink to Dublin one day (16 October 1843) that he scratched the main formulae of the new algebra on a stone bridge that he happened to be passing. Although the legend of quaternion formulae scratched in the stone of the bridge sounds incredible, we note that Hamilton was an obstinate scribbler, carrying his writing materials wherever he went. According to one of his sons, Hamilton would write on his fingernails and on his hardboiled egg at breakfast if there were no 30
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paper at hand. In a letter to his son shortly before his death, Hamilton wrote that the formulae have "as an inscription long since mouldered away" (from the bridge), but a durable notebook where the formulae were written on the day of discovery still exists, kept in the Royal Irish Academy. Hamilton devoted the last 22 years of his life almost exclusively to perfecting the theory of quaternions and applying them to dynamics, astronomy, and the theory of light. Elements of Quaternions, containing more than 750 pages, was published a year after his death (2 September 1865). Let us mention that Hermann Gtinter Grassmann published his Ausdehnungslehre in 1844, independently from Hamilton, and this contained a more general treatise on hypercomplex systems, but these lectures did not gain great popularity because they were too difficult to read. Apart from quaternions, Hamilton made other contributions to pure mathematics. He corrected Abel's proof on the impossibility of solving the general polynomial equation of the fifth degree in radicals and studied properties of the icosahedron and the dodecahedron, inventing the "'Icosian Game" and selling the copyright for 25 pounds. Although the buyer did not make any profit, the game has significant consequences nowadays. It is played on a fiat board with a graph of a dodecahedron drawn with each vertex representing a city. The player has to find a route for travelling along the edges of the graph, passing through each vertex only once and returning to his beginning point. Such a route is known as a Hamilton cycle and the problem as a travelling salesman problem. A great amount of time is spent today in finding Hamilton cycles in different situations (e.g., microchip indust~). Quaternions were born at Brougham Bridge, a stone bridge in Dublin on the Royal Canal at the Broombridge Road, along the railway to Sligo and Galway. A stone plaque at the side of the bridge states that the formula for quaternion multiplication was carved on a stone of the bridge by Hamilton. City busses no. 22 and 22A will take an interested tourist close to the bridge, starting, say, on Canal Street in Dublin city center. The stone plate is badly damaged and the canal around the bridge needs a clean up: the setting itself is otherwise picturesque. Hamilton is buried in Mount Jerome; the inscription on his grave is &v~lp 6~k61rovor K~ ~btkotMl0~q~--"a lover of labour and truth." University of Exeter Mathematics Department North Park Road Exeter EX4 4QE England
Dublin Institute of Technology Mathematics Department Kevin Street Dublin 8 Ireland
Steven H. Weintraub* For the general philosophy of this section, see Volume 9, No. 1 (1987). A bullet (o) placed beside a problem indicates a submission without solution; a dagger (~) indicates that it is not new. Contributors to this column who wish an acknowledgment of their contribution should enclose a self-addressed postcard. Problem solutions should be received by I August 1989.
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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
Film and V i d e o as a Tool in M a t h e m a t i c a l Research Robert L. Devaney
Ever since I was a mathematical "child," I have admired the marvelous films produced by Tom Banchoff and his coworkers at Brown University. What a wonderful way to explain difficult concepts in mathematics to a general audience. As a researcher in dynamical systems, I often thought that such films could be used to great advantage, not only to exhibit dynamical phenomena to non-mathematicians, but also as research tools in their own right. After all, dynamical systems theory and, in particular, bifurcation theory, is the study of how systems move or change in time, so it is only natural to record these events continuously on film. The purpose of this article is to describe some of m y experiences making films and to show how the results (both mathematical and otherwise) have been much more exciting and interesting than I had ever imagined at the outset. My first experience with film came about as an afterthought. I had been investigating the dynamics of entire transcendental functions such as ~ sin z, ~, cos z, and ~,e~. Following the work in the early 1980s of Adrien Douady/John Hubbard [7], Benoit Mandelbrot [8], and Dennis Sullivan [9] on quadratic polynomials, the study of the iteration of entire maps held the promise of exhibiting new and interesting dynamical phenomena as well as interesting computer graphics. The essential singularity at infinity seemed to inject a considerable amount of complexity or "chaos" into the dynamics. To write the necessary computer programs, I recruited the help of three u n d e r g r a d u a t e students, Chris Mayberry, Chris Small, and Sherry Smith. These students formed an ideal team. They were all assistants in the Boston University computer graphics lab, so they came with the necessary programming skills, expertise in graphics, and, best of all, unlimited computer time.
Within weeks they had a marvelous program running to produce the Julia sets (see the box titled "Julia Sets" for definitions) and the associated bifurcation diagrams for such maps as Ke% K sin z, and ~, cos z. The algorithms used were variants of a method suggested by John Hubbard, which in turn was based on my earlier joint work with Michal Krych [6] that characterized the Julia sets of these functions as the closure of the set of points whose orbits escaped to ~. To display a Julia
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set, we simply colored points whose orbit went "too far away" at iteration i with a color that depended on i. Points whose orbits did not escape were left black. The algorithm is explained in the box titled "Hubbard's Algorithm." Thus, colored points on the screen represented points in the Julia set; black points represented the complement of the Julia set; and the colors told us h o w quickly orbits escaped to infinity. I r e m e m b e r vividly the excitement with w h i c h Sherry entered m y office w h e n the first pictures of Julia sets appeared on the screen. "Come upstairs quickly," she said. "You should see the Julia set of sin z!" I was busy at the time, so I told her that I would be up in a moment. Besides, I already knew what the Julia set of sin z would look l i k e - - t h e black region would resemble a collection of "'infinite snowmen" in the plane. "That's right," she said, "but you should see (1 + .1i) sin z." She could hardly conceal her excitement as she described the swirls of color that had replaced the s n o w m e n on the screen. Intrigued, I went upstairs to find an amazing change that occurred in the Julia set of ~, sin z. As the parameter changed from 1 to 1+ ei, the Julia set seemed to "'explode." Color instantaneously replaced regions that had been black; the Julia set seemed to change discontinuously from occupying a very small region in the plane to occupying almost the whole plane. Since that time, this phenomenon has become quite well understood: the Julia sets of entire transcendental functions can and often do "explode." An explosion occurs in a one-parameter family of entire maps Fa at a parameter value )~0 if the Julia set of Fa is nowhere dense for h < ~0, but is the whole plane for h > ho. In general, w h e n the Julia set is nowhere dense, it actually occupies only a small portion of the plane. The following theorem, proved as direct outgrowth of these computer experiments, is typical [3,4]. THEOREM. Let Ea(z) = hez and Ca(z) = ih cos z. Let J(Ea) and J(Ca) denote the Julia sets of Ea and Ca, respectively. Then 1. J(Ea) is a nowhere dense subset of {z E CIRe z > 1} for h ~ 1/e, but J(Ea) = C if ~ > 1/e. 2. J(Ca) is nowhere dense if]hi <~ .67 . . . . but J(Ca) = C if ]hi > .67 . . . .
The students were quited pleased with the program t h e y had written, and rightfully so. The program worked flawlessly, and the images it generated were spectacular. They spent h o u r s over the next few weeks generating all sorts of interesting pictures. Eventually, as always seems to happen in this field, they grew tired of seeing the same old images over and over again, intriguing as they were. They started thinking on a grander scale--Hollywood! Why not make a film of how these Julia sets changed? This 34
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seemed like a fun project to me, although I must admit that I never expected anything of interest mathematically to come of it. It also seemed like an excellent follow-up project for the students to undertake. So we decided to try it. I was to write the "screenplay" for the film and to find the funds for the filming expenses. The students, in turn, would cajole the computer center into giving them enough CPU time to compute the film. This last item was non-trivial. Each frame of the film involved iterating a complex transcendental function up to 35 times on each point of a 512 by 512 grid in the complex plane, a process that consumes 2 - 5 minutes of CPU time on an IBM 3081. With 24 frames per second of film, we were looking for 200 hours of CPU time on Boston University's academic computer in the middle of the school year to produce a three-minute film. Somehow the students managed to "'find" the computer time, and the dean graciously agreed to fund the project and we began. We decided to film two exploding Julia sets. One was the explosion in the family ik cos z, described in the theorem above. The other was the first explosion we had witnessed, the explosion in the family (1 + d) sin z w h e n e = 0. At that time, I did not completely understand w h y this latter explosion occurred but expected that the mechanism would be the same as for the cosine explosion. Computing and filming the Julia sets of these functions was no real problem. The computations were done at night and d u m p e d onto tape, and the images were photographed using 16mm film and a Matrix Instruments camera. The entire process took about six weeks. This would have been fine, except that one of the great nemeses of mathematicians who use student programmers intervened--graduation. Before we finished, two of the students graduated and summer vacation began. This slowed down the process considerably, especially in terms of creating titles for the film and dealing with the processing. Eventually, however, the film, entitled Chaos, was ready. The first screening was to take place at a meeting of the staff of the Academic Computing Center. I was asked to explain briefly how the film was made and what it meant mathematically. I expected that the latter would be no problem, because I thought I knew exactly what I would see. However, as the film ran, I was absolutely dumbfounded by what I saw! By far the most predominant feature of the film was something that I had completely missed as I previewed countless still images from the film. For the family (1 + ei) sin z, the explosion occurred just as I had anticipated: the Julia set seemed to fill the plane with infinitely many spirals of color. But, as the parameter changed, the tails of these spirals swirled around the complex plane rhythmically and s e e m e d to cross neighboring spirals always at the same instant. This
was all the more intriguing because, in the cosine explosion, these spirals again filled the plane, but this time their swirling tails never crossed! Clearly, there was something vastly different between these two explosions, and I was at a loss to explain w h y this was so. See the cover of this issue of the Mathematical Intelligencer and the box tiffed "The Cover." I used the film in a number of seminar talks and colloquia that I gave in 1984, and inevitably the question arose as to w h y the spirals collided for (1 + d) sin z. I concocted a number of different and unsatisfying explanations for this p h e n o m e n o n (ranging from numerical error to "I don't know"), all the while wondering if there really was something mathematical lurking in the background. Eventually, after a number of private screenings of the film, the answer became clear. The "explosion" in the sine family was no explosion after all. As e increased, the family passed through infinitely many parameter values for which the Julia set was indeed the whole plane, but whenever the spirals touched, a small black region appeared, indicating the birth of an attracting periodic orbit. Mathematically, this meant that, as ~ increased, the Julia sets of (1 + d) sin z changed infinitely often from being nowhere dense to being the whole plane, and it was the crossing of the spirals that signalled these transitions. These transitions did not occur in the cosine family, as the above theorem guaranteed, and that was why the spirals did not cross in this family. My experience with this film motivated me to make THE MATHEMATICAL INTELLIGENCER VOL. 1L NO. 2, 1989 3 ~
more, but without m y "kids," I had no way of doing this without an inordinate investment of time learning the graphics and filming process. Luckily, at this time, the National Science Foundation (NSF) had begun to establish several supercomputer centers which would have available all of the computing and technical re36
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sources to make these films. In 1985, however, these facilities were not yet operational, so NSF leased time on various industrial supercomputers, including time on a CRAY X-MP at Digital Productions, Inc. (DP) in Los Angeles. Digital was a leading producer of computer graphics special effects for movies and televi-
sion, so this seemed like an ideal place to make a film. The procedure for getting time on one of the NSF supercomputers is simple: a two-page letter explaining the p r o j e c t to m y p r o g r a m director at NSF, t h e n Richard Millman, was all that was required. Richard carried the ball for me at the Office of Advanced Scien-
tific C o m p u t i n g at NSF, and soon thereafter I was approved to use 75 hours of CRAY time at DP to produce two films. The first film was to describe the changes that occurred in the Julia set of ~, cos z as ~ decreased from -rr to 2.92. I h a d been looking for quite some time at these THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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Julia sets at Boston University and about all I knew was that there was a fascinating and intricate sequence of bifurcations that occurred in this parameter range. On the real line, it was easy to see that this family u n d e r w e n t a "homoclinic bifurcation" to a saddlenode point when ~ - 2.97, and this seemed to control all of the interesting behavior that ensued. But there seemed to be no w a y to put all of the bifurcations into perspective without using a film. As I had anticipated, the technical assistance at DP was superb. I was p u t in touch with Craig Upson, who was to work with me to produce the film. I sent Craig the algorithm to produce the Julia set of ~, cos z and several slides which showed representative resuits. Within a w e e k Craig's program was running and we were ready to begin filming. I found it best to make a quick trip to the supercomputer site at this stage of the process. Using "flipbooks," we previewed the motion of the film and checked whether the window in the complex plane, the speed of the parameter, and other adjustable parameters were right. It took exactly six hours to accomplish all of this, so all that was left was finding a 48-hour period when DP's CRAY was completely free. Thus, after one day in "Hollywood," the film was essentially finished and I went home to await the results. Within two weeks, Craig had computed and shot the entire film, and two weeks later I had the finished product in my hands--Exploding Bubbles. This film contained no major surprises as had the previous film, but it showed that there was a sequence of primary bifurcations, all looking essentially the same. Each of these bifurcations was accompanied by a sequence of secondary bifurcations, which were in turn limits of tertiary bifurcations, and so forth. Later, this phenomenon was explained in a joint paper with Adrien Douady [5J--as the parameter ), decreased from 2.97 to 2.92 in the family ~, cos z, ~, passed through infinitely many Mandelbrot sets: some primary, some secondary, and so forth. The film was a record of the order in which these bifurcations occurred. See the box titled "Exploding Bubbles" and the accompanying photographs. I made one more film at DP in 1986, this time with the able assistance of Stefan Fangmeier. Unfortunately, DP w e n t bankrupt soon thereafter, but in a very fortunate turn of events, both Craig and Stefan reappeared at the NSF-funded National Center for Supercomputer Applications at the University of Illinois. I have since m o v e d m y efforts to this site. With funding from both the Geometric Analysis program at NSF directed by Paul Goodey and the Applied and Computational Mathematics Program at DARPA directed by Helena Wisniewski, I have made several more films. The advent of improved video technology including single frame recording has made it possible to record these images in video format rather than on 38
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16mm film. This eliminates costly film processing and provides more or less instantaneous results: video is ready for viewing as soon as it has been recorded. Moreover, using video, we can record low resolution versions of the films first as an experimental tool, saving the heavy investment of computer time necessary for high resolution films only when the mathematical intricacy warrants it. The results of these efforts have always been mathematically stimulating. Not all of the films have yielded the dramatic surprises as in my first film, but they all give a completely different perspective with which to view bifurcation problems. Usually, this results in a theorem or two. Perhaps the most important consequence of these films will not be the research they inspire. Rather, the films seem to be a particularly effective means of communicating the beauty of mathematics and the excitement of mathematical research to younger students. I have shown the films and lectured on the mathematics behind them to a number of audiences comprised of high school students, undergraduates, and the general public. The students seem fascinated by the intricate patterns that they see on the screen. Nobody ever asks me what this mathematics is good for or what applications I have in mind; rather, the audiences seem to be intrigued that simple dynamical processes can lead to such complicated behavior. Granted, most students are not familiar with the complex sine or cosine. But the realization that not all is known about sine and cosine--the bane of many a high school student's life--or in mathematics in general, is an important message that is easily communicated with film.
