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The Legacy of Paul Erdős
Abstract
Paul Erdős (Erdős Pál, 1913-1996) was one of the most influential mathematicians of the
twentieth century. Erdős’ generation exhibited great talent due in part to the social and cultural changes between 1870 and 1930. In the 1930s, the Jewish Erdős left Hungary due to the worsening political climate, and eventually began a nomadic lifestyle in which he traveled across four continents collaborating with students, teachers, and professors. In popular books, he was described as a wandering ascetic searching only for mathematical truth. Those who knew him best, however, describe him as a compassionate, warm-hearted man. He was addicted to mathematics, but was also aware of current political events and cared deeply for his community of friends. In this paper, I describe the development of mathematical talent in Hungary near the turn of the century, give a biographical sketch of Erdős, and consider his collaborators’ perception of him in light of popular books. Finally, I describe how Hungary continues to nurture mathematical talent, through specialized programs such as the one at Fazekas Gymnasium (Fazekas High School). I also describe the filmmaking projects that I undertook to document these subjects during the past year. Craig Webster: The Legacy of Paul Erdo“s 1. Introduction
2.1 The Development
might be necessary to know a bit of a few languages to communicate. Routinely of Genius
In the summer of 2006, I learned about Tibor Frank’s “The Social Construction During the turn of the twentieth century, of Genius” to describe factors that may geographical and political situation has popular biographies My Brain is Open, special attention to those factors which within the culture. Ultimately, this may by Bruce Schechter, and The Man Who Loved Only Numbers, by Paul Hoffman. section 2.2, I offer a brief biographical of Hungary’s most influential scientists, These books, along with the film N is a Number, were the only sources about from his colleagues and friends as sources. Erdős’ life that were readily available. The books taught me that Erdős was In section 2.3, I consider this portrait in arguably the most prolific mathematician in history, but that he was unable to hold a driver’s license or make a sandwich. He (see Frank, p. 6 for a detailed list). Since not been translated into English. I discuss Toward the end of the nineteenth century, was a man who devoted his entire life to influential figures? This phenomenon, (most significantly the Jews) who had who carried all his belongings in a suit the best proofs are written in the Book, which the almighty Supreme Fascist keeps from humans; who called children epsilons create social and economic opportunities and music noise. In these books, Erdős’ stories that he and others tell about his School) attend specialized programs life. Yet, this year I have spoken to some of his closest friends and read accounts by his collaborators, and they emphasize this past school year, I visited classes and academic world was less suspicious. Many warm-hearted nature, social capabilities, and the Soviets. As a consequence, it has great mathematicians and scientists from that period were Jewish, including Johnny the individual to adapt and assimilate in Wigner, and Paul Turán (Frank 24-30).
To understand Erdős’ life, it is helpful these programs suggests that Hungary diversity of ethnic groups, and therefore has experienced cultural interaction from cultural framework preceding it. A burst the Germans, Slavs, Croats, Romas, contributed to the development of young and many others. This mix has created talent in Budapest. Both innovative rich lingual environments, in which it and traditional ways of teaching were Craig Webster: The Legacy of Paul Erdo“s imported through the Gymnasium Szegő, Pólya, Wigner, Teller, Szekeres, He gained experience and recognition of war in Russia for six years. During that program. Progressive schools encouraged Lovász, Sós, and Bollabas - to name the for his part in developing the hydrogen time, Paul’s mother brought him up. She personal relationships between students most well-known. When asked in 1985 bomb in the Manhattan project, alongside was extraordinarily proud of his abilities and teachers and experimented with about the “great flowering of Hungarian von Neumann. He is considered a in mathematics, languages, and history. new ways of teaching. Prodigies such as mathematics,” Erdős cited Kömal as an Hungarian-born American, since he lived He grew close to his mother, and they Johnny von Neumann encountered the interface between experienced professors more of his life in the U.S. than he did in remained uncommonly close throughout great math educator László Rátz when he and young students. He said that, “It Hungary. But both countries take credit his adult life (Sós 205-207, Simonovits was only ten years old. Immediately, Rátz is where you learned how to solve for his