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More About This Textbook
Overview
Based on a National Magazine Awardwinning article, this masterful biography of Hungarianborn Paul Erdos is both a vivid portrait of an eccentric genius and a layman's guide to some of this century's most startling mathematical discoveries.
Editorial Reviews
James Alexander
Hoffman did not intend to produce a scholarly scientific biography. . . .This book opens doors on a world and characters that are often invisible. . . .Hoffman. . . and others in the book remember the mathematical tidbit that first intrigued them and bound them to this world. —The New York Times Book ReviewScientific American
Hoffman...describes Erdös' life and eccentricities engagingly and deals comprehensively with the great man's mathematical work.Library Journal
Erdö's was a man with no wife, no children, no home, and no job. One of the century's greatest mathematicians, he roamed the world and worked with hundreds of other mathematicians, providing them with challenges, inspiration, and encouragement. Altogether, he authored or coauthored more than 1,500 research papers. His personal eccentricities included a special vocabulary in which men were 'slaves,' women were 'bosses,' and children were 'epsilons.' Hoffman, whose 1987 Atlantic Monthly profile of Erdös won a National Magazine Award for feature writing, provides many anecdotes to illustrate his life and work, but the result is not fully satisfactory. The organization of the material is very loose, and the book is filled out with biographical notes on other mathematicians, many of which are not closely tied to Erdös. Hoffman's volume is an entertaining and useful introduction to Erdös for a general audience, but one hopes that there will be at least one better  and more complete  biography of him in the next few years.  Jack W. Weigel, University of Michigan Library, Ann ArborJames Alexander
Hoffman did not intend to produce a scholarly scientific biography. . . .This book opens doors on a world and characters that are often invisible. . . .Hoffman. . . and others in the book remember the mathematical tidbit that first intrigued them and bound them to this world.  The New York Times Book ReviewKirkus Reviews
An affectionate if impressionistic portrayal of one of the century's greatest and strangest mathematicians. Though little known among nonmathematicians, Erdös, who died in 1996 at age 83, was a legend among his colleagues. According to Hoffman (Archimedes' Revenge, 1988), the Hungarian was so devoted to mathematics that he went without wife, children, steady job, or even a home, preferring to exist as the wandering guest of fellow mathematicians. He lived for math, announcing his visits with a hearty, 'My brain is open,' posing and solving problems while subsisting on amphetamines and coffee ('A mathematician,' Erdös was fond of saying, 'is a machine for turning coffee into theorems'), and forgoing pleasantries like 'Good morning" to jump right in with, 'Let N be an integer.' He published more than 1,500 papers with at least 484 coauthors, who pride themselves on their 'Erdös number of 1' (a figure indicating one's degree of separation from the master). Hoffman, who traveled with and interviewed many of his collaborators, weaves oral histories and clear mathematical explication into a digressive (sometimes too digressive), entertaining whole. Hoffman creates a fullbodied and eccentric character out of hundreds of quotations and anecdotes. Missing are the linear landmarks of conventional biography: Erdös doesn't get born until page 48, a precise account of his death is absent, and his most important mathematical discoveries are nowhere summarized. Though a biography, this book works like the best fiction, finding in a concrete universal to show what mathematics is and who the people are who uncover its truths.Product Details
Meet the Author
Paul Hoffman was president of Encyclopedia Britannica and editorinchief of Discover, and is the author of The Man Who Loved Only Numbers and The Wings of Madness. He is the winner of the first National Magazine Award for Feature Writing, and his work has appeared in the New Yorker, Time, and Atlantic Monthly. He lives in Woodstock, NY.
Read an Excerpt
THE TWOANDAHALFBILLION
YEAROLD MAN
the epitaph Paul Erdos wrote for himself
Tom Trotter
It was dinnertime in Greenbrook, New Jersey, on a cold spring day in 1987, and Paul Erdos, then seventyfour, had lost four mathematical colleagues, who were sitting fifty feet in front of him, sipping green tea. Squinting, Erdos scanned the tables of the small Japanese restaurant, one arm held out to the side like a scarecrow's. He was angry with himself for letting his friends slip out of sight. His mistake was to pause at the coat cheek while they charged ahead. His arm was flapping wildly now, and he was coughing. "I don't understand why the SF has seen fit to send me a cold," he wheezed. (The SF is the Supreme Fascist, the NumberOne Guy Up There, God, who was always tormenting Erdos by hiding his glasses, stealing his Hungarian passport, or, worse yet, keeping to Himself the elegant solutions to all sorts of intriguing mathematical problems.) "The SF created us to enjoy our suffering," Erdos said. "The sooner we die, the sooner we defy His plans."