References 1. P. Blanchard, Complex analytic dynamics on the Riemann sphere, B.A.M.S. II, No. 1 (1984) 85-141. 2. R. L. Devaney, An Introduction to Chaotic Dynamical Systems: Redwood City, CA: Addison-Wesley Co., (1987). 3. R. L. Devaney, The structural instability of Exp(z). Proc. A.M.S. 94 (1985), 545-548. 4. R. L. Devaney, Bursts into chaos. Phys. Lett. 104 (1984), 385-387. 5. R. L. Devaney, and A. Douady, Homodinic points and infinitely many tiny Mandelbrot sets. Preprint. 6. R. L. Devaney, and M. Krych, Dynamics of exp(z), Ergodic Theory and Dynamical Systems 4 (1984), 35-52. 7. A. Douady, and J. Hubbard, ]~tude dynamique des polyn6me Cenplexes, PublicationsMathematiques d'Orsay, 84-102. 8. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco: Freeman & Co. (1982). 9. D. Sullivan, Quasiconformal maps and dynamical systems III. Preprint.
Department of Mathematics Boston University Boston, MA 02215 USA
Mathematicians Sweep 1988 Wolf Prizes Lawrence Zalcman
Mathematics and mathematicians emerged the big winners at the Wolf Foundation Prize ceremonies in Jerusalem on May 12, 1988. The award in Mathematics went to the distinguished mathematicians Friedrich Hirzebruch, Director of the Max Planck Institute for Mathematics in Bonn, and Lars H6rmander, Professor of Mathematics at the University of Lund. Sharing the prize in Physics were Stephen Hawking, Lucasian Professor of Mathematics at Cambridge, and Roger Penrose, Rouse Ball Professor of Mathematics at Oxford. Another mathematician, Raphael Levine, Max Born Professor of Natural Philosophy at the Hebrew University (and formerly Professor of Mathematics at Ohio State University), shared in the Chemistry prize for his work in developing and applying mathematical theories of molecular dynamics. Add to this the dramatic appearance of I. M. Gelfand, the first Wolf laureate in Mathematics, to claim his prize ten years after its initial award, as well as to accept prizes previously awarded to the late A. N. Kolmogorov (1980) and the ailing M. G. Krein (1982), and one has all the ingredients of a genuine mathematical happening. Altogether, eight of the fifteen Wolf Prizes handed out that spring evening in Jerusalem were given for achievements in and of the mathematical sciences. Not bad for a day's w o r k - - a n d a timely reminder of the basic role m a t h e m a t i c s c o n t i n u e s to play in r e s e a r c h throughout the natural sciences. The Wolf Foundation, founded "to promote science and art for the benefit,of mankind," began its activities in 1976 with an initial e n d o w m e n t of ten million dollars, donated principally by Dr. Ricardo Subirana Lobo Wolf (1887-1981) and his wife Francisca. Ricardo Wolf's life story is the stuff of which novels are made. Born in Hannover, Germany, one of fourteen chil-
dren, he completed his studies in chemistry in Germ a n y and emigrated to Cuba before the outbreak of World War I. In 1924 he married Francisca Subirana, the European and world women's tennis champion of the 1920s. For nearly twenty years, Wolf labored to develop a process for recovering iron from the residue of the smelting process; the ultimate success of these efforts made him a rich man. Dr. Wolf was an early and generous supporter of Fidel Castro. In 1961, as a gesture of gratitude, Castro appointed Wolf (at the latter's request) Cuban ambassador to Israel. When Cuba severed diplomatic ties with Israel in 1973, Wolf decided to remain in Israel, where he spent his final years. Since 1978, the Wolf Foundation has awarded, on an annual basis, prizes "'for achievements in the interest of m a n k i n d and friendly relations a m o n g peoples" in the areas of agriculture, chemistry, mathematics, medicine, and physics. In 1981, a prize in t h e arts (which rotates annually among painting, music, architecture, and sculpture) was added. The Wolf Prize in each area consists of a diploma and one hundred thousand dollars, to be shared equally by the recipients in a single area. Winners are selected (without regard to nationality, race, color, religion, sex, or political views) by international committees consisting of from three to five experts; new prize committees are appointed annually. The deliberations of these committees are completely confidential, and
LamremeZa/cman is Lady Davis Professor of Mathematics at Bar-nan University and Editor of Journal d'Analyse Math~natique. He lives in Jerusalem.
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their decisions are final. The official awards ceremony takes place at the Knesset (Israeli Parliament) in Jerusalem, where the winners are handed their prizes by the President of Israel. In certain areas, notably medicine, the Wolf Prize has gained a r e p u t a t i o n as a kind of "pre-Nobel Prize." In fact, six Wolf laureates in medicine have subsequently received the Nobel Prize. Three more Wolf laureates have gone on to win the Nobel Prize in other areas. In 1982 Kenneth Wilson, who shared the 1980 Wolf Prize in Physics with Michael Fisher and Leo Kadanoff "for pathbreaking developments culminating in the general theory of the critical behavior at 40 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989
transitions between the different t h e r m o d y n a m i c phases of matter," was awarded the Nobel Prize in Physics in its entirety, the first Nobel laureate in Physics not to share the prize in more than a decade. 1 One of the recipients of the 1986 Nobel Prize in Chemistry was John C. Polanyi, a 1982 Wolf laureate. Another Wolf laureate of 1982, Leon Lederman, was awarded the Nobel Prize in Physics for 1988. The absence of a Nobel Prize in Mathematics renders the situation there rather different. Indeed, as the richest annual award for outstanding achievement in mathem a t i c s ( a n d lacking a n y r e s t r i c t i o n as to age, nationality, or organizational membership), the Wolf Prize is, arguably, the true "Nobel Prize of Mathematics" (cf. Gelfand's remarks quoted toward the end of this article). These differences are reflected in the age variations among laureates in the different disciplines. Specifically, recipients of the Wolf Prize in Mathematics have tended to be older than winners in the other sciences. Thus, the average age of laureates in mathematics (at the time of the award) stands at 72, in contrast to 61 in chemistry, 58 in physics, and 571//2in medicine. During the first eight years of the awards, only one laureate in mathematics (Gelfand) was less than 69; and the average age of the winners in mathematics in any given year did not drop below 71. More recently, there has been a notable decline in the age of the mathematics laureates; and in 1988 the average age of the recipients actually dipped below 60. Whether this indicates a trend, only time will tell. Equally provocative is the distribution of winners of Wolf Prizes in Mathematics by country: USA 8, France 3, USSR 3, Germany 2, Japan 2, Hungary 1, Sweden 1. Although the p r e d o m i n a n c e of winners from the United States is initially quite striking, it pales in comparison with the statistics from physics, where 16 of the 22 Wolf laureates thus far have been from the United States, or chemistry, where the corresponding figures are 9 out of 16. The distribution in mathematics is much nearer that of medicine, where 8 of 20 Wolf laureates are Americans, with the remaining dozen winners distributed over eight other countries. In any case, before anyone runs off to proclaim the superiority of American mathematics, I note that of the eight USA laureates in mathematics, only one (Hassler
1 Wilson thus became the second winner of the Mathematical Association of America's William Lowell Putnam Mathematical Competition to win a Nobel Prize in Physics; Richard P. Feynman, who received the Nobel Prize in 1965, was the first. Feynman scored in the "top five" in the second Putnam Competition, held in 1939; Wilson did so in the fourteenth and sixteenth competitions, held in 1954 and 1956, respectively. Another Nobel laureate in physics, Murray Gell-Mann (who won the Prize in 1969), received a n honorable mention in the seventh Putnam Competition, held in 1947.
Mark Grigor'evich Krein
Whitney) is native-born and only two (Whitney and Peter Lax) are American-trained.
Typically, the Wolf Prize festivities begin several days before the actual awards ceremony, as the winners arrive in Israel for a week of lectures, receptions, and touring. This year, the excitement starts even earlier, as rumors begin circulating that the USSR has finally agreed to allow 1978 Wolf laureate in mathematics Izrail Gelfand to travel to Israel to receive his prize. (According to the rules of the Wolf Foundation, a laureate must appear in person at the prize ceremony in Jerusalem in order to collect his award. Over the past decade, the Soviet government has repeatedly refused to allow those of its nationals who have been designated Wolf laureates to travel to Israel to accept their prizes, which consequently have been held in trust by the Foundation.) Similar rumors have surfaced in previous years, only to be revealed as unfounded; this time, they prove correct. Gelfand's arrival at BenGurion Airport receives extensive media coverage. He is welcomed not so much as the culture hero he is, but as a long-lost brother. Does his visit herald a new, more generous, emigration policy for Soviet Jews? Inevitably, speculation as to the larger significance, if any, of this gesture by the Soviet Union toward Israel is rife. Gelfand, for his part, resolutely refused to be drawn into the speculation or, indeed, into political discussions of any sort. For him, this is a purely scientific visit, his first, to Israel. It is also an opportunity to make new acquaintances and renew old friendships. Two of Gelfand's favorite (and most famous) students, Joseph Bernstein and David Kazhdan, n o w both pro-
Andrei N. Kolmogorov
fessors at Harvard, are in Israel for the event. The unusual warmth and affection between them and their old teacher are evident for all to see. An aura of good feeling prevails. In a gesture of its own, the Wolf Foundation decides to bend its rules a bit and allow Gelfand to accept on their behalf the prizes awarded his compatriots Andrei Kolmogorov (1980) and Mark Krein (1982). Krein's failing health precludes any possibility of his receiving his award in person; Kolmogorov died in 1987, and his prize is to be forwarded to his family. Less heralded but no less excited, the other Wolf laureates arrive in Israel. For the better part of the next week, they will travel the length and the breadth of this small land, lecturing at universities and other institutions of higher learning. Of the subject areas in which Wolf Prizes are awarded, physics is currently the "hottest," and the physics laureates do not disappoint in their choice of topics. While Hawking regales his audiences with thoughts on The origin of the universe and tales of Black holes and their children--baby universes, Penrose offers ideas on Time asymmetry and quantum gravity and muses on The puzzle of quantum reality. Necessarily, the titles (if not the contents) of the lectures in mathematics are more prosaic. Hirzebruch ties analysis, geometry, and number theory together with talks on Ceby~ev polynomials, special algebraic surfaces and curves, and modular forms and Elliptic functions in topology. H6rmander lectures on The Nash-Moser theorem and paradifferential operators and The Cauchy
problem with small data for fully non-linear perturbations of the wave operator. However, Gelfand draws the largest audiences. He speaks at Tel Aviv and Jerusalem on hypergeometric functions of several variables. By a curious coincidence (or is it rather the well-known THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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stand at the center of the (very different) achievements of Levine and Hawking: Levine used these theories to gain an understanding of h o w molecules collide; Hawking applied them with striking success to issues in general relativity. Of course, the mathematical substrate of general relativity is differential geometry; and it is for groundbreaking applications of geometry and topology to physics that Penrose is receiving his prize. Discoveries in differential geometry and topology, and their applications in analysis and number theory, loom large also in Hirzebruch's contribution to mathematics - - w h i c h brings us full circle. All this calls to mind the words of 1981 Wolf laureate Lars Ahlfors: The more mathematics progresses, the more apparent the essential unity of all mathematics becomes. The central mathematical ideas become more and more alike, in contrast to the techniques, which are becoming increasingly diversified with the regrettable result that many mathematicians can communicate only with their fellow specialists. The progressing algebraization of all mathematics is frequently commented upon. The boundary lines become more and more diffuse. It happens that a modem paper in analysis reads exactly like a paper in topology. To some extent this is due to the growing insistence on a concise and rigorous language which must necessarily be algebraic in character. But there is also a much deeper reason, namely that the number of significant mathematical structures is strictly limited. 4 @
9
9
The awards ceremony itself begins promptly at 5:30 p.m. on Thursday, May 12. Seated on a dais against the impressive backdrop of Chagall tapestries and facing a large audience of academics, members of the diplomatic corps, reporters, and press photographers are the laureates and, to the right, representatives of the Wolf Foundation and the State of Israel. All rise as President Chaim Herzog takes his seat on the platform, and the ceremonies begin. As usual, there is a "'great minds" principle?), this is the very same topic on which 1988 Crafoord Prize laureate 2 Pierre Deligne had lectured at the Hebrew University mathematics colloquium a week or so earlier. 3 The week's rich diet of lectures (and press releases) leads one to ruminate on the diversity of accomplishment of this year's "mathematical" Wolf laureates. Diverse it is, but there are also reassuring interconnections, eloquent testimony to the essential unity of mathematics. Thus, H i r z e b r u c h ' s Riemann-Roch theorem for algebraic vaffeties finds its natural culmination in the Atiyah-Singer index theorem, which provided a major impetus (via work of Seeley and Palais) for the theory of pseudodifferential operators cited in H6rmander's award. The theory of Fourier integral operators, another of HOrmander's major accomplishments, has its roots in the relation between classical mechanics and quantum mechanics. Now, amazingly enough, quantum mechanics and thermodynamics 42
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2 The Crafoord Prizes are awarded by the Royal Swedish Academy of Sciences in several fields not covered by the Nobel Prize: astronomy, biology, geophysics, and mathematics; the prize in mathematics is given every seven years. The first mathematicians to receive the Crafoord Prize were Louis Nirenberg and V. I. Arnold. The 1988 Prize was awarded to Alexander Grothendieck and Pierre Deligne. Grothendieck declined his portion of the prize. 3 1 was witness to a similar coincidence, involving two other Wolf laureates-to-be in mathematics, some twenty years ago. It once happened that Henri Cartan (who shared the 1980 Wolf Prize in Mathematics with Kolmogorov) and Carl Ludwig Siegel (who shared the 1978 Prize with Gelfand) spoke, on successive Thursdays, at the Harvard-MIT-Brandeis Mathematics Colloquium on exactly the same topic. (They both lectured, if I remember correctly, on the Weierstrass Preparation Theorem.) Incidentally, these were the only lectures, out of four years of faithful colloquium attendance, that I understood completely, from beginning to end. 4 Lars V. Ahlfors, Classical and contemporary analysis, SIAM Review 3 (1961), 1-9, p. 1.
scriptural reading (in this case, the famous messianic idyll of Isaiah 11:1-9) and a musical interlude. Then comes the actual presentation of awards. Discipline by discipline, the citations are read and the winners summoned to receive their prizes from President Herzog. After the prizes in a given area have been awarded, one of the recipients offers a response. Hirzebruch speaks on behalf of the laureates in mathematics. His response expresses appreciation for the
award and hope that it will prove an inspiration to further activity. It also addresses, in forthright fashion, the tragic history that links Germans and Jews. This is what he said: Mr. President, Ladies and Gentlemen: For Lars HOrmander and myself, it was a great surprise w h e n we learned at the end of January that the Intemational Committee of the Wolf Foundation THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989 4 3
had chosen us as recipients of the 1988 Prize. We are delighted and deeply honoured for this award. We think with gratitude of Ricardo Wolf and thank the members of his Foundation for all their work and hospitality. We are happy now to belong to an impressive series of prize-winners but, at the same time, also a little ashamed because there are many excellent mathematicians who would have deserved the prize just as much. The prize is a recognition for past work, which extended over a period of several decades; but we hope still to have years ahead of us, during which it
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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
will spur us on to further work in the service of mathematics and mathematicians. Today we are seeing an impressive ceremony in the Knesset; the 12th of May, 1988, and the week we are spending in Israel will certainly remain in our memories as one of the outstanding events of our lives. We have also gained an impression of the great achievements which the people of Israel have accomplished. We hope with all our hearts that this c o u n t r y a n d its n e i g h b o u r s m a y be g r a n t e d a peaceful future.
