I met Paul ErdÃ¶s shortly after his 40th birthday in April 1953 atPurdue University in West Lafayette, Indiana. He was already a livinglegend because of his substantial contributions lớn the theory of numbers,the theory of sets, what is now called discrete mathematics, as well as tomany other areas of mathematics. (For example, although he had littleinterest in topology, his name appears in most topology texts as the firstperson khổng lồ give an example of totally disconnected topological space that isnot zero-dimensional.) I was a 26-year old instructor in my first year atPurdue. Many of my colleagues knew him well. He had been a visitingresearch associate at Purdue for a couple of years during World War II, andhad visited so many universities và attended so many conferences that hewas well known to most of the others. Those that were active in researchadmired his mathematical accomplishments, while others on the faculty wereamused by his eccentricities. What I remember most clearly is hisannouncement khổng lồ everyone that "death begins at 40".
I am not qualified khổng lồ write a biography of ErdÃ¶s, but some backgroundseems necessary. There is an excellently written and accurate obituary ofhim by Gina Kolata in the Sept. 21, 1996 issue of the thủ đô new york Times,beginning on page 1. An interview conducted in 1979 which reveals much ofhis personality appeared in the volume Mathematical People edited byD.J. Albers & G.L. Alexanderson (Birkhauser 1985). The MathematicalAssociation of America (slovenija-expo2000.com) sells two videos of ErdÃ¶s, và RonaldGraham, a long time collaborator, has edited together with Jarik Nesetriltwo volumes on his mathematical work and life. (Both volumes have beenpublished by Springer-Verlag & were available in January 1997. Theyinclude a detailed biographical article by Bella Bollobas.)
ErdÃ¶s was born in Budapest in 1913 of parents who were Jewishintellectuals. His brilliance was evident by the time he was three yearsold. For this reason, & perhaps because two older sisters died of scarletfever shortly before he was born, his parents shielded him almostcompletely from the everyday problems of life. For example, he never had totie his own shoelaces until he was 14 years old, và never buttered his owntoast until he was 21 years old in Cambridge, England. In return for thefreedom to concentrate almost exclusively on intellectual pursuits, he paidthe price of not learning the social skills that are expected of all of usand usually acquired in childhood.
He became internationally famous at the age of trăng tròn when he got asimple proof of a theorem that was originally conjectured byBertrand and later proved by Tchebychev: For every positive integern, there is a prime between n và 2n. Tchebychev"sproof was quite hard! ErdÃ¶s completed the requirements for the Ph.D.at the University of Budapest about a year later, but had no chance ofgetting a position in Hungary because he was a Jew living under a rightwing dictatorship allied with Nazi Germany. He spent some time atCambridge University in 1935. There, his life as a wandering mathematicianbegan. In fact, he had visited Cambridge three times the year before. Heliked traveling and had no trouble working while doing so. He liked people,and except for those who could not tolerate his ignorance of the socialgraces, they liked him. He tried his best lớn be pleasant lớn everyone andwas generous in giving credit và respect khổng lồ his collaborators.
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Almost every number theorist knew of ErdÃ¶s, while few had heard of theyoung Norwegian Selberg. So when the news traveled back to lớn Selberg, itappeared that ErdÃ¶s had claimed all the credit for himself. Theensuing bitterness was not healed by the two of them writing a jointpaper. Selberg later published another elementary proof on his own, andwent on lớn a brilliant mathematical career, eventually becoming a permanentmember of the Institute for Advanced Study in Princeton, the Valhalla formathematicians. ErdÃ¶s had been a visitor there earlier, but was notoffered a membership. Exactly what happened is controversial to lớn this day,and reading the article by Bollobas will shed more light on this matterthan this short summary can.
ErdÃ¶s spent the academic year 1953-54 at the University of Notre Damein South Bend, Indiana. Arnold Ross, the chairman of the MathematicsDepartment, had arranged for him khổng lồ teach only one (advanced) course, andsupplied an assistant who could take over his class if he had the urge totravel khổng lồ talk with a collaborator. ErdÃ¶s had rejected organizedreligion as a young man, and had been persecuted in Roman CatholicHungary. So we teased him about working at a Catholic institution. He saidin all seriousness that he liked being there very much, and especiallyenjoyed discussions with the Dominicans. "The only thing that bothers me",he said, "There are too many plus signs." He came by bus to lớn West Lafayettefairly often for short periods because he had so many friends there andbecause he liked the mathematical atmosphere.
