Humanistic Mathematics Network Journal

Issue 15 Article 7

7-1-1997 Reminiscences of Paul Erdös (1913-1996) Melvin Henriksen Harvey Mudd College

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Recommended Citation Henriksen, Melvin (1997) "Reminiscences of Paul Erdös (1913-1996)," Humanistic Mathematics Network Journal: Iss. 15, Article 7. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss15/7

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Melvin Henriksen HalVey MuddCollege Claremont CA 91711 [email protected] Reprinted from theMathematicalAssociation of America website: htlp:llwww.maa. orgJfeatureslerdos.html I met Paul Erdos shortly after his 40th birthday inApril and perhaps because two older sisters died of scarlet 1953 a t Purdue University in West Lafayette, Indiana. fever shortly before he was born, his parents shielded Hewas already a living legend becau se of his substan­ him almost comp letely from the everyday problems tial contributions to the theory of nu mbers, the theory of life. For example, he never had to tie his own shoe­ of sets, what is now called discrete mathematics, as laces un til he was 14 years old, and never buttered we ll as to many other areas of mathematics. (For ex­ his own toast un til he was 21 years old in Cambridge, ample, although he had little interest in topology, his England . In return for the freedom to concentrate al­ name appears in most topology texts as the first per­ most exclusively on intellectual pursuits, he paid the son to give an exam ple of totally disconnected topo­ price ofno t learni ng the social skills tha t are expected logical space that is no t zero-dimensional.) I was a 26­ of all of us and usually acquired in childhood . year old instructor in my first year at Purd ue. Many of my colleagues knew him well. He had been a visit­ He became intern ationally famous at the age of 20 ing research associate at Purdue for a coup le of yea rs when he got a simple proof of a theo rem that was origi­ during World War II, and had visited so many uni­ nally conjectured by Bertrand and later proved by versities and atten ded so many conferences that he Tchebychev: For every positive integer n, there is a was well known to most of the others. Those that were primebetween n and 2n.Tchebychev's proof was qui te active in research admired his ma thematical accom­ hard! Erdos completed the requirements for the Ph.D. plishments, while others on the facu lty were amused at the University of Bud apest about a year later, but by his eccentricities. What I remember most clearly is had no chance of getting a position in Hungary be­ his announcement to everyone that "death begins at cause he was a Jew living und er a right wing dicta­ 40". torship allied with Nazi Germany. He spent some time at Cambridge University in 1935. There, his life as a I am not qualified to write a biography of Erd os, but wande ring mathematician began. In fact, he had vis­ some background seems necessa ry. There is an excel­ ited Cambridge three times the year before. He liked lently written and accurate obituary of him by Gina traveling and had no trouble working while doing so. Kolata in the Sept. 21, 1996 issu e of the New York He liked people, and excep t for those who could not Times, beginning on page 1. An interview conducted tolerate his ignoran ce of the social graces, they liked in 1979 which reveals much of his personality ap­ him. He tried his best to be pleasant to everyone an d peared in the volume Mathematical People edited by was generous in giving cred it and respect to his col­ D.J. Albers an d G.L. Alexanderson (Birkhauser 1985). laborators. The MathematicalAssociation ofAmerica (MAA) sells two videos of Erdos, and Ronald Graham, a long time I do not know when he first came to the United States, collabora tor, has edited together with [arik Nesetril but he spent the yea rs of World War II here, two of two volumes on his mathematical work and life. (Both them at Purdue. Nor can I give a list of the many uni­ volumes have been published by Springer-Verlag and versities he visited for any substantial length of time. were available in January 1997. They include a de­ By the time I met him, he had written joint papers tailed biographical article by Bella Bollobas.) with many mathematicians most of who m had estab­ lished research reputations before working w it h Erdos was born in Budapest in 1913 of p arents who Erdos. The only Erd os collaborator who worked with were Jewish intellectuals. His brilliance was evident him unwillingly was Atle Selbe rg. In the late 1940s, by the time he was three years old . For thi s reason, both of them, working independently, had obtained