Finally, let me add something which concerns only myself. As a professor at the University of Bonn, I am one of the successors of the famous mathematicians Felix Hausdorff and Otto Toeplitz. Hausdorff committed suicide in 1942, together with his wife, when deportation to a concentration camp was imminent. Toeplitz emigrated to Israel with his family in 1939 and died there the following year. The memory of these mathematicians is with me always on this day.
The ceremony continues. Joshua Jortner and Raphael Levine, the Israeli scientists who share the prize in chemistry, draw special applause. But if it makes sense to speak of a popular favorite among the 1988 Wolf Prize winners, there can be no doubt as to his identity. In his response for the laureates in physics, Stephen Hawking, whose courageous twenty-six year struggle with amyotrophic lateral sclerosis (Lou Gehrig's disease) has gained him worldwide admiration (and major coverage in both Time and Newsweek), addresses the question of human survival. Speaking THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 4 5
from his wheelchair by means of a "talking computer," he propounds the paradox that the very scientific progress that has brought us so tantalizingly close to unraveling the secret of the universe now threatens to put out the lights of civilization before the final synthesis can be attained:
9. . It now looks quite hopeful that we may have a complete theory to describe the universe by the end of the century. The progress of science has shown us that we are a very small part of the vast universe, which is governed by rational laws. It is to be hoped that we can also govern our affairs by rational laws, but the same scientific progress threatens to destroy us all . . . . Let us do all we can to promote peace and so insure that we will survive till the next century and beyond 9
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After the presentations to the 1988 laureates, Gelfand is called up to receive his prize, along with Kolm o g o r o v ' s and Krein's. In his response, he offers some juicy personal observations on the Wolf Prizes and goes on to articulate, if very briefly, the same humanistic vision of science that led Ricardo Wolf to endow his prizes "for achievements in the interest of mankind and friendly relations among peoples 9 Here is what he had to say: I am very happy to be here and to receive personally m y prize. It is much better to be at this ceremony a bit later than to have some substitute, because I regard this prize, the Wolf Prize, as a very great honour for me; and I will explain why. During the first award of the prizes, I received the prize, sharing it with a great mathematician, who was at that t i m e n i n my opinion--the first living mathe-
matician in the world: the German mathematician Carl Ludwig Siegel. I could follow for ten years what happened with the Wolf Prize; and, from my own and m y friends' point of view, it became more and more prestigious. There are two reasons for this as I have said yesterday and will repeat now. The first reason is that the level of this prize is the highest one, of course the same as the Nobel Prize. I can't see what is the difference between them; it may be only the age of the prizes. The second is a personal reason: this is the first time that mathematics is recognized also as a science. But the main reason for me is that I believe that all those who work in science serve human beings and do our best for them. And from this point of view, I regard as a very important thing not only the scientific part of this prize, but also who are the persons
w h o have received this prize. Because I understand you cannot divide science and humanity, and for this reason we must do our best to reduce to a smaller level the aggressiveness in people; we have suffered for this a long time. Most of the Wolf Prize recipients w h o m I know are, from the human point of view, very good people indeed, w h o m I like. They are great not only as scientists, but also as individuals. Moreover, I wish to say that I am very happy to c o n v e y this prize to Mark Krein, w h o m I have known for fifty years or so, a very excellent mathematician who really has no possibility to come here for health reasons. I am also sorry that the great mathematician Kolmogorov died before he received the prize; I am sure he would also have enjoyed very much this ceremony. Thank you very much for all your kindness. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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Izrail M. Gelfand receives his Wolf Foundation Prize from Chaim Herzog, President of Israel. Looking on are Yitzhak Navon, Deputy Prime Minister, Minister of Education and Culture, and Chairman of the Council of the Wolf Foundation (right) and Dr. Moshe Gilboa, Deputy Chairman of the Council (left). As he turns to leave the podium, Gelfand is enveloped in a spontaneous embrace by Yitzhak Navon, Israeli Minister of Education and Culture and Chairman of the Council of the Wolf Foundation (and past President of Israel). Then everyone stands for the singing of Hatikva, the Israeli national anthem; and the awards ceremony is over. Afterwards, the Wolf Prize laureates and their families, together with the ambassadors of their countries, representatives and functionaries of the Wolf Foundation, and a few lucky members of the Israeli scientific establishment gather in the Knesset dining room for a sumptuous feast of veal and chicken. There are toasts; and Fumihiko Maki, 1988 Wolf laureate in architecture, responds eloquently on behalf of the recipients. After coffee and petits fours, the guests file out to go their separate ways, some homeward in Jerusalem, others down to Tel Aviv and Herzlia on the coast. In a few days, the winners will return to their homes 48
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abroad, to Belgium, England, France, Germany, Italy, Japan, Sweden, the Soviet Union, and the United States. For all concerned, May 12, 1988, has been, and will remain, a day to remember. A c k n o w l e d g m e n t . I am grateful for the extremely helpful cooperation of the Wolf Foundation and its Director, Mr. Yaron E. Gruder, during the preparation of this article. In particular, background on the Wolf Prizes and, especially, biographical information on the 1988 laureates have been taken, occasionally verbatim, from material provided by the Foundation.
September 1988 Department of Mathematics and Computer Science Bar-Ilan University 52100 Ramat-Gan Israel
Linear Cellular Automata and the Garden-of-Eden K. Sutner
1. The All-Ones Problem Suppose each of the squares of an n x n chessboard is equipped with an indicator light and a button. If the button of a square is pressed, the light of that square will change from off to on and vice versa; the same happens to the lights of all the edge-adjacent squares. Initially all lights are off. Now, consider the following question: is it possible to press a sequence of buttons in such a way that in the end all lights are on? We will refer to this problem as the All-Ones Problem. A m o m e n t ' s reflection will show that pressing a button twice has the same effect as not pressing it at all. Thus a solution to our problem can be described by a subset of all squares (namely a set of squares whose buttons w h e n pressed in an arbitrary order will render all lights on) rather than a sequence. In fact a set X of squares is a solution to the All-Ones Problem if and only if for every square s the number of squares in X adjacent to or equal to s is odd. Consequently, we will call such a set an odd-parity cover. Trial and error in conjunction with a pad of graph paper will readily produce solutions for n ~ 4. A little more experimentation shows that an odd-parity cover - - s h o u l d one exist--is difficult to construct even for n = 5or6. The brute-force approach to the problem, namely exhaustive search over all subsets of {1. . . . n} x {1. . . . n}, presents 2n2 candidates, and the search becomes infeasible for moderate values of n even with the help of a computer. A less brute-force method would be to try to solve the system
(A + I ) . X =
terpreted as a matrix over GF(2)) and 1 is the vector with all components equal to 1. This method, which involves n 2 equations, again becomes unwieldy for small values of n. For a similar approach to a game related to the All-Ones Problem, see [3]. In any case, Figure 1 shows odd-parity covers for n = 4, 5, 8. Several questions come to mind. For which n does a solution to the All-Ones Problem exist? More generally, how m a n y odd-parity covers are there for an n x n board? What happens if the adjacency condition is c h a n g e d - - s a y , to an octal array (where a cell in the center has eight neighbors)? Can one replace an n x n rectangular grid by some other arrangement of sites and still obtain a solution? To answer some of these questions, we first rephrase the problem in terms of cellular automata.
I
of linear equations over the field GF(2) = {0,1}, where A is the adjacency matrix of the n x n grid graph (inTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2 9 1989 Springer-Verlag New York
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2. Cellular Automata on Graphs A cellular automaton is a discrete dynamical system that consists of an a r r a n g e m e n t of basic c o m p o n e n t s called cells together with a transition rule. Every cell can assume a finite n u m b e r of possible states; the collection of possible states is called the alphabet of the automaton. We will here restrict our attention to the case where the set of possible states is {off, on}, which we represent by {0,1}. The transition rule of a cellular automaton is local in the,sense that the state of cell x at time t + 1 d e p e n d s only on the states of the neighboring cells (including cell x itself) at time t. Traditionally the cells are arranged on a finite or infinite, one- or t w o - d i m e n s i o n a l grid. With a view t o w a r d the AllOnes Problem, however, we will allow arbitrary adjacencies b e t w e e n the sites of our cellular automata. More precisely, let G be a locally finite graph, i.e., a graph such that every vertex v in G is adjacent to only finitely m a n y vertices in G. Let V denote the set of vertices of the graph G, and for any vertex v define the closed neighborhood N v of v by
N~ := { u E V I u adjacent to v } U {v}.
le 10
I## I # I## 0e010
##00|
#e$#D# I#ol o#
O#
oe
leo
JOl
Figure 1. Solutions to the All-Ones Problem for square grids of size 4 x 4, 5 x 5, and 8 x 8. Note that a solution for the 6 x 6 grid is contained in the center part of the 8 x 8 solution.
A pattern of G is a function
X : V--* {0,1} from the collection of all vertices V to the alphabet {0,1}. We let Cc denote the collection of all patterns of G and identify a pattern X : V---~ {0,1} with a subset of the vertex set V, namely the collection of all cells v with X(v) = 1. N o w let v be a vertex of G. A local pattern at v on G is a function X~ : N~---* {0,1}.
Clearly, a n y pattern X : V--~ {0,1} determines a local pattern X~ at v, for each vertex v, by setting
Xdu) := X(u)
3mill IIIInC IlmOc 3mml
(1)
for all u in N~. A local rule p~ associates every local pattern at v with a state: the state of cell v in the next g e n e r a t i o n . G i v e n a c o l l e c t i o n of l o c a l r u l e s (pv : v E V) the c o r r e s p o n d i n g global rule p is obtained by applying the local rules s i m u l t a n e o u s l y - - o r in parallel, to use a m o d e r n t e r m - - t o all the local patterns: p : Cc ~ Cc
metric difference of X and Y. The singletons {v}, v E V, provide a standard basis for Cc in the case of a finite graph G. To keep notation simple we will write v instead of {v}, so that in particular if u # v, then v + u stands for the pattern {u,v}. We will write 1 for the "All-Ones" pattern: l(v) = 1 for all v in V. Similarly, 0 denotes the e m p t y set as an element of Co. A n y pattern Z such that p(Z) = X is called a predecessor of X u n d e r the global rule p. A pattern that fails to have a n y predecessors is frequently called a Garden-of-Eden: once "lost" it remains inaccessible forever; see [1], [2]. To tackle our chessboard problem, define a global rule (y b y d e f i n i n g a collection (yv of local rules as follows: rrv(Xv) := card(Xv) m o d 2.
In other words, (y(X)(v) = 1 iff card(Xv) is odd. A g r a p h t o g e t h e r w i t h rule or will be called a (y-automaton. The predecessors of 1 in a (y-automaton are exactly the solutions of the All-Ones Problem, which can n o w be restated as follows: Given an n x n grid graph G, is I a Garden-of-Eden under rule (y in G?
~(X)(v) := .v(x~) where X~ is defined by (1). Algebraically, the collection of all patterns forms a vector space over {0,1} construed as the two-element field GF(2). This vector space will be called the pattern space. The vector s u m of X and Y in C c is the sym~0
THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989
Before we answer the above question, let us digress briefly. The rule (y is a typical member of the class of "linear rules," which means that (y is linear as a m a p from the pattern space Ca to itself. In fact, (Y(X) = (A +/) 9 X, where pattern X is construed as a column vector a n d the adjacency matrix A of G is construed as
."i,, .,.r ,.'., ...- ...% ...,',",,. ,.-"-,,. ..,......I .,..," I-..,I ;-.,
d::". .,~ ,,,
Table 1. Irreversible n x n grid automata, n ~ 100. Here d, denotes the dimension of Kp.,.,(r.
,:x.
.,,,. -'..
..,'~,.
.~,.".'....:~....."..,._.'...,..':...r
",..
.,:."%. .~ ...... ~.
..<: ~: . . . . .
:..
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Figure 2A. The first 70 steps in the evolution of pattern (0) in P~ with rule or-. B. The first 70 steps in the evolution of pattern (0) in P~ with rule ~. a matrix over the field {0,1}. Another rule very closely related to r is obtained by excluding the vertex v from its o w n neighborhood: or-v(Xv) := card(X v - v) m o d 2. Thus o'-(X) = A 9 X. A n y rule p that is composed of linear local rules Pv is called a linear rule. Despite the simplicity of linear rules like or a n d or-, the cellular automata obtained from t h e m s h o w quite complicated behavior. A classic example is the twosided infinite path graph P| (the vertices of P| are the integers, and v is adjacent to u if a n d only if lu - v I = 1). Figure 2 shows the evolution of the seed pattern {0} on P| using various linear rules. The two-dimensional patterns generated in this fashion have non-integer fractal dimension a n d display such complicated geometric properties as self-similarity. The fractal dimension of the pattern asymptotically generated by rule or is log2(1 + X/5), whereas rule or- generates a pattern of dimension log23. A wealth of information about these and other cellular automata rules as well as m a n y fascinating pictures can be f o u n d in [5] through [8] (rules or a n d orare called rule 150 a n d 90, respectively, in [5]).
3. F i n i t e ( r - A u t o m a t a
O n e of the basic questions of the theory of cellular automata concerns the global reversibility of the transition rule: can pattern X be reconstructed from p(X)?
