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ABOUT PAUL ERDÖS (An overview of
the life of Paul Erdös. This is not an excerpt from the book, FOR OVER half a century, early in the morning or in the middle of the night, mathematicians in Budapest or Berkeley, Prague or Sydney have been summoned from their multi-dimensional dreams by a knock at the door. Their unexpected guest was a short, smiling man wearing thick glasses and an old suit. In one hand he held a small suitcase containing everything he owned, in the other a shopping bag stuffed with papers. It was Paul Erdös, one of the greatest mathematicians of the twentieth century, a man who lived in the space of Platonic Ideals and infinite beauty, who called no place on Earth home. Never one to waste time on formalities with work to be done, Erdös would announce to his host: "My brain is open!" For the next few days, brains open, Erdös and his host, with other mathematicians recruited as needed, would be off on a mathematical journey of problem, conjecture, theorem and proof. The goal of their journey was nothing less than Truth and Beauty. "If numbers aren't beautiful, I don't know what is," Erdös once remarked. While the pursuit of mathematical beauty was Erdös's only goal, his ideas inevitably have found practical applications. One of the small ironies of Erdös's life is that, although he never owned or used a computer, mathematics that he invented is the basis for modern computer science; although he never had a secret, his mathematics is used by those who invent secret codes. Throughout his life Erdös was drawn to areas of mathematics that did not require excessive technical knowledge; many problems that Erdös stated can be easily understood by high school students. The solutions he found often were pure wizardry. He was like a master chef who enjoyed demonstrating his mastery by whipping up magnificent creations from the humblest ingredients. One of Erdös's important books has the beguilingly unassuming title, The Art of Counting. Before long Erdös, who fueled his mathematical creativity with Benzedrine and coffee, would exhaust his hosts. Brain still wide open, he would take off to mathematical congress or visit another colleague, logging hundreds of thousands of miles on his journeys. "Another proof, another roof," he liked to say. "Want to meet Erdös?" mathematicians would ask. "Just stay here and wait. He'll show up." Mathematicians are often solitary individuals. Andrew Wiles, who recently stunned the world by his proof of the famous Fermat Conjecture—a problem that had challenged both amateur and professional mathematicians for two centuries—spent seven solitary years working in his attic office, telling no one of his efforts. Erdös viewed mathematics as a joyous and collaborative activity. He wrote papers with nearly 500 different mathematicians, an unequaled record of intellectual promiscuity. Erdös was the Johnny Appleseed of mathematics, who nurtured mathematics and mathematical talent all around the world. He was a frequent visitor to Budapest, the city of his birth, even during the Cold War, which helps explain the astonishing stream of world-class Hungarian mathematicians. To a large extent, the story of Erdös's life is the story of mathematics in this century. Erdös had a vocabulary all his own: God was the SF, or the Supreme Fascist; Joe and Sam were Joseph Stalin and Uncle Sam, two of his lifelong adversaries; children, whom he loved, were epsilons, which is how mathematicians speak of small quantities; and to leave was to die. Paul Erdös left on September 20, 1996. Dying, in Erdös-speak, meant ceasing to do mathematics. "It's a pity Pósa died at such a young age," he lamented. Pósa, Erdös's favorite child prodigy is alive and well, but at 17 was no longer as consumed with producing mathematical proofs and conjectures as when he was a 13-year-old epsilon. When Erdös left, at the age of 83, he was attending a mathematical conference in Warsaw. "He died with his boots on, in hand-to-hand combat with one more problem," said Ron Graham, director of the information sciences research center at AT&T Laboratories, old friend and collaborator. In a world built on mathematics, mathematicians are strangely ignored. Most people would find it difficult to name a single mathematician, living or dead. Nevertheless, the world's major newspapers noted Erdös's passing. The New York Times ran his obituary on the front page, introducing a whole new public to the story of this eccentric mathematical pilgrim. Erdös received a state funeral and was buried in the Hungarian National Cemetery, in Budapest, amidst the heroes of Hungary's past. Several hundred mathematicians from scores of countries and representatives of the world's major mathematical societies attended. That night and the following day in a hastily arranged seminar, they reminisced about Erdös's contribution to their lives and works. Paul Erdös was born in Budapest on March 26, 1913, the son of two high-school mathematics teachers. His two young sisters, said to have been even smarter than Paul, died of scarlet fever while his mother was in the hospital giving birth to him. Less than two years later the Russians captured his father in an offensive and sent him to Siberia. Paul's mother, Anna, fearful of contagion, educated Paul at home until he went to high school. Even then he only attended school every other year because "Anyuka" kept changing her mind. Paul was a mathematical prodigy. At three he amazed visitors