Humanistic Mathematics NetworkJournal #15 13 "elementary" proofs (meaning: proofs that did not use sphere. complex analysis) of the prime number theorem. The theorem states that the number of primes less than or At that time, Leonard Gillman and I were trying to equal to (a positive real number) x is asymptotically study the structure of the residue class field s of rings equal to x/Iogfr ). This had been conjectured by Gauss of real-valued continuous fun ctions on a topological and Legendre based on empirical data, butit had only space modulo maximal ideals. We had learned quite been proved many years later, by two French math­ a bit about them, but had run into serious set-theo­ ematicians, Jacques Hadamard and Ch arles de la retic difficulties. Erdos had little interest in abstract Vallee Poussin (also working independently). Both algebra or topology, but was a master of set-theoretic proofs depended heavily on complex analysis. What constructions. Without bothering him with our moti­ Selberg and Erdos did in their "elementary" proofs vation for asking them, we asked him a series of ques­ was to avoid using complex analysis (the proofs were tions abou t set theory, which he managed to answer in no sense "easy"). In those pre-email days, the fast­ while we could not. est courier of mathematical news was Paul Erdos. He told anyone who would listen that Selberg and he had He was no t terribly interested when we supplied him dev ised an elementary proof of the prime number with the motivation, and I have often said tha t Erdo s theorem. never understood our paper; all he did was the hard part. This paper by Erdos,Gillman and Henriksenwas Almost every number theorist knew of Erdos, while published in the Annals of Mathema tics in 1955. With­ few had heard of the young Norwegian Selberg. So out any of us realizing it in advance, itbecame one of when the news traveled back to Selberg, it appeared the pioneering papers in nonstandard analysis, and that Erdos had claimed all the credit for himself. The was often credited to Erdos, et al. ensuing bitterness was not healed by the two of them Erdos got an offer allowing him to stay indefinitely at writing a joint paper. Selberg later published another Notre Dame on the same generous basis. His friends elementary proof on his own,and went on to a bril­ urged him to accept. "Paul", we said "how much liant mathematical career, eventuallybecoming a per­ longer can you keep up a life of being a traveling manent member of the Institute for Advanced Study mathematician?" (Little did we suspect that the an­ in Princeton, the Valhalla for mathematicians. Erdos swer was going to tum out to be "more than 40 years.") had been a visitor there earlier, but was not offered a Erdos thanked Ross, but turned him d own. As it member ship. Exactly whathappened is controversial turned out, he would not have been at Notre Dame to this day, and reading the articl e by Bollobas will the next year whatever his answer had been. shed more light on this matter than this short sum ­ mary can. The cold war was in full swing, the United States was in the grip of paranoia about communism, and many Erdos spent the academic year 1953-54 at the Univer­ regarded unc onventional behavior as evidence of dis­ sity of Notre Dame in South Bend, Indiana. Arnold loyalt y. Erdos had never applied for citizenship any­ Ross, the chairman of the Mathematics Depa rtment, where he lived, and had acquired Hungarian citizen­ had arranged for him to teach only one (advanced) ship only by accident of birth. He belonged to no po­ course, and supplied an assistant who could take over litical party, but had a fierce belief in the freedom of his class if he had the urge to travel to talk with a col­ ind ividuals as long as they did no harm to anyone laborator. Erdos had rejected organiz ed religion as a else. All countries who failed to follow this were clas­ young man, an d had be en persecuted in Roman sified as imp erialist and given a name that began with Catholic Hungary. So we teased him about working a small letter. For example, the Ll.S. was samland and at a Catholic institution. He said in all seriousness that the Soviet Union was joedom (after JosephStalin). He he liked being there very much, and especially en­ talked of an organization called the f.b.u-c-a combi­ joyed discussion s with the Dominicans. "The only nation of the EB.I and a .G.p.u (wh ich later became thing that bothers me," he said, "there are too many the K.G.B) and conjectured that their agents were of­ plus signs ." He came by bus to West Lafayette fairly ten interchanged. often for short periods because he had so many friends there and because he liked the mathematical atmo- In 1954, Erdos wanted to go the International Con-