Erdos still didn't see his friends, but his anger dissipatedhis arm dropped to his sideas he heard the highpitched squeal of a small boy, who was dining with his parents. "An epsilon!" Erdos said. (Epsilon was Erdos's word for a small child; in mathematics that Greek letter is used to represent small quantities.) Erdos moved slowly toward the child, navigating not so much by sight as by the sound of the boy's voice. "Hello," he said, as he reached into his ratty gray overcoat and extracted a bottle of Benzedrine. He dropped the bottle from shoulder height and with the same hand caught it a split second later. The epsilon was not at all amused, but perhaps to be polite, his parents made a big production of applauding. Erdos repeated the trick a few more times, and then he was rescued by one of his confederates, Ronald Graham, a mathematician at AT&T, who called him over to the table where he and Erdos's other friends were waiting.
The waitress arrived, and Erdos, after inquiring about each item on the long menu, ordered fried squid balls. While the waitress took the rest of the orders, Erdos turned over his placemat and drew a tiny sketch vaguely resembling a rocket passing through a hulahoop. His four dining companions leaned forward to get a better view of the world's most prolific mathematician plying his craft. "There are still many edges that will destroy chromatic number three," Erdos said. "This edge destroys bipartiteness." With that pronouncement Erdos dosed his eyes and seemed to fall asleep.
Mathematicians, unlike other scientists, require no laboratory equipmenta practice that reportedly began with Archimedes, who, after emerging from his bath and rubbing himself with olive oil, discovered the principles of geometry by using his fingernails to trace figures on his oily skin. A Japanese restaurant, apparently, is as good a place as any to do mathematics. Mathematicians need only peace of mind and, occasionally, paper and pencil. "That's the beauty of it," Graham said. "You can lie back, close your eyes, and work. Who knows what problem Paul's thinking about now?"
"There was a time at Trinity College, in the 1930s I believe, when Erdos and my husband, Harold, sat thinking in a public place for more than an hour without uttering a single word," recalled Anne Davenport, the widow of one of Erdos's English collaborators. "Then Harold broke the long silence, by saying, `It is not nought. It is one.' Then all was relief and joy. Everyone around them thought they were mad. Of course, they were."
*
Before Erdos died, on September 20, 1996, at the age of eightythree, he had managed to think about more problems than any other mathematician in history. He wrote or coauthored 1,475 academic papers, many of them monumental, and all of them substantial. It wasn't just the quantity of work that was impressive but the quality: "There is an old saying," said Erdos. "Non numerantur, sed ponclerantur (They are not counted but weighed). In the old [Hungarian] parliament of noblemen, they didn't count the votes: they weighed them. And this is true of papers. You know, Riemann had a very short list of papers, Godel had a short list. Gauss was very prolific, as was Euler, of course." Even in his seventies there were years when Erdos published fifty papers, which is more than most good mathematicians write in a lifetime. He proved that mathematics isn't just a young man's game.
Erdos (pronounced "airdish") structured his life to maximize the amount of time he had for mathematics. He had no wife or children, no job, no hobbies, not even a home, to tie him down. He lived out of a shabby suitcase and a drab orange plastic bag from Centrum Aruhaz ("Central Warehouse"), a large department store in Budapest. In a neverending search for good mathematical problems and fresh mathematical talent, Erdos crisscrossed four continents at a frenzied pace, moving from one university or research center to the next. His modus operandi was to show up on the doorstep of a fellow mathematician, declare, "My brain is open," work with his host for a day or two, until he was bored or his host was run down, and then move on to another home.
Erdos's motto was not "Other cities, other maidens" but "Another roof, another proof." He did mathematics in more than twentyfive different countries, completing important proofs in remote places and sometimes publishing them in equally obscure journals. Hence the limerick, composed by one of his colleagues:
When Erdos heard the limerick, he wanted to publish a paper in Kurdish but couldn't find a Kurdish math journal.