by multiplying three-digit numbers in his head. At four he made his first mathematical discovery, negative numbers. From that point on he knew he would be a mathematician. As a teenager Erdös began to publish original mathematical results. His first love, to which he would be constant his whole life, was prime numbers. Primes are the basic atoms of the integers, numbers that can be divided only by one and themselves. Discerning the patterns, relationships and distribution of prime numbers has been an obsession of mathematicians since Euclid discovered an elegant proof that the number of primes is infinite. Paul's father showed him this proof when he was 10 and, as he recalls, "I was hooked." Questions about prime numbers are easy to ask and difficult to answer. Is there a prime number between every number and twice that number? Is there a prime between 2,314,947 and 4,629, 814, for example? The answer is yes, it is always true, but this was first proved only in 1850, by Pafnuty Lvovitch Chebyshev, the father of Russian mathematics. Chebyshev's proof was incredibly difficult and long-winded. Ugly. Erdös liked to imagine that God had a book in which he wrote down all the most elegant and beautiful mathematical proofs. "That's one for The Book," was his greatest praise. Chebyshev's proof would definitely not be included in The Book. When he was 17 years, old Paul devised a beautiful new proof of Chebyshev's theorem, short and pithy. One for The Book. Mathematicians outside Hungary began to take notice of the young prodigy from Budapest. The great German mathematician, Issai Shur, dubbed him der Zauber von Budapest, the magician from Budapest. The international legend of Paul Erdös had begun. Growing up a Jew in the increasingly hostile era between the wars, Erdös knew from an early age that he would one day have to leave Hungary. When he was just six years old, faced with increasing anti-Semitism, his mother suggested they convert. "You may do as you wish," said the boy, "but I will remain as I was born." When he got his doctorate, at age 21, he left Hungary to study in Manchester, England. He returned home for holidays, but after Hitler's Anschluss in 1938 he knew he could not return. He fled to the United States where he would spend the next decade, the longest semi-stationary period of his adult life. Then, as he once recalled, "my problems started with Joe and Sam." Although he desperately wanted to see his mother again—his father had died of a heart attack and much of his family had been killed in the Holocaust—he did not want to return to Hungary because of "Joe" (Joseph Stalin). In 1954, however, he was invited to a mathematical conference in Amsterdam. As an alien he would have to apply for a re-entry visa, usually a routine matter. But his extensive correspondence with mathematicians outside the United States, and especially with a number theorist in Communist China, raised the suspicions of the McCarthy era immigration officials. He was a member of the mathematics department at Notre Dame. An immigration agent came to interview Erdös in his office, the last office he would ever call his own. "The immigration officials asked me all sorts of silly questions," Erdös recalled. They asked him what he though of Marx. He had only read the Communist Manifesto and replied, "I'm not competent to judge, but no doubt he was a great man." His visa was denied. Forced to choose between the security of his Notre Dame faculty membership and the freedom to travel, he did not hesitate. He attended the conference and spent most of the following decade in Israel. His requests for a visitor's visa to attend conferences in the United States were repeatedly turned down. In 1958 the State Department granted him a "special visa" to attend a conference in Colorado. During his stay an immigration official escorted him everywhere. In 1962 he wrote to friends that apparently the "U.S. foreign policy is adamant on two points: non-admission of Red China to the U.N. and non-admission of Paul Erdös to the U.S." Erdös is legendary for his near total ignorance of and unconcern with the details of everyday life. Chores such as paying bills, preparing food, and washing clothes were beyond him. "I can make an excellent cold cereal," he remarked, "and I could probably boil an egg, but I've never tried." He buttered his first piece of bread when he was 21. "I had just gone to England to study. It was tea-time and bread was served. I was to embarrassed to admit that I'd never buttered it. I tried. It wasn't so hard." Doing laundry wasn't hard either, but Erdös never did it. His wardrobe—including custom-made socks and underwear made from silk to avoid aggravating his undiagnosed skin condition—was so small that it had to be washed several times a week, a job he left to his hosts. Erdös never married. Indeed, he avoided almost all physical contact with other human beings. He wouldn't shake hands and did not like being hugged. "He hates it if I kiss him,'" says Magda Fredro, a first cousin who was otherwise very close to him. "And he washes his hands fifty times a day. He gets water everywhere. It's hell on the bathroom floor." In 1964 his mother, who was 84, began traveling with him. She hated traveling and barely spoke a word of English, but she wanted to be with her son. They ate every meal together and, according to one profiler, he held her hand every night as she fell asleep. This assertion, to the amusement of his friends, incensed Erdös. In 1971, she died of a