At that time, Leonard Gillman và I were trying lớn study the structure ofthe residue class fields of rings of real-valued continuous functions on atopological space modulo maximal ideals. We had learned quite a bit aboutthem, but had run into serious set-theoretic difficulties. ErdÃ¶s hadlittle interest in abstract algebra or topology, but was a master ofset-theoretic constructions. Without bothering him with our motivation forasking them, we asked him a series of questions about phối theory, which hemanaged lớn answer while we could not.
He was not terribly interested when we supplied him with the motivation,and I have often said that ErdÃ¶s never understood our paper; all hedid was the hard part. This paper by ErdÃ¶s, Gillman và Henriksen waspublished in the Annals of Mathematics in 1955. Without any of us realizingit in advance, it became one of the pioneering papers in nonstandardanalysis, & was often credited to lớn ErdÃ¶s, et al.
ErdÃ¶s got an offer allowing him to lớn stay indefinitely at Notre Dame onthe same generous basis. His friends urged him to lớn accept. "Paul", we said"how much longer can you keep up a life of being a travelingmathematician?" (Little did we suspect that the answer was going to lớn turnout lớn be "more than 40 years.") ErdÃ¶s thanked Ross, but turned himdown. As it turned out, he would not have been at Notre Dame the next yearwhatever his answer had been.
The cold war was in full swing, the United States were in the grip ofparanoia about communism, & many regarded unconventional behavior asevidence of disloyalty. ErdÃ¶s had never applied for citizenshipanywhere he lived, and had acquired Hungarian citizenship only by accidentof birth. He belonged lớn no political party, but had a fierce belief in thefreedom of individuals as long as they did no harm to anyone else. Allcountries who failed to follow this were classified as imperialist andgiven a name that began with a small letter. For example, the U.S. Wassamland và the Soviet Union was joedom (after JosephStalin). He talked of an organization called the f.b.u--a combination ofthe F.B.I and O.G.P.U (which later became the K.G.B) & conjectured thattheir agents were often interchanged.
In 1954, ErdÃ¶s wanted lớn go theInternational Congress of Mathematicians (held every four years), which wasto be in Amsterdam that August. As a non-citizen leaving the U.S. Withplans to lớn return, he had khổng lồ apply for are-entry permit. After being interviewed by an INS agent in South Bend inearly 1954, he received a letter saying that re-entry would be denied if heleft the U.S. He hired a lawyer và appealed only lớn be turned downagain. No reason was ever given, but his lawyer was permitted to lớn examine aportion of ErdÃ¶s" file và found recorded the following facts:He corresponded with a Chinese number theorist named Hua who had left his position at the University of Illinois to lớn return khổng lồ (red) đài loan trung quốc in 1949. (A typical ErdÃ¶s letter would have begun: Dear Hua, Let p be an odd prime ...) He had blundered onto a radar installation on Long island in 1942 while discussing mathematics with two other non-citizens. His mother worked for the Hungarian Academy of Sciences, và had had khổng lồ join the communist tiệc ngọt to hold her position. Khổng lồ ErdÃ¶s, being denied the right khổng lồ travel was lượt thích being denied theright to breathe, so he went khổng lồ Amsterdam anyway. He was confident that hecould easily obtain a Dutch và an English visa. The Dutch gave him a visagood for only a few months, và England would not let him come, likelybecause if they chose lớn deport him, the only country obligated to accepthim was communist Hungary. By then, ErdÃ¶s was a member of theHungarian Academy of Sciences, but he would go to lớn Hungary only if hisfriends could assure him that he would be permitted to leave. At thispoint, he swallowed his pride & obtained a passport from israel (note thepunctuation) which served khổng lồ give him freedom lớn travel anywhere in westernEurope. He was permitted khổng lồ return khổng lồ the United States in the summer of1959 on a temporary visa khổng lồ attend a month long conference on number theoryin Boulder, Colorado. He stopped at Purdue on his way back to lớn Europe togive a colloquium talk. When I picked him up at the airport, what struck mefirst was that he had a suitcase! For many years, he traveled only with asmall leather briefcase containing a change of socks & underwear inaddition to lớn a wash-and-wear shirt, together with some paper & a fewreprints. About a year later, the United States government lost its fear ofErdÃ¶s và gave him resident alien status once more. He never hadtrouble going in or out of the U.S. Again.