4
62 20 16
Locally irreversible s y s t e m s are i n t e r e s t i n g from a t h e r m o d y n a m i c point of view: unlike locally reversible systems, t h e y m a y evolve from u n o r d e r e d to ordered states. Linear rules are locally irreversible in the sense that different patterns can lead to the same state in one particular cell in the next generation. Globally linear rules m a y well be reversible, however. Let us a s s u m e from n o w on that G is a finite graph. For linear rules the situation is simple: rule p is injective if a n d only if it is surjective. Let Ka,p C CG be the kernel of rule p on G and set d(G,p) := dirn(KG, p) = log2(card(Kc, p)). Then the a u t o m a t o n G with rule p is reversible if a n d only if it has no Garden-of-Eden if a n d o n l y if d(G,p) = 0. In particular, the All-Ones Problem can be solved in a n y finite reversible automaton. For rule or the 3 x 3 grid is an example for a reversible automaton; indeed, all (2i - 1) x (2i - 1) square grids are reversible. A polyhedron gives rise to a graph that has the corners of the polyhedron as vertices a n d an edge joining two vertices if and only if an edge of the polyhedron joins the two corresponding corners. Of the five graphs obtained in this fashion from the Platonic solids, only the octahedron is reversible; whereas the tetrahedron, the cube, the dodecahedron, a n d the icosahedron are irreversible. The d-dimensional hypercube 2d is the graph with vertex set {0,1}a a n d there is an edge between v and u if and only if v a n d u have H a m m i n g distance one (i.e., they disagree in exactly one component). The hypercube 2a is THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 ~ 1
t o m a t o n Ps- Notice that Ps is irreversible but nonetheless pattern 1 has a predecessor. For arbitrary n, it is easy to determine all odd-parity covers for P,:
2+4+5~_ 1 ~""$1+2~
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~
1+3 ~)r 1+3+4 2+3+4+5/
~+2+5
1+5_ ~1+2+4+5
~3+4+5 --
.9
2§
Figure 3. The transition diagram GPs" reversible if and only if d is even; for a proof of these results see [4]. Let us agree on some notation for graphs: Pm (Cm) will d e n o t e t h e p a t h (cycle) g r a p h o n m p o i n t s {1. . . . m} a n d Pm,, the rectangular m x n grid graph. Table 1 shows the dimension dn of Kp, n,~ for all irreversible n x n grid automata, n ~< 1001 Note that dim e n s i o n appears to be even for any square grid (it is n o t even in general for rectangular grids). Observe that d2n+l = 2 " d, + 8,, where 8, ~{0,2}. Indeed, the table suggests 82,+1 = 8,. We are not aware of a proof for a n y of these conjectures. N o w s u p p o s e p a t t e r n X has p r e d e c e s s o r Y. By linear algebra the collection of all predecessors of X is the affine s u b s p a c e O-I(X) = Y + KG,p; t h u s the n u m b e r of predecessors of X is either 0 or 2d(G,"). The table provides an explanation for the extra difficulty of constructing an odd-parity cover for the 5 x 5 or 6 x 6 grid compared to the 4 x 4 grid: there are 16 solutions for 4 x 4 grid, 4 for the 5 x 5 grid, but only one for the 6 x 6 grid. O n e can s h o w that in the 4 x 4 board one can pick an arbitrary pattern in the first row and always expand it to an odd-parity cover of the whole
grid. The predecessor relation is best expressed graphically by m e a n s of the transition diagram Go, p of G. Formally ~C,p is a directed graph that has as vertices the patterns of G a n d an edge from X to Y if a n d only if ,(X) = Y. For a linear rule p the in-degree of every point in GG,pis either 0 or 2d(G,p). The out-degree is 1, of course, so the connected components of ~G,p are all unicyclic. I n d e e d , t h e y consist of one cycle a n d a n u m b e r of 2d(c,P)-ary trees a n c h o r e d on that cycle. Clearly, rule p is reversible on G if a n d only if the conn e c t e d c o m p o n e n t s of ~G,p are cycles. The orbit {~ I i I> 0} of a n y pattern X forms a one-generated monoid. Figure 3 shows the transition diagram for the or-au52
THE MATHEMATICAL
INTELLIGENCER VOL. 11, NO. 2, 1989
n n~0(mod3) n~l(mod3) n=2(mod3)
2 + 5 1 + 4 1 + 4 and2
+ + + +
odd-parity covers . . + (n - 4) + (n - 1) 7 + . . + (n - 3) + n 7 + . . + ( n - 4) + (n - 1) 5 + 8 + . . + ( n - 3) + n.
The corresponding pictures m a y be more convincing t h a n algebra; see Figure 4. So Pn is reversible if and only if n ~ 2 (mod 3). For n = 2 (mod 3) there exists exactly one predecessor of 0 (other t h a n 0 itself), namely I + 2 + 4 + 5 + . . + (n - 1) + n; thus d(Pn,or) = 1. Similarly, for the cycle Cn one has the following situation: n n~0(mod3) n=0(mod3)
odd-parity covers 1 1 , 1 + 4 + . . + (n - 2),2 + 5 + . . ( n - 1),3 + 6 + . . + n.
Thus 1 is a fixed point u n d e r or in C, and d(C,,,or) = 2 for n = 0 (mod 3). F i n d i n g o d d - p a r i t y covers for each of the ladder graphs Pn,2, n /> 2 is still straightforward. For even larger grids the search becomes hopeless, as patient readers m a y readily convince themselves. It is n o t clear that an odd-parity cover should exist for arbitrary grids. As we have seen, reversibility is a sufficient but by no m e a n s necessary condition for 1 to have a predecessor. Indeed, we will s h o w that an arbitrary finite g r a p h G possesses an o d d - p a r i t y cover, or, equivalently, the All-Ones Problem has a solution. We do not k n o w of a n y purely geometric (read: graph theoretic) a r g u m e n t to prove this. We will present an algebraic proof.* To this end let v 1. . . . v n be the standard base of the pattern space Ca a n d let C*c be the dual space equipped with the dual base Vl*. . . . vn*. Every vector X =- ~ i V i in Ca gives rise to a linear functional X* := E~ivi* in C'G; define
(z,x):= x~(z). C G is finite dimensional and therefore isomorphic to C*c u n d e r the m a p X ~ X*; we will identify b o t h spaces to keep notation manageable. Z is said to be perpendicular to X, in symbols Z 3_ X, if (Z,X) = 0. Hence Z a n d X are perpendicular if and only if their intersection has even cardinality. Z is perpendicular to a subspace W of C a if and only if Z 3_ X for all X E W. Define the orthogonal c o m p l e m e n t W" of the subspace W by W• = {Z E CG [ Z 3_ W}. Using the notion of orthogonal complement one can n o w characterize * R. TindeUpointed out that there is a purely graph theoretic proof for trees (connected acyclicgraphs). See pages 31-32.
I lel I I# I lel I .,-I Iolllellielll lelllelllolll
. , ,
I lolllell
lel I I01 I I llell]oi I lolllolllell
..-
I IIolllel
Figure 4. Odd-parity covers for P,.
Figure 5. A basis for K~,.,. the patterns that lie in the range of or. 3.1 THEOREM: Let G be an arbitrary finite graph and let
p be one of the rules or or cr-. Then pattern X has a predecessor in G under rule p if and only if X is perpendicular to the kernel of p. Proof: The n e i g h b o r h o o d (and adjacency) matrix of a graph is symmetric. Thus p is selfadjoint and
= <x,p(Y)>.
for a n y two patterns, X,Y. Also note that ( . , .) is nondegenerate in the sense that X = Oiff(Z,X) = O f o r a l l Z i n C c. N o w let W be the range of p. We get Y E W• r r r
for all X in C c (p(X),Y) = 0 for all X in Cc (X,p(Y)) = 0
p(Y) = 0,
i.e., W • = Ka,,. By a well-known theorem of algebra W "• = W. Hence W = Ka, p" and the proof is complete. 9 3.2 THEOREM: The All-Ones Problem has a solution in any finite graph.
Proof: The key observation is that any pattern X such that or(X) = 0 m u s t have even cardinality. To see this, first note that every vertex x in X has o d d degree in the s u b g r a p h G w i t h vertex set X. Second, recall the h a n d s h a k i n g t h e o r e m : the n u m b e r of o d d - d e g r e e points in a n y finite graph is even. Now, (X,1) is the cardinality of X m o d u l o 2, so (X,1) = 0. Hence 1 is p e r p e n d i c u l a r to Kc,,,, a n d has a predecessor u n d e r rule or by Theorem 3.1. 9 Thus I fails to be a Garden-of-Eden u n d e r rule or in a n y finite graph. We note in passing that Theorem 3.2 can be generalized to locally finite graphs. For finite graphs one can c o m p u t e a basis for the affine subspace
of all odd-parity covers in polynomial time, i.e., in a n u m b e r of steps polynomial in the n u m b e r of points of the graph. Surprisingly, the problem of determining an odd-parity cover of minimal cardinality turns out to be computationally difficult: the corresponding optimization problem is NP-hard; see [4]. Thus it is unlikely that a n y efficient (read: deterministic polynomial time) algorithm exists to construct an odd-parity cover of minimal size for a given graph. For certain simple graphs like P4,4, where the kernel of or~ is k n o w n explicitly, T h e o r e m 3.1 provides an easy test of w h e t h e r a given pattern has a predecessor. Figure 5 s h o w s a basis for the kernel of o- on P4,4. It is easy to check that no singleton can have a predecessor. Or consider the path ion, where n - 2 (mod 3) and n > 2. Then in Pn exactly the patterns of the form X = X0 U X 1 , w h e r e X 0 i s a s u b s e t o f l + 2 + 4 + 5 + . . + (n - 1) + n of even cardinality a n d X1 is a subset of 3 + 6 + . . + (n - 2), have a predecessor u n d e r rule or. Hence exactly half the patterns have a predecessor, a n d the other half are Gardens-of-Eden. References
1. E. F. Moore, Machine models of self-reproduction, in: A. W. Burks, Essays on Cellular Automata, University of l]linois Press, 1970. 2. J. Myhill, The converse of Moore's Garden-of-Eden Theorem, in: A. W. Burks, Essays on Cellular Automata, University of Illinois Press, 1970. 3. D. H. Pelletier, Merlin's magic square, Amer. Math. Monthly 94 (1987), 145-130. 4. K. Sutner, Linear automata on graphs, to appear. 5. S. Wolfram, Statistical mechanics and cellular automata, Rev. Modern Physics 55 (1983), 601-644. 6. S. Wolfram, Computation theory of cellular automata, Comm. Math. Physics 96 (1984), 15-57. 7. S. Wolfram, Geometry of binomial coefficients, Amer. Math. Monthly 91 (1984), 566-571. 8. S. Wolfram, Computer software in science and mathematics, Scientific American 251 (1984), 188-203.
Department of Computer Science Stevens Institute of Technology Hoboken, NJ 07030 USA THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989 5 3
How Good Is Lebesgue Measure? Krzysztof Ciesielski*
The problem of determining the distance between two points, the area of a region, and the volume of a solid are some of the oldest and most important problems in mathematics. Their roots are in the ancient world. For example, the formula for the area of a circle was already known to the Babylonians of 2000 to 1600 B.C., although they used as "rr either 3 or 31/K The Egyptians of 1650 B.C. used "rr = (4/3)4 = 3.1604 . . . . The first crisis in m a t h e m a t i c s arose from a m e a s u r e m e n t problem. The discovery of incommensurable magnitudes by the Pythagoreans (before 340 B.C.) created such a great "logical scandal" that efforts were made for a while to keep the matter secret (see [Ev], pages 25, 31, 85, 56).
For a given subset of R", what do we mean by the statement that some number is its area (for n = 2), volume (for n = 3) or, more generally, its n-dimensional measure? Such a number must describe the size of the set. So the function that associates with some subsets of R" their measure must have some "good" properties. How can we construct such a function and what reasonable properties should it have? This question, in connection with the theory of integration, has been considered since the beginning of the nineteenth century. It was examined by such wellknown mathematicians as Augustin Cauchy, Lejeune Dirichlet, Bernhard Riemann, Camille Jordan, Emile Borel, H e n r i Lebesgue, a n d G i u s e p p e Vitali (see [Haw]). Lebesgue's solution of this problem, dating from the turn of the century, is now considered to be the best answer to the question, which is not completely settled even today.
Lebesgue Measure In his solution Lebesgue constructed a family Y of subsets of Euclidean space R" (where n = 1, 2 .... ) and a function m: ~ --* [0,oo] that satisfies the following properties: (a) ~ is a (r-algebra, i.e., ~ is closed under countable unions (if A k E ~ for every natural number k then U {Ak: k E N} E ~) and u n d e r complementation (A E ~ implies R " \ A E ~e); (b) [0,1]" ~ 5~ and m([0,1]") = 1;
* I wish to thank m y colleagues at Bowling Green State University for their hospitality in 1986/87, while writing this article. I am most grateful to V. Frederick Rickey for his kind help. THE MATHEMATICALINTELLIGENCERVOL. tl, NO. 2 9 1989Springer-VerlagNew York
(c) m is countably additive, i.e., for every family {Ak: k E IV} of pairwise disjoint sets from S~,
m
Ak
=
m(Ak);
(d) m is isometrically invariant, i.e., for every isometry i of R" (1) A E ~f if and only if i(A) E ~; (2) re(A) = m(i(A)) for every A in 5~. Recall that a function i: R" --~ R" is an isometry provided it preserves distance, in other words, i is composed of translations and rotations. The properties (a)-(d) seem to capture perfectly our intuitive notion of area and volume. Lebesgue measure is not sensitive to moving sets around without distorting them. To find the measure of a set it suffices to split it up into a finite or countable family of disjoint sets and then to add up their measures (provided, of course, that each piece is measurable). However, we should ask at least one more question: "With h o w many sets does Lebesgue measure deal?" or "'How big is the family ~ ? " In particular, we should decide whether m is universal, i.e., whether ~ is equal to the family ~(R") of all subsets of R". Unfortunately the answer to this question is negative. In 1905 Vitali constructed a subset V of R" such that V is not in S~ (see [Vi], [Haw], p. 123, or [Ru], Thm. 2.22, p. 53). His proof is easy, so we shall repeat it here for n = 1. For x in [0,1] let E x = {y E [0,1]: y - x E Q} and put % = {E=: x E [0,1]}. Clearly % is a nonempty family of non-empty pairwise disjoint sets. Therefore, by the Axiom of Choice there exists a set V C [0,1] such that V n E has exactly one element for every E E %. We prove that V is not in ~e. First notice that if V + q = {x + q: x E V} for q E Q, then V + q and V + p are disjoint for distinct p, q E Q. Otherwisex + q = y + p f o r s o m e x , y E V s o x and y both belong to the same E E %. Then V n E contains more than one element, contradicting the definition of V. N o w if V is in ~, then so is V + q for every q E Q, and re(V) = m(V + q). But U {v + q: q E Q N [0,1]} C [0,2] and so
X
m(V + q) = m ( U { v
+ q: q E Q n [0,1]})
q~Qn[0,1] m([0,2]) = 2. This implies re(V) = m(V + q) = 0 for every q E Q. On the other hand, [0;1] C R = U {V + q : q E Q} which implies m([0,1]) ~ m ( U { v
+ q: q EQ}) =~P m(V + q) = 0, q~Q
which contradicts property (b) of Lebesgue measure. Thus V is not in ~. (For the general case, modify the above proof using V x [0,1]"-1.) Notice that if ~: ~ --~ [0,~] satisfies conditions (a)-(d) (we will call such a function an invariant measure) then the above proof shows that V ~ ~ . Thus, V is not measureable for any invariant measure. In particular, there is no invariant measure ~: ~ --* [0,oo] which is universal, i.e., for which ~ equals ~(R"). Another remark we wish to make is that in Vitali's proof we used the Axiom of Choice (hereafter abbreviated AC). At the beginning of the twentieth century the Axiom of Choice was not commonly accepted (see [Mo]) and Lebesgue had his reservations about Vitali's construction (see [Le] or [Haw], p. 123). Today we accept the Axiom of Choice, so we cannot s u p p o r t Lebesgue's complaints. But was Lebesgue completely wrong? If we do not accept the Axiom of Choice then Vitali's proof will not work. In 1964 Robert Solovay (see [So] or [Wa], Ch. 13) showed something much stronger: we cannot prove that 5~ # ~(R") without AC. More precisely, he p r o v e d that there exists a model ("a mathematical world") in which the usual ZermeloFrankel set theory (ZF) is true and where all subsets of R" are Lebesgue measurable, i.e., ~f = ~(R"). Moreover, although the full power of the Axiom of Choice must fail in this "world," the so-called Axiom of Dependent Choice (DC) is true in the model. In fact, DC tells that we can use inductive definitions and therefore all classical theorems of analysis remain true in this "world." Solovay's theorem has only one disadvantage. Besides the usual axioms of set theory (ZF), Solovay's proof uses an additional axiom: there exists a weakly inaccessible cardinal (in abbreviation WIC, see Lie], p. 28). The theory ZF + WIC is essentially stronger than ZF. For over 20 years m a t h e m a t i c i a n s w o n d e r e d whether it is possible to eliminate the hypothesis regarding WIC from Solovay's theorem, but in 1980 Saharon Shelah s h o w e d that it cannot be done, proving that the consistency of the theory ZF + DC + "~e = ~(R") '" implies the consistency of the theory ZF + WIC (see IRa] or [Wa], p. 209).