bleeding ulcer in Calgary. Erdös would never entirely recover from his loss. Ten years after his mother died a colleague came upon him looking unhappy and asked him what was wrong. "Haven't you heard?" he said. "My mother has left." Erdös threw himself into mathematics more completely than ever before. He began taking pills, lots of them. He started with anti-depressants and then moved on to amphetamines. Whenever he met children—epsilons—Erdös liked to amuse them with a trick. He would fish a bottle of Benzedrine out of his pocket, throw it up in the air and catch it at the last possible moment. Erdös's drug use worried his friends. Once Ron Graham tried to get him to quit by betting him five hundred dollars that he could not stop taking pills for a month. Erdös went cold-turkey for a month. Accepting Graham's check he said: "You've showed me I'm not an addict. But I didn't get any work done. I'd get up in the morning and stare at a blank piece of paper. I'd have no ideas, just like an ordinary person. You've set mathematics back a month." He began taking amphetamines again and mathematics once more progressed at his frenetic pace. Erdös was one of the most prolific mathematicians of all time, writing over 1,500 papers and books. His collaborators, 462 and counting—not all papers he has written have yet been published—could fill a large lecture hall. The phenomenon of Erdös's collaborations has itself become the subject of light-hearted mathematical analysis. Many years ago a mathematical wit invented the concept of The Erdös Number. Erdös himself is assigned Erdös number 0. Mathematicians who have written papers with Erdös—462 at the time of his death, with the number expected to climb as old collaborations are rushed to publication—receive Erdös number 1. Writing a paper with someone having Erdös Number 1 earns the author Erdös Number 2. Currently 4,566 people claim this honor. And so it goes. By the time one reaches Erdös Number 5 almost all mathematicians are included, along with a great many computer scientists, physicists, biologists, economists, social scientists and, perhaps, even baseball players. Carl Pommerance, a long-time Erdös collaborator, recently argued that Hank Aaron should have an Erdös number of 1. Pommerance owns a baseball signed by both Aaron and Erdös, and he believes it should count as a joint publication. Erdös stayed creative until the end of his life, which is remarkable in a field in which one is considered to be over the hill by age 30. The great Indian mathematical genius, Srinivasa Ramanujan, died at 32. Evariste Galois, a great French mathematician, died at 21 in a duel. Perhaps the secret to Erdös's mathematical longevity was the delight he took in seeking out and cultivating young mathematical talent. Hungary has a long tradition of developing young mathematical talent. In 1893, a high-school mathematics teacher named Dániel Arany founded a mathematical monthly for high school students called Középiskolai Matematikai Lapok, or KöMaL for short. KöMaL invites mathematically gifted students to participate in an ongoing creative problem solving competition. The young Paul Erdös was a frequent contributor to KöMaL. In 1896 László Rátz took over as the chief editor of KöMaL; the high school classes of Rátz included the 11-year-old John von Neumann and the 12-year-old Eugene Wigner, who 50 years later remembered him as a "miraculous" teacher. Rátz and KöMaL can take some of the credit for what physicist Otto Frisch called this remarkable "galaxy of brilliant Hungarian expatriates" who drove the development of the atomic bomb, modern computers, the transistor and much more. In addition to von Neuman and Wigner, this era produced Leo Szilard, Edward Teller, Theodor von Kármán, George de Hevesy, and Michael Polanyi. (Physicists, on the other hand, believe Fritz Houtermans explanation: "These people are really from Mars.") Erdös, an ex-prodigy himself, never missed a chance to meet and correspond with talented epsilons. He would talk fondly of their results and follow their progress through the years. For Erdös, who never married, these were his children. Many of the epsilons he encouraged are among today's leading mathematicians. More than mathematics, Erdös taught his young disciples the spirit of collaboration. Although in his personal vocabulary women were "bosses" and men "slaves," and slaves could become "captured" (married) or "liberated" (divorced), some of Erdös's closest friends and collaborators were women; the only prerequisite for working with Erdös was ability. Two of his brightest epsilons once asked why there were so few female mathematicians. He explained that it wasn't a lack of ability: "suppose the slave children (boys) would be brought up with the idea that, if they are very clever, the bosses (girls) will not like them—would there be then many boys who do mathematics?" Both said well, perhaps not so many. The story of Erdös's life is also the story of his collaborators and mathematical friends. One of the closest is Ron Graham, the mathematician from Bell Labs. For over three decades Graham has managed the details of Erdös's life, collecting the stipends and honoraria Erdös receives from all around the world, paying his bills, sending out reprints of his work, and monitoring, as best he could, his general health and well-being. The two men could not be more different. Where Erdös is short, thin and stooped, Graham is six foot two and powerfully built. He put himself through college working