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 to 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 lớn 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 lớn 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 và accurate obituary ofhim by Gina Kolata in the Sept. 21, 1996 issue of the 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 and G.L. Alexanderson (Birkhauser 1985). The MathematicalAssociation of America (nofxfans.com) sells two videos of ErdÃ¶s, và RonaldGraham, a long time collaborator, has edited together with Jarik Nesetriltwo volumes on his mathematical work và life. (Both volumes have beenpublished by Springer-Verlag and 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, và 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 khổng lồ 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 and 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 khổng lồ be pleasant to everyone andwas generous in giving credit và respect lớn 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 khổng lồ 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 khổng lồ 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 khổng lồ 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 lớn 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, & 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, và especiallyenjoyed discussions with the Dominicans. "The only thing that bothers me",he said, "There are too many plus signs." He came by bus 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 and I were trying to 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 to 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 and 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 khổng lồ ErdÃ¶s, et al.
ErdÃ¶s got an offer allowing him lớn stay indefinitely at Notre Dame onthe same generous basis. His friends urged him 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 khổng lồ 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 khổng lồ no political party, but had a fierce belief in thefreedom of individuals as long as they did no harm khổng lồ anyone else. Allcountries who failed to lớn 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 và O.G.P.U (which later became the K.G.B) and conjectured thattheir agents were often interchanged.
In 1954, ErdÃ¶s wanted khổng lồ 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 lớn return, he had to 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 & appealed only khổng lồ be turned downagain. No reason was ever given, but his lawyer was permitted to lớn examine aportion of ErdÃ¶s" file & found recorded the following facts:He corresponded with a Chinese number theorist named Hua who had left his position at the University of Illinois lớn return to lớn (red) đài loan trung quốc in 1949. (A typical ErdÃ¶s letter would have begun: Dear Hua, Let phường 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, & had had to lớn join the communist các buổi party to hold her position. Lớn ErdÃ¶s, being denied the right khổng lồ travel was lượt thích being denied theright lớn breathe, so he went khổng lồ Amsterdam anyway. He was confident that hecould easily obtain a Dutch & 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 to 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 lớn Hungary only if hisfriends could assure him that he would be permitted lớn leave. At thispoint, he swallowed his pride và obtained a passport from israel (note thepunctuation) which served lớn give him freedom khổng lồ travel anywhere in westernEurope. He was permitted khổng lồ return to the United States in the summer of1959 on a temporary visa lớn attend a month long conference on number theoryin Boulder, Colorado. He stopped at Purdue on his way back 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 and underwear inaddition khổng lồ a wash-and-wear shirt, together with some paper và 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 to mouth most of the time until the late1950s. When the Russians sent Sputnik into orbit & the space race began,there was a vast increase in government support of research. This made itpossible for his many friends and co-authors to give him researchstipends. This had little effect on his lifestyle. His suitcase was rarelymore than half full, and he gave away most of his money lớn help talentedyoung mathematicians or to lớn 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 website where you can see a danh sách of all who havean Edos number of 1 (there are 462 of us) & 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 and asked aboutfollow-up work on our paper & progress about solving the mở cửa problems wehad posed. While he devoted his life lớn mathematics, he was widely read inmany areas & I almost always learned a great khuyến mãi talking 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 to 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 khổng lồ say good-bye & 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 vày to the last!
ErdÃ¶s had a special vocabulary that he concocted và 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, and 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 to lớn allow her or him to lớn take a briefpeek. Particularly elegant proofs were described as fit to lớn 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 car out onto Long Island và 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 và released with a warning when they promised never to dosuch a thing again.
Actual version: The car was driven by Arthur Stone (anEnglishman). Hochschild was supposed khổng lồ 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 và 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 lớn be two billion years old because when he was inhigh school, he was taught that the earth was two & a half billion yearsold--but now we know it is four and a half billion years old.
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Because he seemed to lớn be in a state of Brownian motion, it was often hard tolocate him at any given time. ErdÃ¶s visited Claremont twice in the1970s và could often be found at UCLA. For many years the way khổng lồ contacthim was to call 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 và was replaced by Bruce Rothschild. PaulBateman retired. Although Ron Graham himself traveled a great deal, untilthe over he was the person most likely lớn know of ErdÃ¶s"whereabouts.
With ErdÃ¶s" death we have lost one of the great mathematicians andfree spirits of this century và it is hard to lớn imagine that we will seeanyone lượt thích him again. I feel fortunate khổng lồ have had the privilege ofknowing & working with him.
Melvin HenriksenHarvey Mudd CollegeClaremont CA 91711