A Beautiful Mind

A new book sheds light on the brilliant and eccentric life of Budapest-born mathematical genius Paul Erdos, who renounced physical and other pleasures to pursue solutions to impossible problems. Erdos is still revered and missed by many in Israel, where he became a resident - but refused to become a citizen

Dakia Karpel
Dalia Karpel
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Dakia Karpel
Dalia Karpel

Even as a child, Paul Erdos wished he was younger. "I was in a store with my mother and I started to cry," he once said. "It hit me that I was going to die one day. From that moment on, I always wanted to be younger."

Erdos was born in Budapest in 1913 to parents who were both high-school math teachers. By age three, he was multiplying three-digit numbers for the fun of it. He was a prodigy and a mathematical genius, whose witty lectures and sense of humor earned him the nickname - the Bob Hope of mathematicians.

"When Erdos came to Israel for the first time in 1954, he was 41 and already spoke of himself as an old man, and he behaved accordingly," recalls Prof. Dan Amir, the former rector of Tel Aviv University. "In his thick Hungarian accent, he kept saying that he was sick and about to die. His lectures were very popular. He told amusing anecdotes as he discussed mathematical problems and their aesthetic solutions. Even though he repeated these same anecdotes wherever he lectured - and he lectured in 25 countries - they always surprised and entertained his listeners. In 1983, I heard him speak in Houston, Texas. And even though I already knew the anecdote he told, he was just as fascinating as ever."

Hebrew readers now have an opportunity to get to know this extraordinary personality thanks to the newly published Hebrew translation of his biography, "The Man Who Loved Only Numbers" ("Ha'ish She'ahav Rak Misparim," Keter). Author Paul Hoffman is the publisher of "Encyclopedia Britannica" and former host of the PBS television series, "Great Minds in Science." He was also the president and editor in chief of Discover magazine for 10 years. In his book, Hoffman says that Erdos, who died on September 10, 1996 at the age of 83, thought about more mathematical problems than any other mathematician in history.

Erdos, who invented his own particular vocabulary, often spoke of the pranks played by the "Supreme Fascist" or "SF" for short. Erdos felt that God abused him on a regular basis, not just by giving him colds and skin allergies, or hiding his glasses; the worst was the way God tormented him by keeping the most elegant solutions to intriguing mathematical problems hidden. Erdos used to say that the SF possessed a "transfinity" book (transfinity being a mathematical concept even larger than infinity) that contained the best and most elegant proofs to all mathematical theorems.

Prof. Jochanan Schonheim, 81, of the Tel Aviv University School of Mathematical Sciences, was a good friend of Erdos since the 1950s and used to host him in his Ramat Gan home whenever he visited Israel. Schonheim confirmed this week that, already by age 40, Erdos was preparing for his death. "He was always talking about death. He said that death had its hand on his shoulder and that he preferred the title `AD' (for `Almost Dead') to his PhD and other academic degrees."

Schonheim explains that "Erdos did combinatorics and number theory. He worked in what may be the most instinctive branch of mathematics - finite mathematics. He worked mostly on finite problems, which are sometimes more complicated than infinite ones. He was the most prolific mathematician of his time and wrote close to 1,500 papers. You should know that when someone becomes a full professor and has written about 100 papers, that's considered a lot. I wrote about 60.

"Erdos' most important contribution is that he developed particular problems in combinatory mathematics into very general ones with a wide range of applications. He was both a `problem-solver' and a `problem-poser' - in other words, if you spoke with him about any problem, he immediately generalized. He knew everything and worked with all the biggest people in the field.

"He was the stimulator who got us fired up. He had ambitions of solving major problems, such as the Goldbach Conjecture (named for mathematician Christian Goldbach). Goldbach proposed that every even integer greater than two is the sum of two prime numbers (integers are divisible only by themselves and the number 1): For example, 12 equals 5+7, and so on. With the help of computers, mathematicians were able to prove that every odd number up to 100 million may be expressed by the sum of two prime numbers, but they were not able to prove that Goldbach's conjecture is always correct up to infinity. Erdos said that the biggest problems, like Goldbach's hypothesis, would not be solved in our lifetime."

Married to math

Author Hoffman notes that Erdos either wrote or co-wrote approximately 1,475 lengthy and highly substantive papers. This rare accomplishment, in terms of both quantity and quality, was made possible in part because Erdos devoted the great majority of his time to mathematics, having renounced life's physical and other pleasures. He used to say that he was married to mathematics. Apart from his mother, he had no other close family (his father died of a heart attack during World War II).

He had little use for money or property. He considered possessions a nuisance, and gave away much of his money. In 1984, when he won the prestigious Wolf Prize, the biggest award in mathematics, he donated most of the $50,000 prize money to endow a scholarship in Israel in his parents' memory. He kept only $720 for himself. He did not own an apartment and he never held a permanent job. His only possession was a ratty suitcase crammed with papers and notebooks, a good number of medicines and a few changes of clothes. Whatever didn't fit in the suitcase, he carried around in an orange plastic bag that came from a Budapest department store. Schonheim mentions that, in his Hungarian passport, Erdos listed his place of residence as Haifa.