as a circus acrobat and is an accomplished trampolinist and former president of the International Juggler's association. He has bowled two 300 games, is deadly with a boomerang and is the Bell Labs Ping-Pong champion (he took up Ping-Pong seriously when, shortly after their first meeting, Erdös beat him in a game). He edits a score of mathematical journals, is the head of mathematical research at Bell labs and, in his spare time has taken up the piano and mastered Chinese. "It's easy, "Graham says. "There are a hundred and sixty-eight hours in every week." When the government split up Bell Telephone in 1983, Graham's girlfriend, Fan Chung, became the director of Bellcore's Discrete Mathematics group, while Ron stayed at AT&T (now Lucent Technologies). Soon after this corporate divorce, Fan Chung and Graham married and made an important improvement to their home, a separate Erdös Room. "When Paul is in town," Fan Chung says, "it is essential to arrange for Paul sitting. In other words, Paul enjoys to be constantly surrounded by mathematicians. In fact, it is more than an enjoyment, it is a necessity. So we have a stream of mathematicians coming and going, talking and writing." Some mathematicians, often after the age of thirty when they become venerable and wise, stop tinkering with prime numbers and odd problems and begin to build mathematical edifices. Erdös, according to one of his collaborators, Ernst Straus, who also worked with Einstein, "remained the prince of problem solvers and the absolute monarch of problem posers." The problems posed by Erdös are often deceptively easy, but their solutions have generated whole new branches of mathematics. It is almost as if Erdös had already glimpsed the underlying theory and was leading others there with his problems. Unsolved problems sometimes bothered Erdös so much that he would "put a price on their head", offering a cash prize for their solution. These prizes varied from five dollars for simple problems to thousands for ones he did not expect anyone could solve in his lifetime. Over the years Graham has written checks for many of these prizes, but few cash them. They frame them. The press tends to have stressed Erdös's many eccentricities and child-like dependence, painting a picture of a single-minded champion of an arcane world. To a certain extent, of course, this portrait is true, but to his mathematical friends he was far more. His warmth and compassion for his fellow mathematicians are as legendary as the dazzling body of work he left behind. A man who carried all his possessions in one small suitcase, he always found ways to help colleagues, even total strangers. The two times he lectured in India he arranged to have his fees paid to the impoverished widow of the great Indian mathematician, Ramanujan, a woman he had never met. In 1984 he won the $50,000 Wolf prize, by far the most money he ever received at one time in his life. He used $30,0000 to endow a postdoctoral fellowship at the Technion in Israel in the name of his mother and gave most of the rest to needy relatives and colleagues. "I kept only seven hundred and twenty dollars," Erdös said, "and I remember someone commenting that for me even that was a lot of money to keep." The obituary of Erdös that ran in the Washington Post ended abruptly with the painful line "he left no immediate survivors." This bothered Charles Krauthammer, who wrote a touching editorial. In it he told the story of a young mathematician who wanted to go to Harvard but was short of money. "Erdös arranged to see him and lent him $1,000," Krauthammer wrote, "He told the young man he could pay it back when he was able to. Recently, the young man called Graham to say that he had gone through Harvard and now was teaching at Michigan and could finally pay the money back. What should he do? "Graham consulted Erdös. Erdös said, 'Tell him to do with the $1,000 what I did.' No survivors, indeed." "My second great discovery was death," Erdös says (the first was negative numbers). "Children don't think they're ever going to die. I was like that too, until I was four. I was in a shop with my mother and suddenly I realized I was wrong. I started to cry. I knew I would die. From then on I've always wanted to be younger." In the 1970s he began appending the initials P.G.O.M to his name, which stand for Poor Great Old Man. When he turned 60 he added L.D. for Living Dead; at 65 he added A.D. for Archeological Discovery, then L.D. for Legally Dead. When he turned 75 he finished with C.D., which means Counts Dead. This last because the Hungarian Academy of Sciences restricts its membership roll to 200; when you pass 75 you can stay in the Academy and enjoy members privileges, but you no longer count as a member. Often when sitting and working with other mathematicians he would become distracted and mutter aphorisms like, "Soon I will be cured of the incurable disease of life," and "this meeting, like life, will soon come to the end. Except the meeting was much more pleasant." He even wrote his own epitaph: "Finally I am becoming stupider no more," which is pithier in Hungarian, a language that manages to express concepts like "he is becoming more stupid" with just a single word: butul. Now Paul Erdös, a giant of twentieth century mathematics and its soul, has left. Perhaps he will get what he always wanted most: complete access to the Book he had spent his whole life trying to glimpse. © 1998 by Bruce Schechter |
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