Extensions of Lebesgue Measure Let us go back to doing mathematics with the Axiom of Choice. We know that Lebesgue measure is not universal, i.e., Y # ~(R"). So let us examine the problem of h o w we can improve Lebesgue measure. The properties ( a ) - ( d ) of L e b e s g u e m e a s u r e s e e m to be most desirable. Therefore, w e will try to improve m: ~ ---* [0,~] by examining its extensions, the functions p,: ~ ~ [0,oo] such that ~ C ~J~ C ~(R n) and ~(X) = m(X) for all X E ~.
THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 5 5
By Vitali's theorem, if it is an invariant measure then 2J~ # ~(R'). In this context the following question naturally appears: " H o w far can we extend Lebesgue measure and what properties can such an extension preserve?" This question has been investigated carefully by members of the Polish Mathematical School and all but one of the results presented here were proved by this group. Let us first concentrate on the extensions that are invariant measures. The first result in this direction is due to Edward Szpilrajn (who later changed his name to Edward Marczewski). In 1936 he proved that Lebesgue measure is not a maximal invariant measure, i.e., that there exists an invariant measure that is a proper extension of Lebesgue measure. In connection with this result, Waclaw Sierpifiski (in 1936) posed the following question "'Does there exist any maximal invariant measure?" (see [Sz]). Let us notice that by p r o p e r t y (b) a n y such measure should extend Lebesgue measure.
it([O,1]n) ~ it(Rn) = it
it(Nj) = O.
~ j=l
Therefore the invariant measure v as in (ii) is a proper extension of it and so Wis not maximal. At this stage of our discussion we are able to give a partial answer to the question of our title. If we restrict our search to invariant measures, then Lebesgue measure is not the richest. However, any other invariant measure has the same defect. This means that if we would like to compare invariant measures only by the size of their domain then the best solution to the measure problem simply does not exist. On the other h a n d , if we deal only with the subsets of R" constructed without the Axiom of Choice but using the induction allowed by DC (this is the case in all constructions in classical analysis) then all sets we are interested in are Lebesgue measurable. Thus, in view of the above a r g u m e n t s a n d the n a t u r a l n e s s of Lebesgue's construction, the Lebesgue measure is the u n i q u e reasonable candidate to be a canonical invariant measure. H o w f a r can w e e x t e n d Lebesgue m e a s u r e a n d Another idea to improve Lebesgue measure was to w h a t p r o p e r t i e s can such an e x t e n s i o n pre- extend it to a universal measure it: !~(Rn) ~ [0,oo] by serve? w e a k e n i n g some part of properties (a)-(d), t h u s avoiding Vitali's argument. The most popular approach is to drop isometric invariantness and to conThis question was examined by several mathemati- sider as measures the functions it: ~ff~~ [0,oo] that satcians under different assumptions. The answer was isfy conditions (a), (b), and (c). This definition, of meaalways the same: there is no maximal invariant mea- sure is common today mostly because we can easily sure. The firstresult was noticed by Andrzej Hulanicki generalize it from R" to an arbitrary set. To answer the in 1962 (see [Hu]) under the additional set-theoretical question of when there exists a universal measure exassumption that the continuum 2 ~ is not R V M (this tending L e b e s g u e measure, let us consider the folassumption will be discussed laterin this article).This lowing sentence: "There exists a measure it: !~(R) result was also obtain by S. S. Pkhakadze (see [Pk]) [0,oo] satisfying (a)-(c) such that m({x}) = 0 for all using similar methods. In 1977 A. B. Harazi~vili (from x E R . " If this sentence holds, we say that 2'* is RVM, Georgia, USSR) got the same answer in the one-di- i.e., on the continuum there is a real-valued measure. mensional case without any set-theoretical assumpIt is well known that if 2`0 is RVM, then there exists a tion (see [HarD. Finally, in 1982 Krzysztof Ciesielski universal extension of Lebesgue measure (the conand Andrzej Pelc generalized Harazi~vili's result to all verse implication is obvious). We shall s[<etch the n-dimensional Euclidean spaces (see [CP] or [Ci]). proof of this theorem for the one-dimensional case by The idea of Ciesielski and Pelc's proof is due to Hafinding the measure it: ~([0,1]) --~ [0,oo] extending Lerazi~vili. Using the algebraic properties of the group of besgue measure. So let v: @(R) --* [0,oo] be a measure isometries of R n they constructed a family {Nj: j = such that v({x}) = 0 for x E R. For every X C R there 0,1,2,3 .... } of subsets of R ~ with following properexists X0 C X such that v(X0) = 1/2 v(X) (for the proof ties: see [Je], Ex. 27.3 and Thm. 66, p. 297). So by an easy induction we can define the family of sets {Xs: 's is a (i) R" = (.J {Nj:j = 0,1,2 .... }; finite sequence of 0s and ls} such that: v(Xo) = 1 (ii) if it: ~ ---> [0,oo] is an invariant measure then (f~ is considered to be a sequence of length 0), XsA0, Nj ~ ~ implies it(Ni) = O; (iii) for every invariant measure it: ~ --> [0,oo1 and X~I C Xv and v(X~/u) = 1/2 v(X~) where s^i means an for every natural number j there exists an invariant extension of sequence s by digit i. For A C [0,1] we defined it(A) by measure v: ~ ~ [0,oo] extending it such that Nj E ~. This result easily implies the nonexistence of a maximal invariant measure. If it: ~ --) [0,oo] is an invariant measure then, by (ii), there exists a natural number j such that Nj e ~ff~,because otherwise we would have 56
THE M A T H E M A T I C A L INTELLIGENCER VOL. 11, NO. 2, 1989
X,4 where a(n) is the sequence of the first n digits of the
binary representation of a. It is not difficult to see that this )~ really extends Lebesgue measure on [0,1]. The systematic investigation of the statement "20) is RVM'" was started in 1930 by Stanislaw Ulam, when he proved that it implies the existence of a weakly inaccessible cardinal (i.e., axiom WIC), so it cannot be proved under the usual axioms of set theory (see [U1] or Ue], Thin. 66, p. 297). Although this result was fundamental for one of the most interesting parts of modern set theory, namely the theory of large cardinals, for our discussion it is only a disadvantage. It means that in order to have a universal countably additive measure extending Lebesgue measure we not only lose isometric invariantness, we must also assume a very strong additional axiom that is usually not accepted by mathematicians.
Finitely Additive Extensions of Lebesgue Measure Let us consider another approach to the universal extension of Lebesgue measure by asking: "Is it really necessary to assume that measure is countably additive? Isn't it enough to deal with finitely additive measures?" More precisely, let us consider the properties: (a') Sp is an algebra, i.e., S~ is d o s e d under unions (if A , B E ~ then A U B E S~) and under complementation (A E S~ implies R " \ A E S~); (c') m is finitely additive, i.e., for every disjoint pair A and B from S~, m ( A U B) = m(A) + m(B).
A function m: ~/~ --) [0,o0] satisfying the properties (a'), (b'), (c') and (d) will be called a finitely additive isometrically invariant measure. Does there exist a universal finitely additive isometrically invariant measure extending Lebesgue measure? Stefan Banach (see [Ba] or [Wa]) proved in 1923 that such a measure exists on the plane and on the line (i.e., for n = 2 and n = 1). This beautiful result seems to be the only reasonable improvement of Lebesgue measure. But what is going on with n-dimensional Euclidean spaces for n _-__3? The answer, due to Stefan Banach and Alfred Tarski (1924; see [BT] or [Wa]), is surprising: there is no universal finitely additive isometrically invariant extension of Lebesgue measure for n I 3. But undoubtedly more surprising is the result that leads to this conclusion: the Banach-Tarski Paradox. To state this paradox let us introduce the following terminology. We say that a set A c Rn is congruent to B C R" if we can cut A into finitely many pieces and rearrange them (i.e., transform each of these pieces using some isometry of R n) to form the set B. The Banach-Tarski Paradox is the theorem that a ball B with volume I is congruent to the union of two similar
disjoint balls B1 and B2 each having the same volume 1I There is even more: we can do this by cutting the ball B into only five pieces. This result is so paradoxical that our discomfort with it can probably be compared only with the Pythagorean "logical scandal" connected with the discovery of incommensurable line segments. This is also one of the strongest arguments against the use of the Axiom of Choice (see [Mo], p. 188). But what about our measures on R3? It is easy to see for every universal finitely additive isometrically invariant measure ~ that if X C R" is congruent to Y C R", then ~(X) = )L(Y). In particular ~(B) = la(Bt U B2). But for 3-dimensional Lebesgue measure m(B) = 1 # 2 = re(B1 U B2), s o ~ doesn't extend Lebesgue measure m . C o n n e c t e d w i t h this p a r a d o x is a nice o p e n problem, the Tarski Circle-Squaring Problem from 1925, about which Paul ErdOs wrote (see [Ma], p. 39): It is a very beautiful problem and rather well known. If it were my problem I would offer $1000 for it--a very, very nice question, possibly very difficult. Tarski noticed that although in 3-dimensional Euclidean space all bounded sets with nonempty interior are congruent, this is not the case on the plane where sets with different Lebesgue measure are not congruent. But what about subsets of the plane with the same measure? In particular he formulated his CircleSquaring Problem: "Is a square with unit measure congruent to a circle with the same measure?" This problem seems to be so difficult that Stan Wagon wrote about it (see [Wa], p. 101): "The situation seems not so different from that of the Greek geometers w h o considered the classical straight-edge-and-compass form of the circle-squaring problem."
Conclusion So, what is the answer to the question " H o w good is Lebesgue measure?" In the class of invariant measures, Lebesgue measure seems to be the best candidate to be a canonical measure. In the class of countably additive not necessarily invariant measures, to find a universal measure we have to use a strong additional set-theoretical assumption and this seems to be too high a price. Thus the best improvement of Lebesgue measure seems to be the Banach construction of a finitely additive isometrically invariant extension of Lebesgue measure on the plane and line. However, such a measure does not exist on R" for n 1 3, and to keep the theory of measures uniform for all dimensions we cannot accept the Banach measure on the plane as the best solution to the measure problem. From this discussion it seems clear that there is no reason to depose Lebesgue measure from the place it has in modern mathematics. Lebesgue measure also THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 ~ 7
has a nice topological p r o p e r t y called regularity: for e v e r y E ~ ~e a n d e v e r y e > 0, there exists an o p e n set V D E and closed set F C E such that m ( V \ F ) < ~. It is n o t difficult to p r o v e that L e b e s g u e m e a s u r e is the richest countably additive measure having this property (see [Ru], T h m . 2.20, p. 50).
References S. Banach, Sur le probl6me de la mesure, Fund. Math. 4 (1923), 7-33. [BT] S. Banach and A. Tarski, Sur la d6composition des ensembles de points en parties respectivement congruents, Fund. Math. 6 (1924), 244-277. [Ci] K. Ciesielski, Algebraically invariant extensions of tr-finite measures on Euclidean spaces, to appear. [CP] K. Ciesielski and A. Pelc, Extensions of invariant measures on Euclidean spaces, Fund. Math. 125 (1985), 1-10. [Ev] H. Eves, An Introduction to the History of Mathematics, Saunders College Publishing, 1983. [Har] A.B. Harazi~vili, On Sierpifiski's problem concerning strict extendability of an invariant measure, Soviet. Math. Dokl. 18 no. 1 (1977), 71-74. [Haw] T. Hawkins, Lebesgue Theory of Integration, The Univ. of Wisconsin Press, 1970. [Hu] A. Hulanicki, Invariant extensions of the Lebesgue measure, Fund. Math. 51 (1962), 111-115. Ue] T. Jech, Set Theory, Academic Press, 1978. [Le] H. Lebesgue, Contribution a l'6tude des correspondances de M. Zermelo, Bull. Soc. Math. de France 35 (1907), 202-221. [Ma] R . D . Mauldin, The Scottish Book, Birkh/iuser, Boston, 1981. [Mo] G. Moore, Zermelo's Axiom of Choice, SpringerVeflag, New York, 1982. [Pk] S.S. Pkhakadze, K teorii lebegovskoi mery, Trudy Tbilisskogo Matematiceskogo Instituta 25, Tbilisi 1958 (in Russian). [Ra] J. Raisonnier, A mathematical proof of S. Shelah's theorem on the measure problem and related results, Isr. J. Math. 48 (1984), 48-56. [Ru] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Company, 1987. [So] R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56. [Sz] E. Szpilrajn (alias E. Marczewski), Sur l'extension de la mesure lebesguienne, Fund. Math. 25 (1935), 551-558. [Ul] S. Ulam, Zur Mass-theorie in der allgemeinen Mengenlehre, Fund. Math 16 (1930), 140-150. [Vi] G. Vitali, Sul problema della mesure dei gruppi di punti di una retta, Bologna, 1905. [Wa] S. Wagon, The Banach-Tarski Paradox, Cambridge Univ. Press, 1985. [Ba]
Dept. of Mathematics, Mechanics and Computer Science Warsaw University Warsaw, Poland and Dept. of Mathematics The University of Louisville Louisville, KY 40292 USA 58 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989
B Kpyry ~Ipysefi 3a npaaJInnqnbIM CTOJIOM MbI BbIXOJI KHHr~Inamefi aaMeqadivi. Jlepxca B pyKax xpycT~mnfi SOBbtfi TOM, Bce OT ;IytuH Hac C H3efi no3JIpaBa~an. ~ a , nacTynna Ham JIoaroxoannbIfi ,~ac, H nax:e CKeIITHKMpaqHbIfi ynbt6nyac~. O H n o J I o I I I e J I , q T O S b I no3JIpaBnTb Mac, A s OT pa~OCTH.., npocuyaca. Ha cna M. F. Kpefina B H o a o r o a m o I o HOqb 1963 r. H o CBItJIeTeJIbCTByI/I. C. HOXBIa~oaa.
A r o u n d the festive table all our friends H a v e c o m e to mark our n e w book's publication. The n e w and shiny v o l u m e in their hands, T h e y offer Is and m e congratulation. O u r long-awaited h o u r has come at last. The sourest skeptic sees h e was mistaken A n d smiling comes to cheer us like the rest; A n d I a m so d e l i g h t e d . . . I awaken. From M. G. Krein's dream, N e w Year's 1963, as r e p o r t e d b y I. S. Iokhvidov. English version b y C h a n d l e r Davis. (The b o o k d r e a m t of, Introduction to the theory of linear nonselfadjoint operators by I. C. G o h b e r g and M. G. Krein, was published only in 1965.)
Strings Yu. I. M a n i n
This paper is dedicated to the memory of Vadik Knizhnik.
Introduction Recently I got an invitation to a conference on strings and superstrings to be held in E1 Escorial, Spain. Enclosed was a poster (see p. 61) representing as a Riemann surface of genus 17 with 4 cusps the famous San Lorenzo Monastery built by Felipe II. The double symbolism of a monstrous castle-surface hanging by a cord had been skillfully rendered by an artist. The rope and prongs were reminiscent of the Inquisition and the sadistic monarch, while at the same time they were a commonplace visualization of the new toys of theoretical physicists--classical and quantum strings. Well, not quite new. The foundations of string theory were laid back in the 1960s when Veneziano discovered his remarkable dual amplitude in strong interaction physics. It was soon understood that the Veneziano model describes quantum scattering of relativistic one-dimensional objects, i.e., strings, instead of common point-particles. This model was in qualitative agreement with experimental data on the partonlike b e h a v i o u r of strong interactions. O n e could imagine a meson as a tube of colour flux with quarks attached to its ends. The string length scale then should be roughly 10 -13 cm. Because a string has intemal excitation modes, ,this could explain the proliferation of strongly interacting particles. However, the hadronic interpretation of dual string theory was plagued by m a n y quantitative disagreements. To quote just one, it so happened that the quantum theory of relativistic strings seemed to be
consistent only in 26-dimensional space-time, while hadrons apparently lived in our 4-dimensional world! Meanwhile the quantum chromodynamic, i.e., the theory of quantized Yang-Mills fields, gained mom e n t u m as the correct theory of strong interactions, and strings became outdated. The modern Renaissance of the string theory is based upon its reinterpretation s u g g e s t e d in 1974 by J. Scherk a n d John Schwarz.