ErdÃ¶s had lived from hand khổng lồ mouth most of the time until the late1950s. When the Russians sent Sputnik into orbit and the space race began,there was a vast increase in government tư vấn of research. This made itpossible for his many friends and co-authors khổng lồ give him researchstipends. This had little effect on his lifestyle. His suitcase was rarelymore than half full, & he gave away most of his money lớn help talentedyoung mathematicians or khổng lồ offer cash prizes for solving research problemsof varying degrees of difficulty. (The cash prizes were not as costly as hehad expected. The winners would often frame his checks without cashingthem. Solving a $1000 problem would make you internationally famous, andbeing able lớn say that you solved any of his prize problems enhanced yourreputation.) Around 1965, Casper Goffman concocted the idea of anErdÃ¶s number. If you had written a joint paper with him, yourErdÃ¶s number was 1. If you had written a joint paper with someonewith ErdÃ¶s number 1, your ErdÃ¶s number is 2, và so oninductively. There is now an ErdÃ¶s NumberProject trang chủ page on the web where you can see a list of all who havean Edos number of 1 (there are 462 of us) and 2 (all 4566 of them,including Albert Einstein). All in all, ErdÃ¶s wrote about 1500research papers, and 50 or so more will appear after his death.
While we did no more joint research, we often met at conferences or when wewere both visiting the same university. Sometimes I could hardly talk tohim because he was surrounded by mathematicians eager to ask him questions,but when I could, he inquired about mutual friends & asked aboutfollow-up work on our paper & progress about solving the xuất hiện problems wehad posed. While he devoted his life lớn mathematics, he was widely read inmany areas & I almost always learned a great giảm giá khuyến mãi talking to lớn him aboutmany non-mathematical ideas. I saw him last in Budapest last Sept. 4. Heattended the first half of a talk I gave about separate vs. Jointcontinuity. He apologized in advance about having lớn leave early because hehad made an appointment he could not break before he knew I would bespeaking. Even then, he made two helpful comments while present. Before Ileft the Academy of Sciences, I stopped to say good-bye and saw him goingover a paper with a young Hungarian mathematician. He died in Warsaw of aheart attack on Sept. 20. He worked on what he loved to bởi vì to the last!
ErdÃ¶s had a special vocabulary that he concocted & usedconsistently in his speech. Some samples are:Children are Epsilons Women are Bosses Men are Slaves Married Men have been Captured Alcoholic Drinks are Poison God is The Supreme Fascist or SF Music is Noise.Examples:
I asked Louise Piranian (President of the League of Women Votersin Ann Arbor, Michigan in the early 1950s) "When will you bosses take thevote away from the slaves?" Answer :"There is no need; we tell them how tovote anyway."
"Wine, women, and song" becomes "Poison, bosses, & noise".
ErdÃ¶s said that the SF had a Book containing elegant proofs of all theimportant theorems, và when a mathematician worked very hard, the SF couldbe distracted long enough lớn allow her or him lớn take a briefpeek. Particularly elegant proofs were described as fit to be placed in theBook.
There are many ErdÃ¶s stories that were embellished over the years andmade more delightful than the truth. For example, consider the story aboutblundering into a radar installation in 1942:Embellished version: ErdÃ¶s, Hochschild (a German) andKakutani (a Japanese) drove a oto out onto Long Island and held an animatedmathematical conversation in German. They walked onto a radar installationand were apprehended by a guard who was convinced that he had caught agroup of foreign spies. They were questioned closely by militaryintelligence & released with a warning when they promised never khổng lồ dosuch a thing again.
Actual version: The car was driven by Arthur Stone (anEnglishman). Hochschild was supposed to come, but did notbecause he had a date. They were speaking English because it wastheir only western language understood by Kakutani. The guardwas satisfied as soon as they presented proper identification,and they were visited individually & briefly a few days laterby military intelligence agents. ErdÃ¶s liked to tell many stories about himself. In particular, when hegrew older, he claimed to be two billion years old because when he was inhigh school, he was taught that the earth was two và a half billion yearsold--but now we know it is four & a half billion years old.
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Because he seemed khổng lồ be in a state of Brownian motion, it was often hard tolocate him at any given time. ErdÃ¶s visited Claremont twice in the1970s and could often be found at UCLA. For many years the way lớn contacthim was to điện thoại tư vấn Ron Graham of Bell Labs on the east coast, Paul Bateman ofthe University of Illinois, or Ernst Strauss at UCLA to find out where hewas. Strauss died in 1983 and was replaced by Bruce Rothschild. PaulBateman retired. Although Ron Graham himself traveled a great deal, untilthe end he was the person most likely khổng lồ know of ErdÃ¶s"whereabouts.
With ErdÃ¶s" death we have lost one of the great mathematicians andfree spirits of this century & it is hard lớn imagine that we will seeanyone lượt thích him again. I feel fortunate lớn have had the privilege ofknowing và working with him.
Melvin HenriksenHarvey Mudd CollegeClaremont CA 91711