*
Erdos first did mathematics at the age of three, but for the last twentyfive years of his life, since the death of his mother, he put in nineteenhour days, keeping himself fortified with 10 to 20 milligrams of Benzedrine or Ritalin, strong espresso, and caffeine tablets. "A mathematician," Erdos was fond of saying, "is a machine for turning coffee into theorems." When friends urged him to slow down, he always had the same response: "There'll be plenty of time to rest in the grave."
Erdos would let nothing stand in the way of mathematical progress. When the name of a colleague in California came up at breakfast in New Jersey, Erdos remembered a mathematical result he wanted to share with him. He headed toward the phone and started to dial. His host interrupted him, pointing out that it was 5:00 A.M. on the West Coast. "Good," Erdos said, "that means he'll be home."
When challenged further in situations like this, Erdos was known to respond, "Louis the Fourteenth said, `I am the state'; Trotsky could have said, `I am society'; and I say, `I am reality.'" No one who knew him would disagree. "Erdos had a childlike tendency to make his reality overtake yours," a friend said. "And he wasn't an easy houseguest. But we all wanted him aroundfor his mind. We all saved problems up for him."
To communicate with Erdos you had to learn his language. "When we met," said Martin Gardner, the mathematical essayist, "his first question was `When did you arrive?' I looked at my watch, but Graham whispered to me that it was Erdos's way of asking, `When were you born?'" Erdos often asked the same question another way: "When did the misfortune of birth overtake you?" His language had a special vocabularynot just "the SF" and "epsilon" but also "bosses" (women), "slaves" (men), "captured" (married), "liberated" (divorced), "recaptured" (remarried), "noise" (music), "poison" (alcohol), "preaching" (giving a mathematics lecture), "Sam" (the United States), and "Joe" (the Soviet Union). When he said someone had "died," Erdos meant that the person had stopped doing mathematics. When he said someone had "left," the person had died.
*
At five foot six, 130 pounds, Erdos had the wizened, cadaverous look of a drug addict, but friends insist he was frail and gaunt long before he started taking amphetamines. His hair was white, and corkscrewshaped whiskers shot out at odd angles from his face. He usually wore a gray pinstriped jacket, dark trousers, a red or mustard shirt or pajama top, and sandals or peculiar pockmarked Hungarian leather shoes, made especially for his flat feet and weak tendons. His whole wardrobe fit into his one small suitcase, with plenty of room left for his dinosaur of a radio. He had so few clothes that his hosts found themselves washing his socks and underwear several times a week. "He could buy more," one of his colleagues said, "or he could wash them himself. I mean, it takes zero IQ to learn how to operate a washing machine." But if it wasn't mathematics, Erdos wouldn't be bothered. "Some French socialist said that private property was theft," Erdos recalled. "I say that private property is a nuisance."
The only possessions that mattered to him were his mathematical notebooks. He filled ten of them by the time he died. He always carried one around with him, so that he could record his mathematical insights on a moment's notice. "Erdos came to my twins' bar mitzvah, notebook in hand," said Peter Winkler, a colleague of Graham's at AT&T. "He also brought gifts for my childrenhe loved kidsand behaved himself very well. But my motherinlaw tried to throw him out. She thought he was some guy who wandered in off the street, in a rumpled suit, carrying a pad under his arm. It is entirely possible that he proved a theorem or two during the ceremony."
All of his clothes, including his socks and custommade underwear, were silk, because he had an undiagnosed skin condition that was aggravated by other kinds of fabric. He didn't like people to touch him. If you extended your hand, he wouldn't shake it. Instead, he'd limply flop his hand on top of yours. "He hated it if I kissed him," said Magda Fredro, a first cousin who was otherwise very close to him. "And he'd wash his hands fifty times a day. He got water everywhere. It was hell on the bathroom floor."
Although Erdos avoided physical intimacy, and was always celibate, he was friendly and compassionate. "He existed on a web of trust," said Aaron Meyerowitz, a mathematician at Florida Atlantic University. "When I was a graduate student and we had never met before, I gave him a ride. I didn't know the route and asked him if he wanted to navigate with a map. He didn't want to [and probably didn't know how to]. He just trusted that I, a total stranger, would get him there."