THE MATHEMATICAL INTELLIGENCER VOL. I1, NO. 2 9 1989 Springer-Veflag New York 5 9
String theory is n o w considered as a candidate theory of elementary particle physics at the Planck scale (-10 -33 cm) rather than at the hadronic scale. This romantic leap of twenty orders of magnitude makes the situation in m o d e m theoretical physics extremely bizarre and poses new problems of relating theory to the phenomenology of low energy (former high energy) physics. Psychologically this leap was prepared for by a decade of Grand Unification models based upon Yang-Mills fields with a large gauge group and a bold extrapolation of the high-energy behaviour of coupling constants of strong and electro-weak interactions. A n o t h e r essential i n g r e d i e n t of m o d e r n string theory, also created in the 1970s, is supersymmetry, i.e., a mathematical scheme that allows us to incorporate bosons and fermions in a multiplet mixed by a symmetry supergroup. At the classical level, this involves an exciting extension of differential and alge-
The q u a n t u m theory of relativistic strings seemed to be consistent only in 26-dimensional space-time, while hadrons apparently lived in our 4-dimensional world! braic geometry, Lie group theory, and calculus by introducing anticommuting coordinates that represent the half-integer spins of fermions. A string endowed w i t h s u c h f e r m i o n i c c o o r d i n a t e s is called a superstring. In a sense, supersymmetry implies general covariance a n d t h e r e b y requires unification with gravity. The quantum theory of superstrings is consistent in 10-dimensional space-time. Because this is still far from our four dimensions, it was suggested, as a revival of an old Kaluza-Klein idea, that the extra six dimensions should be compactified at Planck scale. More specifically, our space-time presumably has a structure of a product M 4 x /<6, where M 4 is the Minkowski space of special relativity w h i l e / ~ is a compact Riemannian space of diameter - 1 0 - ~ cm, i.e., a point for all practical purposes. For theoretical purposes, however, it is not a point by any means. In a fantastic paper [16], Ed Wit-ten and collaborators proposed that (in a vacuum state)/~ is a Calabi-Yau complex manifold with a complicated topology responsible for such exotic properties of our universe as the existence of three (or four) generations of elementary constituents of matter, i.e., leptons and quarks. In the beginning of the 1980s, Michael Green and John Schwarz discovered that requirements of consistent quantization (the so-called anomaly cancellation) place severe restrictions on the possible gauge groups of the superstring theory. It seems now that a specific 60
THE MATHEMATICALINTELLIGENCER VOL. 11, NO. 2, 1989
superstring model, called the Es x E8 heterotic superstring, could eventually become a "Theory of Everything." Such are Great Physical Expectations. Mathematically, the (super)string theory is no less interesting. As the Moscow physicist Vadik Knizhnik once remarked, unification of interactions is achieved through unification of ideas. Physics papers devoted to m a n y facets of string theory are now filled with homotopy groups, Kac-Moody algebras, moduli spaces, Hodge numbers, Jacobi-Macdonald identities, and modular forms. A researcher trying to find his or her way in this mixture of seemingly disparate structures and techniques soon discovers that a physicist's intuition often transcends the purely mathematical one.
Some Physics Let me r e v i e w the physical c o n t e n t of m o d e r n quantum field theory before proceeding to its mathematical scheme. In the 1920s, fundamental physics consisted of four principal theories: electromagnetism, general relativity (i.e., gravity theory), quantum mechanics, and statistical physics. Broadly speaking, the first three dealt with "elementary" phenomena while the fourth dealt with "collective" phenomena and their general laws. The scale of elementary phenomena was defined by four fundamental constants: e (electron charge), G (Newton's constant), c (light'velocity), h (Planck's constant). The group of dimensions generated by them essentially coincides with the group generated by three classical physical observables: mass, length, and time. In other words, starting from G, c, h one can define the "natural," or Planck, units:
Mpl = ( ~cG-X)~ - 1 0 - s g, epl = h Mpl-'C -1 --10 -33 cm, tpl = epf-' - 1 0 - O s e c . The trouble is, we know of no elementary processes of Planck scale. Indeed, m o d e m accelerators allow us to probe space-time at scales down to 10 -x6 cm and 10 -26 sec only. On the other hand, Mpl is the mass of a macroscopical drop of water about 0.2 m m diameter, and elementary particles of such a large mass do not exist in our world. This incompatibility of the three fundamental theories was long considered as evidence of a need for a d e e p e r t h e o r y combining all of them, for short, a (G,c, fi)-theory. In fact, two approximations had been discovered: general relativity, which could be considered as a (G,c)-theory, and quantum electrodynamics, i.e., a (c, fi)-theory. No one has yet succeeded in developing a consistent (G, h)-theory, i.e., quantum gravity.
Invitation to a conference on strings and superstrings.
61
The actual history of physics in our century followed an alternative course: since the discovery of radioactivity and the subsequent construction of the first accelerators, the list of elementary particles and forces tended to grow; immense efforts of several generations of physicists were devoted to the development of quantum field theory explaining the variety of observed phenomena. In the 1960s we had to be content with the following picture. There are several sorts of matter particles, seemingly point-like ones, i.e., without discernible internal structure; stable matter consists of quarks and electrons. All matter particles are fermions, i.e., they obey Fermi statistics and have spin 1/2. There are also quanta of four fundamental forces: photons (electromagnetic force), gluons (strong force), vector mesons (weak force), and graviton (?) (gravity). They are bosons, meaning that they obey Bose-Einstein statistics and have spin 1 (or 2 for graviton).
This romantic leap of twenty orders of magnitude makes the situation in modem theoretical physics extremely bizarre and poses new problems of relating theory to the phenomenology of low energy (former high energy) physics. Although elementary particles are point-like, they do have internal degrees of freedom. Mathematically, this means that a wave-function of a quark, say, is not a scalar function on space-time but a section of a vector bundle associated with a principal G-bundle, where G is a Lie group called the gauge group. (The choice of G ideally should have been governed by fundamental laws of nature but in the practice of the 1960s was model dependent). Similarly, a wave function of a quantum of fundamental force is a connection on the corresponding vector bundle, i.e., a matrixvalued differential form describing a parallel transport of internal state vectors along paths in space-time. A (second-quantized) theory of this kind is generally called a Yang-Mills theory. The highest achievements of this epoch were the socalled standard model describing electroweak and strong interactions by means of the Yang-Mills fields with the gauge group SU(3) x SU(2) x U(1), and several projects of Grand Unification based upon some large (preferably simple) group G containing SU(3) x SU(2) x U(1). This large group should be a symmetry group of a fundamental theory at a higher energy, which is broken by some mechanism at lower energy leading to the effective Lagrangians of our present day physics. In all these developments gravity was neglected be62
TIlE MATHEMATICAL INTELL1GENCER VOL. 11, NO. 2, 1989
cause the gravitational interaction between elementary particles is many orders of magnitude weaker than, say, the electromagnetic one (this is another way of saying that Planck mass is very large). In fact, the overall picture of our universe is determined by diff e r e n t forces at different scales. At the scale about 10-13 cm, quarks are bound into protons and neutrons by the strong force. The atomic nucleus consists of protons and neutrons bound by residual forces. The strong interaction, being a short-distanced one, dies out at atomic scale, and the electromagnetic interaction takes its part binding electrons and nuclei into a neutral atom. The electromagnetic interaction is longdistanced and very strong in comparison with gravity, but for some reason positive and negative electric charges cancel each other out in large lumps of matter like stars and planets. Gravitational charge, i.e., mass, never cancels, but only adds up, so that at astronomical scale gravity becomes the main force. The residual electromagnetic forces, in the form of light and radio waves, serve as a source of energy and information for our kind of living matter. (This hierarchy of scales reflected in the hierarchy of physical theories is a characteristic trait of our modern understanding of nature. Any future unified theory will have to explain it.) Thus, all observed effects of gravity are in fact collective ones. They can become discernible at an elementary interaction level only in sufficiently excited matter, e.g., if an elementary particle is accelerated to the energy ~Mp~"a, which is far outside the range of any conceivable laboratory. However, such conditions existed in the very early universe, and the physics of these extremal states probably determined the universe's later fate on a cosmological scale. Given this background, let us now reconsider some features of string models. Their first striking property is a prediction of a definite dimension of the spacetime at Planck scale: 26 for bosonic strings and 10 for superstrings. This source of embarrassment in dual models of hadrons now becomes one of the major predictions of the theory. However, this prediction is not directly testable and it poses the problem of explaining the apparent four-dimensionality of the low-energy world. Very loosely speaking, one can imagine that 26 - 10 = 16 dimensions somehow conspire to accommodate the internal degrees of freedom of fundamental particles (16 being the rank of the gauge groups Es x Es and SO(32)), while the remaining 10 - 4 = 6 dimensions "spontaneously" compactify at the Planck scale early in the course of cosmological evolution. Below we shall have more to say about the mathematical origin of these critical dimensions, 26 and 10. It suffices to mention here that their appearance is a pure quantum field theoretical effect. A second property of string models is the unification of the four known forces, including gravity, in an
effective Lagrangian of low-energy approximation to the full theory, which itself is much richer. String models place very tight constraints on the possible Grand Unification gauge group. It may be Es x E8, one factor being responsible for common matter, with the other one responsible for the socalled " d a r k m a t t e r , " w h i c h interacts w i t h the common one only through gravity. Finally, string models incorporate supersymmetry into the framework of fundamental physics. One must add that such a theory actually does not exist yet. We have an ideal picture, a puzzle some of w h o s e f r a g m e n t s have marvellously f o u n d their proper places while others still remain a challenge. Besides, all of this may prove someday to be just wishful t h i n k i n g - - a s physics. Fortunately, mathematics is less perishable.
In the following pages I shall try to make explicit some mathematical structures of f u n d a m e n t a l physics, stressing specific properties of string models. A model of a physical system starts with a description of a set of "virtual classical paths" and an action functional S: --~ R. Generally speaking, ~ is a function space, e.g., a space of maps of manifolds f: M ~ N, or a product of such spaces. Maps may be subject to some boundary conditions; N may be a bundle over M and may consist of sections of this bundle, etc. Usually, M and/or N are (pseudo)Riemannian manifolds (with fixed or variable metrics) and S is obtained by integrating a natural volume form over M or N. Three examples follow. (i) General relativity. Here M is a fixed 4-dimensional C| @ = space of Lorenz metrics g = gabdXadxb on M (i.e., sections of S2(TM) --~ M with positivity conditions), and f
S(M,g) = -(16~rG) -1 J., R volg (1)
where G is the Newton constant, R = Ricci curvature, volg = the volume form of g. (ii) Massive point-particle propagating in a space-time (M,g). Here ~ is the set of maps ~/:[0,1] ~ M, /. 1
S(~/)
=
-m lo ds,
vol~(g)
(Nambu's action)
(2)
where m = mass, ds 2 = ~/*(g), the induced metric. The image of [0,1] is the particle's virtual world-line. Off) String propagating in a space-time (M,g). Here ~ is the set of maps cr:N ~ M, where N is a surface whose image is the string's world-sheet,
(3)
where T is the so called "string-tension" of dimension (length) -2 and cr*(g) is the induced metric. We shall always measure time in length units and action in Planck units (or, as physicists say, put ~/= c = 1).
Classical equations of motion. These are equations for stationary points of S: 8S -- 0. Finding their solutions or exploring their qualitative properties is the main task of classical mathematical physics. Quantum expectation value and partition function. These are given by the following formal expressions (Feynmann integrals):
(0) = Z -1 fh O(p)e~S(p)Dp,
Mathematical Structure of Quantum Field Theory
(Hilbert-Einstein action)
S(cr) = - ~
f
Z = J~ ds(p)Dp,
(4)
(5)
where 0:~ --~ R is an observable and Dp is a formal measure on ~. Most problems of q u a n t u m field theory can be thought of as problems of finding a correct definition and a computation method for some Feynmann path integral. From a mathematician's viewpoint almost every such computation is in fact a half-baked and ad hoc definition, but a readiness to work heuristically with such a priori undefined expressions as (4) and (5) is necessary in this domain. There are several standard tricks. First, one tends to work with the so-called "Euclidean" versions of (4) and (5), where eisO') turns into e -sO'). Besides "better convergence" (whatever that means), this makes explicit a basic analogy between quantum field theory in (D space, 1 time) dimensions and statistical physics in D + 1 space dimensions, allowing us to use rich technical tools and insight on collective p h e n o m e n a . Second, one tries to reduce (4) and (5) to finite-dimensional integrals using group invariance and/or approximations. Third, one tries to reduce (4) and (5) to Gaussian integrals, whose theory is the only developed chapter of infinite-dimensional integration, e.g., by means of an appropriate perturbation series. Here is, for example, a standard heuristic explanation of the correspondence between classical and quantum laws of motion. Assume for simplicity that (with given boundary conditions) the equation 8S = 0 has only one solution p = Po. Assume also, by analogy with the finite-dimensional case, that the stationary phase approximation is valid, i.e., the quantum expectation value {0) = f@0(p) eiS(p)Dp coincides with O(po) up to a universal factor and a small error. This means THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 6 3
that quantum observables practically have their classical values on the classical path. A necessary condition for the validity of the stationary phase approximation is that S/tibe large on @. This is in accordance with the discovery of early quantum theory that the classical regime corresponds to h---* 0. A successful calculation of a path integral often involves one or more limiting processes that differ from the Archimedes-Newton-Lebesgue prescription of adding up an infinity of infinitesimal contributions. In fact, such a calculation usually gives a finite value as a difference or a quotient of two (or more) infinities.
The electromagnetic interaction is long-distanced and very strong in comparison with gravity, but for some reason positive and negative electric charges cancel each other out in large lumps of matter like stars and planets. I believe that there is a message in this observation. Each level of reality we become aware of is but a flimsy foam on the surface of an infinitely d e e p ocean, usually called a vacuum state. It is a state of lowest energy, but its energy is infinite. We are divided by a thin film from an eternal fire, whose first tongues are the flames of the nuclear age. Will a mature string theory start a new auto-da-fd? Returning to mathematics, it sometimes happens that the infinities of a concrete model apparently do not reduce to a finite number. A notorious example is the Einstein gravity theory (1), which for this reason is called unrenormalizable. Only after an extension to a larger picture, hopefully a stringy one, can gravity become finite. If a theory is renormalizable (or, as happens with super-symmetric models, even finite), the indeterminacies in choosing "infinite constants" are resolved by experimental d a t a - - v a l u e s of various charges and coupling constants. Ideally, nothing like this should be allowed: a perfect theory must predict everything.
Operatorapproach.