achievements. Erdős, as always, is and Sós ix-xii). asked Johnny’s father, Max, if he could problems,” but added that “such things much more difficult to pin down. He left tutor Johnny privately. Max agreed, and have more than one reason” (Oláh 3-29). Hungary when he was 21, but returned In his late teens, Erdős developed his often to his best friends in Budapest – mathematical talent by solving problems of charge. By the time Johnny was 17, he The intersection of social and cultural even during the Cold War. Erdős seemed from Kőmal. Through the journal, had been at the university for a few years, changes in Budapest helped to create a to return to Hungary more than any other many of the best mathematicians got to and many of the professors considered burst of talent in the emerging capital. place, especially toward the end of his know each other by face and name from him their peer. The encouragement Hungary’s history of foreign domination life. Therefore, he would be considered a distance. When these young students that Rátz gave was paramount to von gradually produced the side effect of Hungarian. It might be argued that moved to Budapest to study at Pázmány Neumann’s early success. The informal a creative problem solving ability in Hungary claims certain famous scientists Péter University (now Eötvös Loránd relationship between von Neumann and Hungarian culture. Toward the end of the who lived abroad much of their life. University), they created an informal Rátz exemplifies the degree to which nineteenth century, Hungary assimilated Nevertheless, the development of their and wandering group. The group, accelerated learning can help bright the Jewish community. Jews were drawn talent is unambiguous: Kömal, the Eötvös consisting of Erdős, Gallai, Grünwald, students. Rote learning was still common to academic circles, where conversion competitions, and personal instruction Klein, Szekeres, Turán, Vázsonyi, and in literature and language, but the to Christianity was less often seen as a nurtured top students. Nowhere were others, would take excursions through emergence of industrialization provided social maneuver. New forms of education these talents developed more acutely the city and trips through the park, impetus for innovation in the sciences imported from Germany propelled than in Paul Erdős, one of the greatest discussing mathematics, politics, and talented students. Kömal and the Eötvös mathematicians of the twentieth century. In 1894, the secondary school journal minds, while increased cultural value 2.2 Biographical Sketch
Középiskolai Matematikai Lapok (Kömal) was placed on scientific achievement and the Eötvös math competitions due to industrialization. The result was a of Paul Erdős
began. The journal and the competitions flourishing of mathematical and scientific intended to stimulate young students talent. death of his sisters had a great effect on of the journal contained problems that Yet as Miklós Dezső, the current head of students were encouraged to solve. They the Rényi Institute points out, it could be systems. Eventually, they married. Erdős physics teachers, and they recognized his would send in their answers, and correct argued that many famous “Hungarian” talents immediately. Before the age of five, solutions were published, sometimes scientists owe their achievements as he could multiply three digit numbers in with a picture of the solver. Indeed, many much to the United States or England great Hungarian mathematicians and as to Hungary. For instance, Edward scientists had their photos printed in Teller left Hungary at the age of 18 and Paul’s father was captured as a prisoner Kömal, including Erdős, von Neumann, lived most of his life in the United States. Craig Webster: The Legacy of Paul Erdo“s of the Hungarian Dénes Kőnig. A few century does one mathematician work impression that Erdős was some sort of to anything else. He writes, “[Erdős] years later he gave an elementary proof seriously in so many fields (Halász et. surreal genius, but in fact he was a solid renounced physical pleasure and material of Chebeychev’s theorem: that there is al. 55-60). His success was partially due person, with likes and dislikes, human possessions for an ascetic contemplative always a prime between n and 2n (Aigner to his ability to remain open to others’ frailties, and many lovable qualities.” life, a life devoted to a single narrow and Ziegner 7-13). He finished his Ph.D. ideas and problems. He could strike up Much of the biographical information mission: uncovering mathematical truth” at 21 and moved to Manchester where mathematical friendships where ever available to the public is written by (Hoffman 25). In talking with Professor he studied for four years. He kept in he went with students, teachers, and outsiders to his life. Paul Erdős and his Miklós Simonovits, a collaborator good contact with friends and family in professors. Rather than build theorems, Mathematics, however, is written by those