What little money Erdos received in stipends or lecture fees he gave away to relatives, colleagues, students, and strangers. He could not pass a homeless person without giving him money. "In the early 1960s, when I was a student at University College London," recalled D. G. Larman, "Erdos came to visit us for a year. After collecting his first month's salary he was accosted by a beggar on Euston station, asking for the price of a cup of tea. Erdos removed a small amount from the pay packet to cover his own frugal needs and gave the remainder to the beggar." In 1984 he won the prestigious Wolf Prize, the most lucrative award in mathematics. He contributed most of the $50,000 he received to a scholarship in Israel he established in the name of his parents. "I kept only seven hundred and twenty dollars," Erdos said, "and I remember someone commenting that for me even that was a lot of money to keep." Whenever Erdos learned of a good causea struggling classical music radio station, a fledgling Native American movement, a camp for wayward boyshe promptly made a small donation. "He's been gone a year," said Graham, "and I'm still getting mail from organizations he gave donations to. Today I got a postcard from an Israeli girls' home."
In the late 1980s Erdos heard of a promising high school student named Glen Whitney who wanted to study mathematics at Harvard but was a little short of the tuition. Erdos arranged to see him and, convinced of the young man's talent, lent him $1,000. He asked Whitney to pay him back only when it would not cause financial strain. A decade later Graham heard from Whitney, who at last had the money to repay Erdos. "Did Erdos expect me to pay interest?" Whitney wondered. "What should I do?" he asked Graham. Graham talked to Erdos. "Tell him," Erdos said, "to do with the thousand dollars what I did."
*
Erdos was a mathematical prodigy. At three he could multiply threedigit numbers in his head, and at four he discovered negative numbers. "I told my mother," he recalled, "that if you take 250 from 100, you get 150. My second great discovery was death. 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 1970, I preached in Los Angeles on `my first two and a half billion years in mathematics.' When I was a child, the Earth was said to be two billion years old. Now scientists say it's four and a half billion. So that makes me two and a half billion. The students at the lecture drew a timeline that showed me riding a dinosaur. I was asked, `How were the dinosaurs?' Later, the right answer occurred to me: `You know, I don't remember, because an old man only remembers the very early years, and the dinosaurs were born yesterday, only a hundred million years ago.'"
Erdos loved the dinosaur story and repeated it again and again in his mathematical talks. "He was the Bob Hope of mathematics, a kind of vaudeville performer who told the same jokes and the same stories a thousand times," said Melvyn Nathanson at a mathematical memorial service for Erdos in Budapest. "When he was scheduled to give yet another talk, no matter how tired he was, as soon as he was introduced to an audience, the adrenaline (or maybe amphetamine) would release into his system and he would bound onto the stage, full of energy, and do his routine for the thousand and first time."
In the early 1970s, Erdos started appending the initials P.G.O.M. to his name, which stood for Poor Great Old Man. When he turned sixty, he became P.G.O.M.L.D., the L.D. for Living Dead. At sixtyfive he graduated to P.G.O.M.L.D.A.D., the A.D. for Archeological Discovery. At seventy he became P.G.O.M.L.D.A.D.L.D., the L.D. for Legally Dead. And at seventyfive he was P.G.O.M.L.D.A.D.L.D.C.D., the C.D. for Counts Dead. In 1987, when he was seventyfour, he explained: "The Hungarian Academy of Sciences has two hundred members. When you turn seventyfive, you can stay in the academy with full privileges, but you no longer count as a member. That's why the C.D. Of course, maybe I won't have to face that emergency. They are planning an international conference for my seventyfifth birthday. It may have to be for my memory. I'm miserably old. I'm really not well. I don't understand what's happening to my bodymaybe the final solution."
Erdos outlived most of his friends and, to his dismay, watched some of them lose their minds. His college thesis adviser, Leopold Fejer, one of the strongest mathematicians in Hungary, was burned out by the age of thirty. "He still did very good things, but he felt that he didn't have any significant new ideas," said Erdos. "When he was sixty, he had a prostate operation and after that he didn't do very much. Then he was on an even keel for fifteen or sixteen years, and then he became senile. There was some disturbance of the circulation. It was very sad because he knew he was senile and he said things like, `Since I became a complete idiot....' He was very well kept in the hospital but died of a stroke in 1959."