14 Humanistic Mathematics Network Journal #15 gress of Mathematicians (held every four years), which government lost its fear of Erdos and gave him resi­ was to be in Amsterdam that August. As a non-citi­ dent alien status once more. He never had trouble zen leaving the Ll.S. with plans to return, he had to going in or out of the Ll .S. again. Erdos had lived from apply for a re-en try permit. After being in terviewed hand to mou th most of the time until the late 19505. by an INS agent in South Bend in early 1954, he re­ When the Russians sen t Spu tnik into orbit and the ceived a letter saying that re-en try would be denied if space race began, there was a vast increase in govern­ he left the Ll.S. He hired a lawyer and appealed only ment support of resea rch. This made it possible for to be turned down again. No reason was ever given, his many friend s and co-authors to give him research but his lawyer was permitted to examine a portion of stipends. This had little effect on his lifestyle. His suit­ Erdos' file an d found record ed the following facts: case was rarely more than half full, and he gave away most of his money to help talented young mathema­ • He corresponded with a Chinese nu mber theorist ticians or to offer cash prizes for solving research prob­ named Hua who had left his position at the Unl­ lems of varying degrees of difficulty. (The cash prizes versi ty of Illinois to return to (red) China in 1949. were not as costly as he ha d expected. The winners (Atyp ical Erdos letter would have begun: Dear Hua, would often fram e his checks wi thout cashing them, Let p be an odd prime.i.) Solving a $1000 problem would make you interna­ tionally famous, and being able to say tha t you solved • He had blundered onto a rad ar installation on any of his prize problems enhanced your reputation.) Long island in 1942 while d iscussing mathematics Around 1965, Casper Goffman concocted the idea of with two other non-citizens. an Erdos nu mber. If you had wri tten a joint paper with him, your Erdos number was 1. If you had written a • His mother worked for the Hungarian Academy joint paper with someone with Erdos number 1, your ofSciences, and had had to join the communist party Erdos number is 2, and so on inductively.There is now to hold her position. an Erd Os Number PrQject home page on the web wh ere you can see a list of all who have an Ed os nu m­ To Erdos, being denied the right to travel was like ber of 1 (there are 462 of us) and 2 (all 4566 of them, being denied the right to breathe , so he went to including Albert Einstein). All in all, Erdos wrote about Amsterdam anyway. He was confident that he could 1500 research papers, and 50 or so more will appear easily obtain a Dutch and an English visa. The Dutch after his death. gave him a visa good for only a few mon ths, and En­ gland would not let him come, likely becau se if they While we did no more joint research, we often met at chose to deport him, the only country obligated to conferences or when we were bo th visiting the same accept him was communist Hungary. By then, Erdos university. Sometimes I could hardly talk to him be­ was a member of the Hungarian Academy ofSciences, cause he was surrounded by mathematicians eager but he would go to Hungary only if his friend s could to ask him questions, but when I could, he inqu ired assure him tha t he would be permitted to leave. At about mutual friend s and asked about follow-up work this point, he swallowed his pride and obtained a on our paper and progress about solving the open passport from israel (note the punctuation ) which problems we had posed. While he devo ted his life to served to give him freedom to travel anywhere in mathematics, he was widely read in many area s and I western Europe. He was permitted to return to the almost always learned a great deal talking to him United States in the summer of 1959 on a temporary about many non-mathematical ideas. I saw him last visa to attend a month long conference on number in Budapest last Sept. 4. He attended the first half of a theory in Boulder, Colorado. He stopped at Purdue talk I gave about separate vs. joint contin uity. He on his way back to Europe to give a colloquium talk. apologized in advance about having to leave early When I picked him up at the airport, what struck me because he had made an appointment he could not first was that he had a suitcase! For many years, he break before he knew I would be speaking. Even then, traveled only with a small leather briefcase contain­ he made two helpful comments while present. Before ing a change of socks and underwear in addition to a rleft the Academy of Sciences, I stopped to say good­ wash-and-wear shirt, together with some paper and bye and saw him going over a paper with a young a few reprints. About a year later, the United States Hungarian mathematician . He died in Warsaw of a

HumanisticMathematics Network Joumal#15 15 heart atta ck on Sept. 20. He worked on what he loved installation and were apprehended by a guard who to do to the last! was convinced that he had caught a group of for­ eign spies. They were questioned closely by mili­ Erdos had a special vocabulary that he concocted and tary intelligence and released with a warning when used consistently in his speech. Some samples are: they promised never to d o such a thing again.

»Children are Epsilons - Actual version: The car was driven by Arthur - Women are Bosses Stone (an Englishman). Hochschild was su pposed - Men are Slaves to come.but did not becau se he had a date. They - Married Men have been Captured were speaking English because it was their on ly - Alcoholic Drinks are Poison western language understood by Kakutan i. The -God is The Supreme Fascist or SF guard was satisfied as soonas they presented proper «Music is Noise. identification, and they were visited individually and briefly a few days later by military intelligence Examples: agents.

I asked Barbara Piranian (President of the League of Erdos liked to tell many stories about himself.In par­ Women Voters in Ann Arbor, Michigan in the early ticular, when he grew older, he claimed to be two bil­ 1950s) "When will you bosses take the vote away from lion years old because when he was in high school, the slaves?" Answer "There is no need; we tell them he was taught tha t the earth was two and a half bil­ how to vote anyway." lion years old- bu t now we know it is four and a half billion years old. "Wine, women, and song" becomes "Poison, bosses, and noise". Because he seemed to be in a sta te of Brownian mo­ tion, itwas often hard to locate him at any given time. Erdos said that the SF had a Book containing elegant Erdos visited Claremont twice in the 1970s and could proofs of all the important theorems, and when a often be found at UCLA. For many years the way to mathematician worked very hard, the SF could be contact him was to call Ron Graham of Bell Labs on distracted long enough to allow her or him to take a the east coast, Paul Bateman of the University of Illi­ brief peek. Particularly elegant proofs were described nois, or Erns t Strauss at UCLA to find ou t where he as fit to be placed in the Book. was. Strauss died in 1983 and was replaced by Bruce Rothschild . Paul Bateman retired. Although Ron Gra­ There are many Erdos stories that we re embellished ham himself traveled a grea t deal, until the end he over the years and made more delightful than the was the person most likely to know of Erdos' where­ tru th. For example, consider the story about blunder­ abouts. ing into a rad ar installation in 1942: With Erdos' death we have lost one of the great math­ - Embellished version:Erdos, Hochschild (a Ger­ ematicians and free spirits of this century and it is hard man) and Kakutani (aJapanese)drove a car out onto to imagine that we will see anyone like him again. I Long Island and held an animated mathematical feel fortunate to ha ve had the privilege of knowing conversation in German. They walked onto a radar and working with him.

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