of Erdős and graph theorist at the Budapest, visiting three times a year and he attacked problems, searching for who knew him best, although the two mathematical research center in Budapest continuing to collaborate with Turán. their underlying structure. He was volume set is hard to obtain and is steeped (called the Rényi Institute), I learned that Their feverish correspondence between renowned for posing problems suited in mathematical jargon. The first 100 Erdős enjoyed fine dining, and would 1934 and 1940 gives a sense of Erdős’ style to the individual with whom he worked. pages of volume one fill out the existing go out with his friends when he had in mathematics and life. His conjectures, In his lectures, he announced general impression of Erdős by adding the stories the chance. In Simonovits’ estimation, proofs, and problems are interspersed problems and offered cash rewards for and opinions of his collaborators. The he was not an ascetic. In fact, he was with his political and personal concerns solutions. The real reward, however, lay common theme is that the caring nature well-connected to the world around him. about the growing anti-Semitic sentiment in discovery of the groundbreaking results and generous attitude of Erdős stood out Erdős kept up with current political events that solutions often elicited. During his and was well-versed in world history. The Erdős returned to Hungary in 1938. The collaborators and produce nearly 1500 The construction of Erdős’ popular recalls that “[he] had an abiding interest political climate was so bad that it forced papers. image has not relied on a combination in political issues in countries all over the Erdős to leave to the United States. He of perspectives from people that knew world.” In 1980, while at a conference him well. The books My Brain is Open in Europe, Erdős stayed up all night to 2.3 Remembering
and The Man Who Loved Only Numbers, watch the U.S. Presidential election, fellowship was not renewed. This is when His Legacy
and the film, N is a Number, were created while Bateman – the American – went to he began his “nomadic” lifestyle. During by non-mathematicians outside of his bed. The next morning, Erdős knocked the next fifty years, he traveled all across Popular accounts of Erdős concentrate community. These works determine on Bateman’s door and said, “Reagan is the world, giving lectures, participating on his quirky anecdotes and witty how the public perceives Erdős. In many your president” (Halász et. al. 9-13). in conferences and workshops, and remarks. He would recall funny stories collaborating with other mathematicians in interviews, so much that they became excite the public about mathematics and Erdős was also devoted to his friends. canonized. For instance, in almost every entertain the reader with stories about Both RK Guy and Joel Spencer mention detailed account of him, he describes the Erdős’ character. They describe his that he paid attention to the names of When Erdős was 50 (in 1963), situation of when he was 21 in England amazing mathematical ability in a way their family members and cared about celebratory and was faced for the first time to make his that the layperson can understand. These them on a personal level. During the paper that surveyed some of Erdős’ own sandwich. This anecdote illustrates works also recount the events of Erdős’ time of Soviet rule in Hungary, János accomplishments. A striking fact about his his mother’s early pampering, but perhaps life, exploring the difficulties that he Surányi recalls how Erdős would bring work was that he could make significant misleads the public to believe that his encourntered. However, there are certain back special medications from the West progress in many areas: number theory, eccentricities were his most exceptional points that could be represented more to friends and their families behind the probability theory, graph theory and attribute. There is no doubt that Erdős’ Iron Curtain (Halasz et. al. 21-25, 47-51) asymptotical combinatorics, constructive life was much different than the norm, Indeed, Erdős took special care of those theory of functions, set theory, and but popular accounts tend to exaggerate In The Man Who Loved Only Numbers, close to him. set-theoretical topology, theory of series, these details. His collaborator, Paul and geometry. Rarely in the twentieth Bateman has said, “Some writers give the his entire life to math and was oblivious Craig Webster: The Legacy of Paul Erdo“s In N is a Number, Erdős talks about Once I learned about the views of formal aspect of Hungarian culture. Most from Magyar Television. Ultimately, I his aversion to sexual stimulation and Erdős’ collaborators, I wanted to create of the film is a close up of Erdős as he talks would like to make the film available for explains that he never had a serious a documentary to explain the situation. and tells stories. It is a sit down discussion, distribution. Since the film is only 50 relationship