When Paul Turan, his closest friend, with whom he had written thirty papers, died in 1976, Erdos had an image of the SF assessing the work he had done with his collaborators. On one side of a balance the SF would place the papers Erdos had coauthored with the dead; on the other side the papers written with the living. "When the dead side tips the balance," Erdos said, "I must die too." He paused for a moment and then added, "'It's just a joke of mine."
Perhaps. But for decades Erdos vigorously sought out new, young collaborators and ended many working sessions with the remark, "We'll continue tomorrow if I live." With 485 coauthors, Erdos collaborated with more people than any other mathematician in history. Those lucky 485 are said to have an Erdos number of 1, a coveted code phrase in the mathematics world for having written a paper with the master himself. If your Erdos number is 2, it means you have published with someone who has published with Erdos. If your Erdos number is 3, you have published with someone who has published with someone who has published with Erdos. Einstein had an Erdos number of 2, and the highest known Erdos number of a working mathematician is 7. The great unwashed who have never written a mathematical paper have an Erdos number of [infinity].
"I was told several years ago that my Erdos number was 7," Caspar Goffman at Purdue wrote in 1969. But "it has recently been lowered to 3. Last year I saw Erdos in London .... When I told him the good news that my Erdos number had just been lowered, he expressed regret that he had to leave London that same day. Otherwise an ultimate lowering might have been accomplished."
With Erdos's death, the No. 1 Club's membership will hardly grow, except for the admission of a few stragglers who had joint papers with him in the works that should be published soon. "When these papers come out," said Graham, "we'll scrutinize them carefully to make sure no one is pretending to have worked with Erdos." And those who could have worked with him but didn't are having regrets. "One evening in the seventies," recalled MIT mathematician GianCarlo Rota, "I mentioned to Paul a problem in numerical computation I was working on. Instantly, he gave me a hint that eventually led to the complete solution. We thanked him for his help in the introduction to our paper, but I will always regret not having included his name as a coauthor. My Erdos number will now permanently remain equal to two."
The mathematical literature is peppered with tongueincheek papers probing the properties of Erdos numbers, and Jerrold Grossman at Oakland University in Rochester, Michigan, runs an Internet site, called the Erdos Number Project, which tracks the coveted numbers. One of Erdos's specialties was graph theory. By a graph, mathematicians don't mean the kind of chart Ross Perot waved at the TV cameras. They mean any group of points ("vertices" is the lingo) connected by lines ("edges"). So a triangle, for example, is a graph with three vertices and three edges. Now take Erdos's 485 collaborators and represent them by 485 points on a sheet of paper. Draw an edge between any two points whenever the corresponding mathematicians published together. The resulting graph, which at last count had 1,381 edges, is the Collaboration Graph.
Some of Erdos's colleagues have published papers about the properties of the Collaboration Graph, treating it as if it were a real mathematical object. One of these papers made the observation that the graph would have a certain very interesting property if two particular points had an edge between them. To make the Collaboration Graph have that property, the two disconnected mathematicians immediately got together, proved something trivial, and wrote up a joint paper.
"I wrote a paper once about the Collaboration Graph," said Graham, "that filled, I claimed, a muchneeded gap in the mathematical literature. Well, if the gap was much needed, I shouldn't have written the paper!" There is a tradition of writing these papers under pseudonyms. "I've used the name Tom Odda," said Graham. Tom Odda? "Look it up in Maledicta, the Journal of Verbal Aggression," said Graham. "You'll find it under Mandarin terms of abuse. Tom Odda means your mother's______, where the blank is too unmentionable even for Maledicta to fill in."
*
Though he was confident of his skill in mathematics, outside that arcane world Erdos was very nearly helpless. After his mother's death, the responsibility of looking after him fell chiefly to Ronald Graham, who spent almost as much time in the 1980s handling Erdos's affairs as he did overseeing the seventy mathematicians, statisticians, and computer scientists at AT&T Bell Labs. Graham was the one who called Washington when the SF stole Erdos's visa; and during Erdos's last few years, he said, "the SF struck with increasing frequency." Graham also managed Erdos's money, and was forced to become an expert on currency exchange rates because honoraria from Erdos's lectures dribbled in from four continents. "I signed his name on checks and deposited them," Graham said. "I did this so long I doubt the bank would have cashed a cheek if he had endorsed it himself."