We described earlier the Lagrangian approach to quantization. There is an alternative approach which in various contexts is called the Hamiltonian, canonical, or operator approach; it takes the following form. On the space of solutions P of the classical equations of motion 8S = 0 there is a natural Poisson structure, i.e., a Lie bracket on a space F of functionals on P. For example, in classical mechanics the space of classical paths P can be identified with the phase space, because a classical motion is defined by its initial values of position a n d m o m e n t u m . The c o r r e s p o n d i n g 64
THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989
Poisson structure is defined by the well-known symplectic form. (This basic example suggests taking for P an appropriate space of boundary values, which is often done in string theory.) A unitary representation of a subalgebra of F in a Hilbert space H defines an operator quantum description of the system. Of course, such a representation is rarely unique, and the nonuniqueness corresponds to the indeterminacy of path integrals. The so-called "geometric quantization'" is a method of constructing such representations in functions on P, or sections of a bundle on P, or in appropriate cohomology. An expectation value (0) of an observable 0 ~ F calculated by means of a path integral should coincide with an operator expectation value of the kind (vac I ~(0) I vac) (or (0~ ] ~(0) I 4) for appropriate state vectors vac, ~, ~ ( H ) defined by a representation ~. In order that this make sense one should ensure that 0 can be considered as a functional on 0~ uniquely defined by its restriction to P C ~. In general, the path integral quantization and operator quantization should be considered as complementing each other rather than being strictly equivalent. This complementarity inherited from classical mechanics reappeared in the guise of the Schroedinger and Heisenberg approaches to quantum mechanics. In string theory, comparison of the two approaches poses some intriguing problems on the connections between modular forms on Teichmueller spaces and moduli spaces of vector bundles, on the one hand, and r e p r e s e n t a t i o n t h e o r y of the Virasoro, KacMoody, and similar Lie algebras on the other hand. Before the advent of q u a n t u m field theory in this mathematical domain only genus one modular forms appeared as character series of representations, and even that seemed a mystery. Symmetries. The basic structure (@, S) is often complemented by an action of a group G on ~ leaving S invariant (or, infinitesimally, by an action of a Lie algebra 9.I which in an infinite-dimensional situation may well be non-integrable). Mathematical aspects of such a picture may have various physical interpretations of which we mention several. Classical symmetries of a fiat space-time acting upon give rise to the e n e r g y - m o m e n t u m operators. Noether's theorem on conservation laws is reflected in the structure of the m o m e n t u m map VL:P--~ ?,l*, where P is the space of classical motions with an invariant symplectic structure. Local gauge symmetries in the theory of Yang-Mills fields and diffeomorphisms of the space-time of general relativity give rise to physically indistinguishable states. Thus, in this case one should actually call @/G the space of virtual paths, choose various G-invariant subspaces as spaces of quantum states, etc. String theory also provides such phenomena.
Path integral (or operator) quantizarion may break up classical G-invariance of the picture due to the indeterminacies in the regularization scheme. A precise description of the resulting non-invariance is the subject-matter of the theory of anomalies. In recent years it became clear that the essential aspect of anomalies reflects cohomology properties of G. The disappearance of quantum anomalies is considered to be an important metatheorerical criterion of the consistency of a quantum model. This disappearance led to the discovery of critical dimensions and preferred gauge groups. Finally, a few words should be said about the conformal group, which is the group of local rescalings of a metric of space-time or string world sheet: gab--+ efg~b. The conformal invariance of a physical model implies the absence of a natural scale (length, mass, or energy). In the context of statistical physics this happens in the vicinity of phase transitions. One can conjecture that fundamental physics is governed by conformally invariant laws. Anyway, conformal invariance plays an essential role in string theory.
Correspondence principles. Historically, the correSpondence principle is a loosely formulated prescription for d e r i v i n g classical laws of p h y s i c s from quantum ones. Modern fundamental physics is a conglomerate of theories, or models, each of which is applicable at an appropriate scale or is a simplified version of a more adequate but too complex theory. All informal rules of patching together these models at the outskirts of their validity ranges may reasonably be called correspondence principles. If, as I believe, this openness of physics is its essential characteristic, then correspondence principles themselves may be elevated in status and could eventually become considered as physical laws acting in transition periods as rites of passage. For example, 26, 10, and 4 may have been successive steps of the formarion of our space-time from an infinite-dimensional quantum chaos during the first 10 -~ seconds of creation.
Reading Suggestions Entering string theory is not an easy task for mathemaricians. Two recent publications may help to get a general perspective and to choose a particular topic for deeper study: the monograph-textbook [1] and the anthology [2]. Two ICM Berkeley talks [3] and [4] were at least partly devoted to strings. Witten's talk is a beautiful initiation to quantum field theory. The Belavin and Knizhnik discovery [5] together with previous works by Quillen and Faltings led to i m p o r t a n t progress in the t h e o r y of d e t e r m i n a n t
bundles [6]--[10], extending Grothendieck's Riemann-Roch theorem. Much remains to be done in this domain which is simultaneously "the component at infim'ty" of arithmetical geometry [11], [12]. Path integrals of string theory could eventually be reinterpreted in arithmetical terms via a version of Siegel-Tamagawa-Weil theory. A beautiful recent result by Shabat and Voevodsky, drawing upon previous work by Grothendieck and Bely, shows that natural lattice approximations in string theory are inherently arithmetical ones [13]. For a rich representation-theoretic part of string theory see [14], [15], [10] and many pages of [1] and [2]. Good luck, with a few strings attached!
References 1. M. B. Green, J. H. Schwarz, E. Witten, Superstring Theory, in 2 vols, Cambridge: Cambridge University Press (1987). 2. J. H. Schwarz, ed., Superstrings: The First Fifteen Years of Superstring Theory, in 2 vols, Singapore: World Scientific (1986). 3. E. Witten, Physics and geometry, Berkeley ICM talk (1986). 4. Yu. I. Manin. Quantum strings and algebraic curves, Berkeley ICM talk (1986). 5. A. A. Belavin and V. G. Knizhnik, Algebraic geometry and the geometry of quantum strings, Phys. Lett. 168B (1986), 201-206. 6. D. S. Freed, Determinants, torsion, and strings, preprint, MIT (1986). 7. J.-M. Bismut, H. Gillet, and C. Souh~, Analytic torsion and holomorphic determinant bundles, preprint, Orsay 87-T8 (1987). 8. P. Deligne, Le d~terminant de la cohomologie, preprint, Princeton (1987). 9. A. A. Beilinson and Yu. I. Manin, The Mumford form and the Polyakov measure in string theory, Comm. Math. Phys. 107 (1986), 359-376. 10. A. A. Beilinson and V. V. Schechtman, Determinant bundles and Virasoro algebras, submitted to Comm. Math. Phys. 11. G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 118 (1984), 387-424. 12. Yu. I. Manin, New Dimensions in Geometry, Springer Lecture Notes in Math. 1111 (1985), 59-101. 13. V. A. Voevodsky and G. B. Shabat, Equilateral triangulations of Riemann surfaces and curves over algebraic number fields, preprint (1987). 14. G. B. Segal, Unitary representations of some infinite dimensional groups, Comm. Math. Phys. 80 (1986), 301-342. 15. B. Feigin and B. Fuchs, Representations of the Virasoro algebra, Seminar on supermanifolds, 5 (D. Leites, ed.), Stockholm: Dept. of Math., Stockholm Univ., N 25 (1986). 16. P. Candelas, G. Horowitz, A. Strominger, and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B258 (1985), 46-90. Steklov Math. Institute 42 Vavilova, Moscow 117966 GSP-1, USSR THE M A T H E M A T I C A L INTIELLIGENCER VOL. II, NO. 2, 1989 65
Chandler Davis*
Descartes' Dream: The World According to Mathematics by Philip J. Davis and Reuben Hersh Harcourt Brace Jovanovich, 1986, 400 pp.
Reviewed by Robert Osserman The title, Descartes" Dream, derives from both a literal dream of Descartes and its subsequent transformation into a figurative dream: that of the unification and explanation of all knowledge through the method of reason. In the formulation of the authors' subtitle, the dream of Descartes is to reconstitute the world according to mathematics. The avowed goal of the book is to sound an alarm: Descartes' dream may evolve into our nightmare. The a u t h o r s of this book are two of the most thoughtful and articulate commentators on the current mathematical scene. Their earlier book, The Mathematical Experience, does a superb job of conveying to a general audience a sense of the richness and scope, the beauty and the blemishes of the mathematical endeavor. In this book, their sequel, the central theme is not one likely to leave the reader feeling uplifted or optimistic. An appropriate title might have been "'The Mathematical Experience: The Down Side.'" Not that the earlier book s i d e - s t e p p e d the d o w n side, as the various headings under the rubric "Underneath the Fig Leaf" indicate. In fact, part of the unique quality and charm of that book was its refusal to adopt a Madison Avenue approach, selling mathematics as the unsullied Queen of the Sciences. Here, however, we are exposed at book length to the underside of the fig leaf, and we may be left feeling more assaulted than enlightened.
* C o l u m n editor's address: Mathematics Department, University of Toronto, Toronto, Ontario M5S 1A1 Canada
In their preface, the authors lay out their goals and concerns. Here is a brief excerpt. We are concerned with the impact mathematics makes when it is applied to the world that lies outside mathematics itself; when it is used in relation to the world of nature or of human activities. This is sometimes called applied mathematics. This activity has now become so extensive that we speak of the "mathematization of the world." We want to know the conditions of civilization that bring it about. We want to know when these applications are effective, when they are ineffective, when beneficial, dangerous, or irrelevant. We want to know how they constrain our lives, how they transform our perception of reality. Some of the chapter and section headings give a further indication of the authors' concerns: Are we drowning in digits? The social tyranny of numbers. Cognition and computation. Computer graphics and the possibility of high art. Why should I believe a computer? Loss of meaning through intellectual processes: Mathematical abstraction. Mathematics and imposed reality. Mathematics and the end of the world. Other topics and subtopics that they engage range from c o n c r e t e e x a m p l e s - - c o m p u t e r d a t i n g , IQ testing, military applications--to broad issues--the philosophy of mathematics and computing, ethics, religion. The presentation is, on the whole, clear, eloquent, and forceful. Every reader is likely to find at least some of the sections to be of unusual interest. Having said that, I must admit that my overall reaction upon finishing the book was one of disappointment. I had somewhat the feeling that one experiences w h e n trying to make a dinner out of the assorted snacks at a cocktail party; no matter how high the quality of the individual items, the total effect is not one of satisfaction or nourishment. What we are offered here is a smorgasbord of essays, new and re-
66 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2 9 1989Spfinger-VerlagNew York
printed, along with transcribed lectures, dialogues, and interviews. The same kaleidoscopic approach was used with resounding success in The Mathematical Experience. There, however, it was ideally suited to the goal of the b o o k - - t o convey a sense of the richness and variety of mathematical experience. It is less appropriate in this case, where there is a unifying theme, and one looks for a sustained development. A sizeable component of the book is not concerned with mathematics at all, but with computers, which is fair enough in the broader context of the mathematization of our society. And a large portion of the computer-oriented material comes in the form of taped and edited interviews or dialogues between Philip Davis and Charles Strauss. The format of the interview does inject a degree of liveliness into the subject matter, but it also has certain drawbacks. It is first of all not an ideal w a y to convey technical information, although that is a prerequisite for any serious discussion of the underlying issues. For example, in the section entitled "Metathinking as a way of life," I found the technical discussion more allusive than explanatory, seemingly directed at a reader already familiar with the inner workings of a computer. Perhaps even more problematic was the difficulty I found in situating myself in the shifting level of seriousness of the various dialogues, w h o s e tone alternates b e t w e e n light banter and heavy philosophizing. Having no expertise in computers myself, I could not tell how literally to interpret the responses to diverse questions. The problems with the format were highlighted for me in another interview, between Philip Davis and Joan Richards, an historian of science at Brown U n i versity. I should like to focus in some detail on this exchange, in part to illustrate the difficulties with the interview format, but more to examine one of the central issues: the ways in which a particular development inside mathematics--in this case the development of non-Euclidean geometry--is part of a transformation of the contemporary views and values of the broader community. This theme is central to one of the principal arguments of the book: that mathematical research does not take place, as is sometimes depicted, in isolation from the outside world, but is a social and community activity that affects and is affected by the surrounding environment. Surely, if one wishes to make that case, few topics in mathematics lend themselves as well as non-Euclidean geometry. Since Euclidean geometry played for so long the role of the one rock of certainty amid the ever-changing s y s t e m of beliefs in science and religion (to say nothing of politics and, social mores) it would not be surprising if the introduction of non-Euclidean geometry fostered doubts at the deepest level about our ability to be utterly sure about anything. What Joan Richards points out is that identifying cause and effect can be a delicate enterprise; the will-
The Mathematician by Diego Rivera, 1918. Reprinted with permission of the Olmedo Foundation. ingness to see non-Euclidean geometry as having broader applicability outside of mathematics seems clearly related to a shift in world views, such as a growing rejection of an authoritarian interpretation of knowledge. The question is: did the discovery of nonEuclidean geometry act as a catalyst for that shift in world views, or rather, did the shift occur first and determine, or at least influence, the response to nonEuclidean geometry? Richards seems to feel: some of both. As an example of the latter, she contrasts Cayley's apparent indifference to non-Euclidean geometry in 1865 with Sylvester's view of it in 1869 as of prime importance in the larger scheme of things. She concludes: "'We can't evade the question of w h y nonEuclidean geometry was virtually ignored from 1830 to 1870. The time has to be ripe." The implied premise here is that nothing happened internally in mathematics to explain the change in view, and so we must look to external causes. But the subject did not remain static between 1830 and 1870. In particular, it was precisely the interval between Cayley's remarks in 1865 and Sylvester's in 1869 that witnessed the publication of two major landmarks: THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989
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Riemann's fundamental re-thinking of the nature of geometry, placing non-Euclidean geometry within a much broader context (first published in 1867) and Beltrami's decisive discovery of a model for hyperbolic geometry in the unit disk (which appeared in 1868). Out of this model came the conclusion that the centuries' long effort to find a contradiction in non-Euclidean geometry was doomed to failure unless Euclidean geometry itself contained a contradiction. (Recall that although Bolyai and Lobachevsky became personally convinced that their new geometry was a viable one, they had no proof that a contradiction could
The avowed goal of the book is to sound an alarm: Descartes" dream may evolve into our nightmare. not be found.) And, as Richards has herself pointed out elsewhere [5], it was Riemann's work that led Helmholtz and Clifford to the vigorous espousal of non-Euclidean geometry. Under those circumstances, I see no reason to invoke the additional hypothesis that "the time has to be ripe." A more extreme form of the same thesis, and one touching more directly on the central nerve of mathematical activity, is expressed later in the same interview. Here, in slightly abbreviated form, are both the question and the response: PJD: Since non-Euclidean geometry actually lay close to the surface, why wasn't it discovered hundreds of years earlier? Could you argue that what prevented it from emerging was the desire for an authoritarian basis for knowledge? Had Euclid been wrapped in the mantle of Authority? JR: I don't think so. Starting in the early eighteenth century, there was a sharp acceleration of interest in asking what is the nature of mathematical truth. The context of that acceleration was the aftermath of the new epistemologles that came out of the scientific revolution of the seventeenth century. In that context, and only in that context, non-Euclidean geometry was, first, generated, and second, interpreted. Here, both question and answer are directed toward a different and more radical notion of the interaction between mathematics and the outside world: not the ways in which inner developments of mathematics may influence the broader world view, nor even the effect of a general intellectual stance on the interpretation of mathematics, but rather, whether mathematical progress itself can be hindered by a prevailing attitude in the outside community. The question is well worth pondering, especially in view of the current debates over the sources and modes of financial support for mathematicians. However, the answer given here, in the context of non-Euclidean geometry, seems to run counter to the actual course of events. Non-Euclidean THE MATHEMATICAL