with a woman. This raises But after speaking with some of his and there is little attempt to spice it up. minutes, there would need to be more an ethical question. Is it appropriate to former colleagues at the Rényi Institute, I Consequently, the film may require some material on the DVD. Kardos created discuss the sex-life of a mathematician in a experienced resistance. Sós, Erdős’ closest patience, but in return we see Erdős 70 other films that would be new to the documentary film? What if he talks about living friend, told me that in her writings responding to questions spontaneously. English-speaking world. Vera Sós also it on camera? Naturally, humans want to she had said everything that she wanted While many accounts of him focus on indicated that there are other Hungarian know the intimate details of others’ lives. to say about Paul. Twelve years after what happened in his life, this account also interviews of Erdős yet to be translated. But it seems that this detail may distract his death, this story did not need to be focuses on what is happening. It depends Thus, there are many possibilities to fill revisited by another foreigner who didn’t less on Erdős retelling stories and more out this DVD, and even create a series on Erdős expressing his opinion. In one about Hungarian-born mathematicians During Erdős’ life, he used coffee, I understood the wisdom of this attitude instance, Erdős discusses the possibilities or based on Kardos’ work. caffeine pills, and Benzedrine to stay and decided that in fact creating my awake. In The Man Who Loved Only documentary might not help the situation. This topic is relevant since we are living The project would require backing Numbers, Hoffman introduces this by While talking to Miklós Simonovits, he in the future that Erdős talks about. The to distribute. I have written the NSF writing, “At five foot six, 130 pounds, told me about an interview of Erdős shot film is also unique because it features with regard to their Informal Science Erdős had the wizened, cadaverous face of by a Hungarian filmmaker named Kardos Erdős speaking in his mother tongue. It is Education program. This program a drug addict, but friends insist that he was István (1942-2006). I had not heard of it. the only documentation that I am aware in provided partial funding for N is a Number frail and gaunt before he started taking Did this interview exist in English? which Erdős speaks Hungarian. Despite and its values seem closely aligned to amphetamines” (Hoffman 7). Erdős took its cultural and historical interest, the the benefits of releasing Kardos’ films. pills on and off throughout his life, and After speaking with the Magyar film could be criticized in a few ways. For The Templeton Foundation held a it was not something that he flaunted. Television archive and the National instance, the questions asked are at times conference in October of 2007 about Consequently, the discussion may deserve Széchenyi Library, I learned that the general or speculative. Some may argue Hungarian mathematical talent around more sensitivity when introduced. When 50 minute piece on Erdős was not in that it is not “new” enough, since some the turn of the century. I have seen the Hoffman asked him, Erdős said, “You English. Moreover, Kardos István had information in the film could be found video documentation of lectures from shouldn’t have mentioned the stuff about created 70 similar pieces about influential in existing materials. But the Hungarian the conference via webcast. Perhaps this Benzedrine. It’s not that you got it wrong. Hungarian mathematicians and scientists, nature of the production adds a candid footage could be combined to create It’s just that I don’t want kids who are including 13 Nobel prize winners. portrait of Uncle Paul that cannot be a documentary about mathematical thinking about going into mathematics Professor Simonovits gave me a copy of education, and then released with some of to think that they have to take drugs to this compilation of interviews of Erdős During my time in Hungary, I have been Kardos’ films. This work would provide succeed” (Hoffman 268). In N is a Number, – one shot in 1978, the other in 1988. In working to make this film available to a Hungarian perspective to the existing his use of Benzedrine is not included and the film, Kardos and another reporter English speakers. The most effective material on Erdős, and therefore has in My Brain is Open, it is mentioned in a ask Erdős questions about his life and translation, I believe, is subtitles. Subtitles significant cultural value. more discreet way. The way in which mathematics. He recounts his experience Erdős’ use of Benzedrine is discussed can as a young man, reflects upon the meaning would allow the viewer to hear Erdős’ shade the perception of his character. of his work, discusses his philosophy on There is a fine line between telling the collaborating, and gives his perspective on complete truth and exploiting the details the future of computers. The austere set, long pauses, and slow pacing reflect the help of the Fulbright commission, I am working to gain copyright permission Craig Webster: The Legacy of Paul Erdo“s Sándor, sat patiently at the front of the in the U.S. for significant periods of On April 4th, 2008, this year’s competition 3. Math in the