On the wall of Graham's old office, in Murray Hill, New Jersey, was a sign: ANYONE WHO CANNOT COPE WITH MATHEMATICS IS NOT FULLY HUMAN. AT BEST HE IS A TOLERABLE SUBHUMAN WHO HAS LEARNED TO WEAR SHOES, BATHE, AND NOT MAKE MESSES IN THE HOUSE. Near the sign was the "Erdos Room," a closet full of filing cabinets containing copies of more than a thousand of Erdos's articles. "Since he had no home," Graham said, "he depended on me to keep his papers, his mother having done it earlier. He was always asking me to send some of them to one person or another." Graham also handled all of Erdos's incoming correspondence, which was no small task, because many of Erdos's mathematical collaborations took place by mail. He sent out 1,500 letters a year, few of which dwelt on subjects other than mathematics. "I am in Australia," a typical letter began. "Tomorrow I leave for Hungary. Let k be the largest integer...."
Graham had less success influencing Erdos's health. "He badly needed a cataract operation," Graham said. "I kept trying to persuade him to schedule it. But for years he refused, because he'd be laid up for a week, and he didn't want to miss even seven days of working with mathematicians. He was afraid of being old and helpless and senile." Like all of Erdos's friends, Graham was concerned about his drugtaking. In 1979, Graham bet Erdos $500 that he couldn't stop taking amphetamines for a month. Erdos accepted the challenge, and went cold turkey for thirty days. After Graham paid upand wrote the $500 off as a business expenseErdos 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 promptly resumed taking pills, and mathematics was the better for it.
In 1987 Graham built an addition onto his house in Watchung, New Jersey, so that Erdos would have his own bedroom, bathroom, and library for the month or so he was there each year. Erdos liked staying with Graham because the household contained a second strong mathematician, Graham's wife, Fan Chung, a Taiwanese emigre who today is a professor at the University of Pennsylvania. When Graham wouldn't play with him, Chung would, and the two coauthored thirteen papers, the first in 1979.
Back in the early 1950s, Erdos started spurring on his collaborators by putting out contracts on problems he wasn't able to solve. By 1987, the outstanding rewards totaled about $15,000 and ranged from $10 to $3,000, reflecting his judgment of the problems' difficulty. "I've had to pay out three or four thousand dollars," Erdos said then. "Someone once asked me what would happen if all the problems were solved at once. Could I pay? Of course I couldn't. But what would happen to the strongest bank if all the creditors asked for their money back? The bank would surely go broke. And a run on the bank is much more likely than solutions to all my problems." Now that he's gone, Graham and Chung have decided to pay the cash prizes themselves for Erdos's problems in graph theory, about which they have published a book. More than one hundred graph theory problems have a contract on them, for a total of more than $10,000. Andrew Beal, a Dallas banker and amateur mathematician, has offered to help bankroll Erdos's problems in other fields.
Graham and Erdos would seem an unlikely pair. Although Graham is one of the world's leading mathematicians, he did not, like Erdos, forsake body for mind. Indeed, he continues to push both to the limit. At six foot two, with blond hair, blue eyes, and chiseled features, Graham looks at least a decade younger than his sixtytwo years. He can juggle six balls and is a past president of the International Jugglers Association. He is an accomplished trampolinist, who put himself through college as a circus acrobat. ("Trampolining is just like juggling," said Graham, exhibiting a mathematician's tendency to generalize, "only there's just one object to juggleyourself.") He has bowled two 300 games, is vicious with a boomerang, and more than holds his own at tennis and PingPong.
While Erdos could sit for hours, Graham is always moving. In the middle of solving a mathematical problem he'll spring into a handstand, grab stray objects and juggle them, or jump up and down on the superspringy pogo stick he keeps in his office. "You can do mathematics anywhere," Graham said. "I once had a flash of insight into a stubborn problem in the middle of a back somersault with a triple twist on my trampoline."