INTELLIGENCER VOL. 11, NO. 2, 1989
geometry was not generated in the context of a broader questioning about the nature of mathematics, science, and knowledge. Precisely the reverse was the case. It was generated by attempts, in the words of Saccheri, to "'vindicate Euclid of every flaw.'" The method was to be by contradiction. But as one mathematician after another tried to push the argument further, the suspid o n gradually grew that the outcome would not be a contradiction, but a new kind of geometry: a consistent alternative to Euclid. Taken together, these responses by Richards dearly reflect an underlying belief in the importance of outside influences on the development of mathematics. Taken at face value, they seem hard to justify. The format of the interview does not lend itself to supporting such statements with the necessary evidence, and it leaves us uncertain of h o w seriously to take them as definitive versions of the author's views. Seen in a broader context, the title of the DavisRichards interview-dialogue ("Non-Euclidean geometry and ethical relativism") suggests the extent to which this book may be seen as part of a larger debate that is currently taking place. A recent article by Tzvetan Todorov [7] under the heading "Is Western culture really in decline?" discusses the underlying
This theme is central to one of the principal arguments of the book: that mathematical research does not take place, as is sometimes depicted, in isolation from the outside world, but is a social and community activity that affects and is affected by the surrounding environment. theme in a spate of recent books, including The Closing of the American Mind by Allan Bloom, that would answer "Yes!". (The final section of Davis and Hersh says "'The human institutions in the sense of Vico are changing, eroding," where "the human institutions, according to Vico, are the family, law, government, myth, religion, language, art, poetry, song.") Todorov says: The substance of the argument runs as follows. We no longer know how to tell the good from evil, the beautiful from the ugly, the elevated from the moronic, the essential from the ridiculous. The fault lies with the relativism of values currently in vogue, with the rejection of all guiding principles other than tolerance. This relativism is the final consequence of the Cartesian revolution, which placed man at the center of the universe and transformed technical thinking, An the broader sense of the term, into the norm for all thought. Todorov goes on to compare this argument with that used earlier by Heidegger [3] in a similar vein. A collection of essays by Jacques Maritain [4] whose title
essay, appropriately enough, is The Dream of Descartes, offers a s o m e w h a t different variation of the same theme. What all these works have in common is a belief that what they see as a Cartesian type of rationalism has led on the intellectual level to a trivialization or superficiality of thought, and on the broader human level to a rejection, or at least a neglect, of humanistic and moral concerns. Most extreme is the attempt in the present book to impute to mathematics and science a role in the holocaust (p. 290): "In my mind it is no accident that the great evils of the period 1933-1945 were perpetrated in a country that was the world leader in theoretical science and mathematics." More specifically, they suggest (p. 291) that " a d v a n c e d mathematization, through abstraction and subsequent loss of meaning, played a role.'" As I listened recently to the histrionics and virulence of an old speech by Hitler, I could think of nothing further removed from mathematical abstraction. It is especially ironic that the greatest exponent of mathematical racism of the Nazi era, Ludwig Bieberbach, railed against some of the same tendencies toward excess formalism that are bemoaned in the present book; he extolled the "Germanic" style of mathematics, which he conceived as a more humanistic variety, involving imagery and intuition [1]. As must be obvious by now, this book is an intensely personal one. The issues dealt with are not a matter of purely academic interest to the authors, but of keenly felt dilemmas to be engaged head on. A formalistic approach to mathematics that turns the subject into a game stripped of external meaning is taken a s b o t h an example and a symbol of a far larger problem: a (peculiarly?) twentieth-century tendency to move up the ladder of abstraction, away from the immediate reality of the senses and emotions. In the visual arts there was a long evolution away from content into abstract and non-representational art. In literature and even more in literary criticism, old-fashioned concerns with "'character" and "plot" were supplanted by a focus on form, conventions, and "literary theory." Needless to say, the authors of this book are not the first to inveigh against the dehumanizing effect of this trend. The debate has raged at least since the beginning of the century when cubism in art and atonality in music were viewed as a rejection of all that was central to their respective art forms. If anything, the book's main weakness may be its failure to acknowledge the strong countermovements that have taken hold in the past decade or two. The predominant position that abstract art held in the forties and fifties has long since given w a y to a variety of representational forms. Parallel movements have taken place in literature, theater, and film, as well as in music. Similarly in science, we have seen leading physicists, such as Richard Feynman and C. N. Yang, calling repeatedly
u p o n n e w entrants in the field to b e w a r e of the dangers of empty mathematization, divorced from a firm grasp on physical content. Within mathematics itself, the heyday of purity and abstraction--the reign of Bourbaki--is clearly over; current graduate students are much more likely to be excited by the more intuitive parts of geometry and analysis, not to say theoretical physics and applications. So w h y all the fuss? Presumably because, despite these countertrends, the broad picture remains a s e e m i n g l y inexorable march of m a t h e m a t i z a t i o n throughout our culture. But what is the real danger of the mathematization of just about everything? Not that one does a mathematical analysis of a situation, which is no more likely to point the wrong way than a hunch or a ball-park estimate. It is that one commits the fallacy of equating a quantitative answer with a definitive one. One would like to think that for a sophisticated audience of mathematicians and scientists it would be scarcely necessary to point out the fallacy; we are well aware that the process of quantifying any real-world situation involves a number of approximations and the total omission of certain not easily quantifiable factors. Unfortunately, I would have to agree with the authors that these obvious facts are often overlooked, and superficial mathematical analyses are given more weight than they deserve. On the other hand, for an unsophisticated audience there is an equal and opposite danger: that abuse of quantitative methods leads to their total rejection. They then become easy prey to the tobacco-industry spokesman: "Smoke all you want. The so-called links between cigarettes and lung cancer are 'purely statistical.' " Perhaps the distinction that the authors are trying to make is one that would these days be characterized as that b e t w e e n mathematics use and mathematics abuse. Basically, they a p p l a u d and encourage the trend away from abstraction and toward contact with the real world. But aside from the type of abuse mentioned above, there is also the inescapable fact that the real world to which mathematics may be applied is
Within mathematics itself, the heyday of purity and abstraction--the reign of Bourbaki-is clearly over. also the world of competitiveness, greed, and aggression. Most of us are only dimly aware of the degree to which mathematics is carried on outside the ivory tower. I must admit to having been startled when I saw not long ago a reference to the National Security Agency as the largest employer of mathematicians. Whether one takes such news as a good or a bad sign, I think that it is clearly something to be aware of and to ponder. Similarly, whether we agree or disagree THE MATHEMATICALINTELLIGENCER VOL. 11, NO. 2, 1989 6 9
with the views expressed by Davis and Hersh, we should be grateful that they have raised the issues in a lively and provocative fashion and have not shrunk from expressing their own clear convictions.
References 1. Ludwig Bieberbach, Die v61kische Verwurzelung der Wissenschafl (Typen mathematischen Schaffens), Sitzungsberichte der Heidelberger Akademie der Wissenschaft, 1940, 5. Abhandlung, pp. 1-31. See also Allen Shields, "Klein and Bieberbach: Mathematics, race, and biology," Mathematical Intelligencer Vol. 10, No. 3 (1988), 7-9. 2. Philip J. Davis and Reuben Hersh, The Mathematical Experience, Houghton Mifflin, paperback (1982). 3. Martin Heidegger, "Modem science, metaphysics, and mathematics," What is a Thing?, Chicago: Henry Regnery Co. (1967). 4. Jacques Maritain, The Dream of Descartes, New York: Harper & Row (1964). 5. Joan Richards, "The reception of a mathematical theory: non-Euclidean geometry in England, 1868-1883," Natural Order: Historical Studies of Scientific Culture, Beverly Hills: Sage Publications (1979), 143-166. 6. Gregor Sebba, The Dream of Descartes, Carbondale: Southern Illinois University Press (1987). 7. Tzvetan Todorov, ' H e philosopher and the everyday," The New Republic (Sept. 14 & 21, 1987), 34-37. Postscript. The literal dream of Descartes continues to exert a strong fascination. The books by Maritain and Sebba listed above both contain detailed descriptions of the dream and discuss its influence on Descartes' life and philosophy. The latter book embarks on a lengthy interpretation of the dream itself and includes references to a full-scale Jungian analysis of the dream, as well as Freud's response to a request that he analyze it. Department of Mathematics Stanford University Stanford CA 94305 USA
The P61ya Picture Album: Encounters of a Mathematician By George P61ya Edited by G.L. Alexanderson Birkh/iuser, Boston-Basel, 1987. 160 pp; US $35. Reviewed by Lee Lorch George P61ya (1887-1985) was one of the most original, productive, respected, and versatile mathematicians of this century. Possessed of a warm and friendly disposition and a gentle h u m o u r , he attracted widespread acquaintanceships in his long life. He had broad social interests, encouraged by his wife of 67 years (who survives him), herself a personality of intelligence, independence, courage, and compassion. Among the P61yas' hobbies were the collection of stamps centred about science and scientists, and, what 70
THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 2, 1989
Gabor Szeg6 and George P61ya taking the manuscript of their Aufgaben und Lehrsittze aus der Analysis to Springer for its first publication.
is now partially before us, a huge assemblage of photographs of mathematicians. Their albums were well known and often displayed to guests. Now, thanks to G.L. Alexanderson, a former student of P61ya and a good friend of the P61yas, and also to Stella P61ya, 130 of these photographs are made public. Included are a few that P61ya acquired from his older colleague in Zfirich, Adolf Hurwitz, and others inherited from his father-in-law, the physics professor Robert Weber. The book is in an attractive black-and-white format. None of the colour prints of the P61ya collection are reproduced, but these constitute a minority of the albums. A two-page introduction and Alexanderson's gracious and informative 13-page biographical sketch precede the pictures. A more detailed obituary with extensive commentary on P61ya's main scientific and pedagogical contributions has appeared in The Bulletin of the London Mathematical Society, vol. 19, part 6, November 1987, pp. 559-608, written by G.L. Alexanderson and L.H. Lange and various specialists. Both are well worth reading. The photos depict 146 identified individual mathematicians (including a handful of theoretical physicists such as the young Albert Einstein, shown on the cover and as the first entry, performing a violin duet with Lisi Hurwitz while her father, Adolf Hurwitz, plays at conducting). A number of unidentified individuals appear in various group photos. The reader could try to see how many can be identified for future printings. My contribution is Mrs. Gabor Szeg6, unidentified on page 73. (Peter Lax and Peter Szego have made independently the same observation.) The index records the 146 scientists depicted. It does not list Ferdinand Springer (of Springer-Veflag), page 54, nor the wives of 14 mathematicians, including Stella P61ya. The entries give dates of birth and death (omitting Szeg6's death which, like P61ya's, occurred in 1985) and the countries of birth and principal em-
ployment or refuge. To avoid listing practically the entire UN, it prudently associates Paul Erd6s only with Hungary. It could be a d d e d that Stefan Bergman worked also in the USSR, I believe his first refuge from the Nazis, and Jean Dieudonn6 also in the US. A few others could be supplemented similarly. There are only three large group photos. Most depict individuals alone or in small groups. Many are informal "candid camera" shots. Associated with each of the photos is some comment, most taken from off-the-cuff remarks by P61ya himself, taped as he talked to guests in his home. The remarks reproduced are thus rather random. They reflect P61ya's pleasure in concise witticism and so quite a few reproduce brief humorous comments of a gentle nature about the respective individuals, or recall some of their harmless foibles or humorous comments, or a bit about mathematics. Sometimes they relate P61ya's specific connection with an individual and other personal comments. About Harald Bohr, page 67, he goes deeper: "He was a very kind person . . . . [W]e can all be kind. You talk to a boring student because you feel it your duty 9. . Or you talk to a nasty c o l l e a g u e . . , because you don't wish to collide with him, so you are kind out of duty or self-interest. But Harald Bohr was naturally k i n d . . , an inborn instinct." The Nazi accession to power in Germany and the horrible consequences were deeply ingrained in the consciousness of those already adult a half-century ago. P61ya would have regarded it as pedantic to remind visitors that this was the reason that so many (at least 34) of the persons photographed had to leave their posts in Europe or, if then young enough, to be employed in several different countries. By exception, it is specified that Otto Blumenthal died in a concentration camp, also (page 86) that Alfred Pringsheim (Thomas Mann's father-in-law), being Jewish, "could not stay in Germany at that time." Commenting on Ludwig Bieberbach, he states (page 58) that his " n a m e today largely recalls his famous conjecture on sc-hlicht functions (. ~. proved in 1984 by Louis de Branges)" and forbears to remark on Bieberbach's infamy as a Nazi. Courses in the history of mathematics could ~use this album to interest students in the impact of society on the lives, attitudes, and structure in the mathematical community of that period. This would refer also to absences. All of the mathematicians portrayed are white; only six are female. This reflects the world in which P61ya lived. It was, as the student could easily discover, not one of which he approved. P61ya was always anxious to encourage w o m e n and minority students. In the 1950s, as Visiting Lecturer for the Mathematical Association of America, he took initiative to i n d u d e Historically Black Institutions in his itinerary. He had been
The "conductor" is Adolf Hurwitz, the violinists are Albert Einstein and Hurwitz's daughter Lisi.
P61ya expounding a point to a dubious Littlewood.
one of the first to sign a petition to the governing bodies of the American Mathematical Society and the Mathematical Association of America to reorganize (southern) meetings so that Black colleagues could participate on terms of full equality. (The specific issues that precipitated such petitions are discussed in Appendix 2, Black Mathematicians and their Works, Virginia Newell et al., editors, 'Dorrance & Co., Bryn Mawr, PA, 1980, unfortunately not easily accessible. Cf. also Science, August 10, 1951, pp. 161-162.) Students and the public generally view mathematics as remote from humanity, as something which merely is, not as something developed by human beings as part of humanity's quest to understand the world and to shape it, by people selected and shaped by society. Perhaps such an attractive book as this, by helping students and others see what m a n n e r of mortals mathematicians be, can help dispel the false and pedagogically dangerous view that humanity is passive with respect to mathematics, and humanize this component of humanity's work. Professor Alexanderson is to be thanked, as is Mrs. P61ya, for the labour of love that brings The P61ya Picture Album so attractively to general view. Department of Mathematics York University North York, Ontario M3J 1P3 Canada THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 2, 1989 7 1
by Robin Wilson*
One of the most remarkable sets of mathematical stamps is a set of ten Nicaraguan stamps issued in 1971, featuring "the ten mathematical formulas that
changed the face of the earth." Each stamp has a descriptive paragraph on the reverse side, and the featured equations are:
(Napier's law and Newton's law have been featured in previous columns.)
If you are interested in mathematical stamps, you are invited to subscribe to Philamath. Details may be obtained from the Secretary, Estelle A. Buccino, 135 Witherspoon Court, Athens, GA 30606 USA.
* C o l u m n editor's address: Faculty of M a t h e m a t i c s , T h e O p e n University, Milton K e y n e s M K 7 6 A A E n g l a n d
72 THE MATHEMATICALINTELLlGENCERVOL. I1, NO. 2 9 1989Springer-VerlagNew York