time, and he talked about the differences took place, and I was there to film the Secondary School
between Hungary and the U.S. in event. The beginning was relaxed and full math education. In Hungary, there is a of playful excitement. As time ran out, the In the 1950s, the International through the window. Twenty minutes sense of community spirit, and talented pressure and tension increased. Frantic Mathematics Olympiad (IMO) was later, Sándor strolled the room quietly, students are encouraged to go deeper participants speed walked to the judges, introduced, bringing together the top peeking at his students’ work and helping into mathematics, instead of speed ahead. delivering their solutions. The event talent from six Soviet countries including when necessary. Sándor’s style is to allow Whereas in the U.S., the community is brought with it a real sense of enjoyment. Hungary. Now, more than ninety his students to grapple with the problems, larger and less closely connected, and the Fazekas brought the NYU computer countries participate. Students from each in order to encourage independent math teachers are not trained especially scientist Joel Spencer, as a guest lecturer. country take two exams. They have four thinking. This practice is designed to for their job. I hope that the documentary He gave his lecture in Hungarian to a and a half hours to solve three difficult prepare them for working independently will bring out these points, so that the crowd of over 90. Most of the audience problems. Hungary faces giants such at the IMO. They need to develop a U.S. can learn from Hungary’s example. as Russia, China, India, and the U.S. self-driven, critical thinking capacity. attentively as Spencer talked about several Nevertheless, Hungary repeatedly placed Eventually, Sándor would ask his students The sense of a small, active community problems involving asymptotes and graph within the top ranks of the competition to share their solutions on the chalkboard. was evident in the “Fazekas Napok” theory. It seemed that everyone in the and many Hungarian students walk away The students explained their discoveries, (Fazekas Days), an informal math audience wanted to attend the two hour competition involving over 100 students. session, as classes had been cancelled for Teams of seven are given 21 questions, the occasion.
Fazekas Gymnasium, located in the Hungarian mathematics education is a and they have 90 minutes to solve as many 8th District in Budapest, is one of the century old tradition, which continues problems as they can. A correct answer I may have stumbled upon yet another secondary school which has specialized today. More than 15,000 students each wins the team between 20 to 40 points. reason that high level mathematics is math programs for talented students. year participate in the competitions The team loses 10 points for an incorrect so good in Hungary. Not only is the Toward the end of February, I began hosted by the Bolyai Society (including answer. Pastries and soft drinks are given community small, but its members to attend the informal preparations the former Eötvös competition). Many to the teams, who sit at circular tables and are friendly and helpful. Professor for the IMO. I could understand only students still greet the new publication of work intensely on the problems. Once a Simonovits introduced me to the head bits of Hungarian, but could see how Kőmal with excitement and send answers team has a solution, the team’s runner coach of Hungary’s IMO team – József comfortable these 17 and 18 year-olds each month. The Fazekas program is delivers it to a cadre of judges, mostly Pelikán – who then introduced me to were when they solved problems at especially active, and similar programs first or second year students from the Sándor Dobos. Teachers such as Sándor the chalkboard. I had rarely seen such exist throughout Hungary. I am currently university (many of whom went to Fazekas are relentless in their generosity. He has confidence at my high school in the creating a documentary film about the year before). Each answer is a single introduced me to other teachers and U.S., even when we solved much easier secondary school math education, paying integer between 0 and 10,000 written on a educators, who in turn have invited me for problems. Of course, the Hungarian math particular attention to the development small slip of paper. The judges grade them an interview or into their classrooms to students who participated in the voluntary of bright students. I interviewed Dobos on the spot. The scoreboard changes in observe and film. Each person that I talk to Friday afternoon