"If you add up Ron's mathematical theorems and his double somersaults," one of his colleagues said, "he'd surely have a record." Graham, in fact, does hold a world recordone no less peculiar. He was cited in The Guinness Book of World Records for coming up with the largest number ever used in a mathematical proof. The number is incomprehensibly large. Mathematicians often try to suggest the magnitude of a large number by likening it to the number of atoms in the universe or the number of grains of sand in the Sahara. Graham's number has no such physical analogue. It can't even be expressed in familiar mathematical notation, as, say, the number 1 followed by .a zillion zeroes. To cite it, a special notation had to be invoked in which exponents are heaped on exponents to form a staggering leaning tower of digits.
Besides staying on the cutting edge of mathematics and acrobatics, Graham found time to learn Chinese and take up the piano. Neither his wife nor his coworkers understand how he does it. "It's easy," Graham said. "There are a hundred and sixtyeight hours in every week."
Erdos and Graham met in 1965 in Boulder, Colorado, at a conference on number theory, and immediately began collaborating, writing twentyseven papers and one book together. That meeting was also the first of many spirited athletic encounters the two men had. "I remember thinking when we met that he was kind of an old guy," Graham said, "and I was amazed that he beat me at PingPong. That defeat got me to take up the game seriously." Graham bought a machine that served PingPong bails at very high speeds and went on to become PingPong champion of Bell Labs. Even when Erdos was in his eighties, they still played occasionally. "Paul loved challenges," said Graham. "I'd give him nineteen points and play sitting down. But his eyesight was so bad that I could just lob the ball high into the air and he'd lose track of it."
In later years Erdos came up with novel athletic contests at which he'd seem to have more of a chance, though he invariably lost. "Paul liked to imagine situations," Graham said. "For example, he wondered whether I could climb stairs twice as fast as he could. We decided to see. I ran a stopwatch as we both raced up twenty flights in an Atlanta hotel. When he got to the top, huffing, I punched the stopwatch but accidentally erased the times. I told him we'd have to do it again. `We're not doing it again,' he growled, and stormed off.
"Another time, in Newark Airport, Erdos asked me how hard it was to go up a down escalator. I told him it could be done, and I demonstrated. `That was harder than I thought,' I said. `That looks easy,' he said. `I'm sure you couldn't do it,' I said. `That's ridiculous,' he said. `Of course I can.' Erdos took about four steps up the escalator and then fell over on his stomach and slid down to the bottom. People were staring at him. He was wearing this ratty coat and looked like he was a wino from the Bowery. He was indignant afterward. `I got dizzy,' he said."
Erdos and Graham were like an old married couple, happy as clams but bickering incessantly, following scripts they knew by heart though they were baffling to outsiders. Many of these scripts centered on food. When Erdos was feeling well, he got up about 5:00 A.M. and started banging around. He'd like Graham to make him breakfast, but Graham thought he should make his own. Erdos loved grapefruit, and Graham stocked the refrigerator when he knew Erdos was coming. On a visit in the spring of 1987, Erdos, as always, peeked into the refrigerator and saw the fruit. In fact, each knew that the other knew that the fruit was there.
"Do you have any grapefruit?" Erdos asked.
"I don't know," Graham replied. "Did you look?"
"I don't know where to look."
"How about the refrigerator?"
"Where in the refrigerator?"
"Well, just look."
Erdos found a grapefruit. He looked at it and looked at it and got a butter knife. "It can't be by chance," Graham explained, "that he so often used the dull side of the knife, trying to force his way through. It'll be squirting like mad, all over himself and the kitchen. I'd say, `Paul, don't you think you should use a sharper knife?' He'd say, `It doesn't matter,' as the juice shoots across the room. At that point I give up and cut it for him."
Graham was not the only one who had to put up with Erdos's kitchen antics. "Once I spent a few days with Paul," said Janos Path, a fellow Hungarian emigre. "When I entered the kitchen in the evening, I was met with a horrible sight. The floor was covered by pools of bloodlike red liquid. The trail led to the refrigerator. I opened the door, and to my great surprise saw a carton of tomato juice on its side with a gaping hole. Paul must have felt thirsty and, after some reflection, decided to get the juice out of the carton by stabbing it with a big knife."
In mathematics, Erdos's style was one of intense curiosity, a style he brought to everything else he confronted. Part of his mathematical success stemmed from his willingness to ask fundamental questions, to ponder critically things that others had taken for granted. He also asked basic questions outside mathematics, but he never remembered the answers, and asked the same questions again and again. He'd point to a bowl of rice and ask what it was and how it was cooked. Graham would pretend he didn't know; others at the table would patiently tell Erdos about rice. But a meal or two later Erdos would be served rice again, act as though he'd never seen it, and ask the same questions.
Erdos's curiosity about food, like his approach to so many things, was merely theoretical. He never actually tried to cook rice. In fact, he never cooked anything at all, or even boiled water for tea. "I can make excellent cold cereal," he said, "and I could probably boil an egg, but I've never tried." He was twentyone when he buttered his first piece of bread, his mother or a domestic servant having always done it for him. "I remember clearly," he said. "I had just gone to England to study. It was teatime, and bread was served. I was too embarrassed to admit that I had never buttered it. I tried. It wasn't so hard." Only ten years before, at the age of eleven, he had tied his shoes for the first time.
His curiosity about driving was legendary in the mathematics community, although you never found him behind the wheel. He didn't have a license and depended on a network of friends, known as "Uncle Paul sitters," to chauffeur him around. But he was constantly asking what street he was on and questioning whether it was the right one. "He was not a nervous wreck," Graham said. "He just wanted to know. Once he was driving with Paul Turan's widow, Vera Sos. She had just learned to drive, and Paul was doing his usual thing, `What about this road?' `What about that road?' `Shouldn't we be over there?' Vera was distracted and she plowed into the side of a car that must have been going forty or fifty miles an hour. She totaled it, and vowed that she would never drive with Erdos again."
But outside mathematics, Erdos's inquisitiveness was limited to necessities like eating and driving; he had no time for frivolities like sex, art, fiction, or movies. Erdos last read a novel in the 1940s, and it was in the 1950s that he apparently saw his last movie, Cold Days, the story of an atrocity in Novi Sad, Yugoslavia, in which Hungarians brutally drowned several thousand Jews and Russians. Once in a while the mathematicians he stayed with forced him to join their families on nonmathematical outings, but he accompanied them only in body. "I took him to the Johnson Space Center to see rockets," one of his colleagues recalled, "but he didn't even look up." Another mathematician took him to see a mime troupe, but he fell asleep before the performance started. Melvyn Nathanson, whose wife was a curator at the Museum of Modern Art in New York, dragged Erdos there. "We showed him Matisse," said Nathanson, "but he would have nothing to do with it. After a few minutes we ended up sitting in the Sculpture Garden doing mathematics."
Table of Contents
Interviews & Essays
Before the live bn.com chat, Paul Hoffman agreed to answer some of our questions:
Q: What were the circumstances of your initial meeting of Paul Erdös? Did you know immediately that you were dealing with someone special?
A: I met Paul Erdös in 1986. At our first meeting, Erdös barely noticed me. He was headdeep in one mathematical problem or another. I decided then that I was going to follow him around for a few weeks as he traveled and showed up unannounced on his colleagues' doorsteps, eager to skip the small talk and plunge right into mathematics. He would declare, "My brain is open," and wait for his colleagues to serve up challenging problems.
I knew immediately that he was special. He spoke in aphorisms ("A mathematician is a machine for turning coffee into theorems") and had a language all his own: "epsilons" for children, "bosses" for women, "slaves" for men.
Q: Who would you consider your literary influences?
A: I like the clarity and simplicity of John McPhee's writing, and I've been inspired by the novelistic detail in the case studies written up by Freud and Oliver Sacks.
Q: Have you read any book over the past year that you would strongly recommend?
A: Sherwin Nuland's How We Die.
Q: If you had to pick one thing about Mr. Erdös that surprised you the most, what would it be? Why?
A: I was pleasantly surprised that Erdös defied the stereotype of the asocial loner. He was a very sociable genius who made mathematics into a social activity. And despite doing mathematics 20 hours a day for a few decades, he knew about many other subjects: politics, history, music, science.