preparation were no Sándor and some of his students, and real-time, as teams turn in answers, and speaks highly of another, and this friendly ordinary students. They were some of the filmed both advanced and regular-track the teams battle each other until the very spirit is conducive to collaboration. As last minute. The competition encourages the Fazekas teacher Pátaki János told collaboration and cross-checking between me, the United States may be good at After giving a few problems to the researcher in combinatorics and a students. Regardless of whether they win, creating strong mathematical institutions students, the teacher (and deputy coach member of the first ever specialized math of the Hungarian IMO team), Dobos program at Fazekas in 1962. He has lived Craig Webster: The Legacy of Paul Erdo“s how to create a community spirit, where Paul Erdős was one of the greatest math students. The tight-knit math Bibliography the best students are exposed to their mathematicians of the twentieth century. community and tradition of Kőmal and I have used quotes and ideas from interviews with Laszlo Lozász, peers so that they can learn and develop His character has been recorded in other competitions bring bright students Miklos Simonovits, and Joel Spencer, which were conducted together. Given this trend, the legacy of popular biographies by American authors together throughout their development. in 2008 at Eötvös Loránd University, the Rényi Insitute and developing talented youth in mathematics who accentuate funny stories about him Perhaps more attention should be paid Fazekas Gymnasium. will continue, independent of political more than his genuine kindness. While he problems or lack of funding. The was addicted to math, he also cared about in mathematics. Teachers such as Sándor Albers, DJ and Alexanderson, GJ. “Interview with P. Erdős.” Hungarian math community remains the people around him and paid attention Dobos encourage women to participate, Mathematical People, Birkhauser, Boston: 1985. strong and healthy despite the radical to history and politics. When I spoke with changes of the past century, and appears his former collaborators, many approved mathematics as a very “womanly” subject. Aigner, Martin and Ziegler, Günter. Proofs from the Book. These social morays may change, but the Springer, New York: 2004. legacy of outstanding mathematical talent of Erdős and his contemporaries will Csicery, George. N is a Number. Zala Films, 1993.
4. Conclusion
István shows Erdős talking in Hungarian about his life. I hope that a translation Frank, Tibor. “The Social Construction of Genius,” from the During this past year I learned about of the film would provide a fresh, new Conference “Budapest: the Golden Years,” Princeton, 2007.
the golden age of Hungary. The sharp increase of talent in math and science perspective that is less adulterated than Halász, G., Lovász, L., Simonovits, M. and Sós, V.T. (Eds.) Paul was in large part due to a confluence of only documentation (that I am aware Erdős and his Mathematics, Springer: Budapest, 2002.
cultural and social developments linked to the newly emerging capital Budapest, of) in which Erdős speaks in his mother Hoffman, Paul. The Man Who Loved Only Numbers. New York, innovative educational practices imported the calm and formal nature of Hungarian from Germany, and the assimilation of the Jewish community. The Kömal journal culture. Moreover, the Kardos interview Oláh, V. “Centennial Issue of Kömal,” János Bolyai and the Eötvös competitions encouraged giving a chance for Erdős to express his students to solve challenging questions and rewarded young talent. Due to the opinion about modern issues such as how Schechter, Bruce. My Brain is Open. Touchstone, New York: political climate in Hungary around I am working on gaining copyright World War II, many of these scientists were forced to leave for the United States. permission to add English subtitles, while I create a subtitle track. I am also finding a Simonovits, M. and Sós, V. T. Recent Trends in Combinatorics. It could be said that some of the great way for the film to be distributed. Forward. “Paul Erdős: The Man and the Mathematician,” CUP, scientists are as much American as they are Hungarian, since their most celebrated work occurred in the U.S. But the debt Simultaneously, I am creating a Sós, V. T. “Paul Erdős”, Aequationes Math. 54 (1997) 205 – 220.
they owe to Hungary is undeniable. It is clear that the innovative education and secondary school education in Hungary exciting competitions in Hungary helped that should be finished in May. I have to develop the talent of many young observed classes, interviewed students, students, some of which went on to do teachers, and professors, and filmed great works. This legacy continues today competitions. The end product will through specialized secondary school show some of the keys to Hungary’s programs such as Fazekas.

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