Hoffman writes that for the last 25 years of his life - after his mother died - Erdos spent 19 hours a day working on mathematics, keeping himself going with amphetamines like Benzadrine or Ritalin, and hefty doses of espresso and caffeine pills.

"A mathematician is a machine for turning coffee into theorems," Erdos once told his worried friends. And just to reassure them, he said, "I'll have plenty of time to rest once I'm in the grave." He once made a bet with a friend that he could go an entire month without taking any stimulants. Erdos won the bet, but said that it had been an especially unproductive period. "I set mathematics back a month," he told his friend.

This week, Schonheim said that Erdos had ways of getting prescriptions for amphetamines and recalled that he had accompanied him to the pharmacy on more than one occasion. "He took the pills discreetly and never talked about it, though he didn't keep it a secret either. In the last years of his life, he stopped using those pills because of heart problems. He had a pacemaker at the end."

Erdos had a unique way of expressing himself. When he asked someone, "When did you get here?" he meant, "When were you born?" He referred to children as Epsilon, the Greek letter used in mathematics to represent small quantities. Women were "bosses," men were "slaves," to be married was to be "captured" and to be divorced was to be "liberated." If you remarried, you were "recaptured." Music was "noise," alcohol was "poison" and "to preach" meant "to give a math lecture." He called the United States "Sam" and the USSR "Joe." When Erdos said that someone had died, he meant that the person had stopped doing mathematics. When someone really died, Erdos said that he'd "left."

Hoffman says that Erdos collaborated in writing papers with more than 500 different mathematicians from all over the world. The lucky ones who worked directly with him are referred to by a code name: Erdos Number 1. To be designated Erdos Number 2 means that the writer published a paper with someone who published a paper with Erdos. Erdos Number 3 is the next step removed. The highest number on this scale is currently Erdos Number 17.

Prof. Schonheim wrote four papers with Erdos, the last of which was published in Hungary in 1993, when Erdos was 80. Schonheim said this week that after every one of Erdos' visits to Israel, a flood of papers co-authored by him appeared, as well as original papers that were inspired by him. Over 20 Israeli mathematicians, including Ron Aharoni and Noga Elon (the full list can be found in the book) can lay claim to the sought-after Erdos Number 1 designation.

Prof. Marcel Herzog of Tel Aviv University, also an Erdos Number 1, says that Erdos had a phenomenal memory. "In lectures, he would cite precise references about who wrote what where and in which year. He wasn't always easy to understand. When he covered new material, I didn't always understand him thoroughly. He was sloppy at times. I guess that can happen when you're talking about anything for the hundredth time. In general, he would plant an idea of a way to a solution and others would ponder it while he was already running on to the next problem.

"In a way, Erdos was like a kind of postal system. Sometimes, people thought about a certain problem in one place and then he would talk about it in another place. And so a fertile connection and cooperation was created between people who didn't know each other. Erdos maintained that the most beautiful proofs are written in God's book. But he did manage to find beautiful proofs. He was also unique in awarding cash prizes for the solution of mathematical problems."

Starting in the 1950s, Erdos offered monetary prizes for solutions to problems in graph theory that he had been unable to solve, Hoffman writes. By 1987, he had given out $15,000 in prize money. An appendix to the Hebrew version of the book lists all the recipients of the Anna and Lajos Erdos Prize in Mathematics, awarded annually, since 1977, to an outstanding Israeli mathematician.

Obsession with prime numbers

Paul Erdos was born in Budapest in March 1913. On the day that his mother, a high-school math teacher, gave birth to him, her two daughters, ages five and three, contracted septic scarlet fever and died. According to Hoffman, Erdos once said that his mother used to refer to the two girls as the smarter ones in the family. His mother never recovered from the tragedy. Like many other assimilated Jewish families, the family changed its name from "Englander" to "Erdos" (which means "from the woods" in Hungarian).

When Paul was a year-and-a-half and World War I was raging, his father was taken prisoner and was sent to Siberia for six years. His mother put him in the care of a German nurse while she taught school. As a toddler, he was already making mathematical calculations such as how many days were left until his mother had a vacation. He amused family friends by asking them how old they were so he could calculate exactly how many seconds they had been alive. As soon as he learned to read, his mother brought him piles of books on medicine, which he devoured. Fearing that the schools were sources of contagion, his mother decided to keep him at home. He studied with a private tutor until he was ready for high school. Even then, he only went to school every other year, because his mother was still very dubious about it.

In March 1919, a Jew from Transylvania named Bela Kun fomented a revolution and transformed Hungary into a Soviet republic. During Kun's brief regime, Anna Erdos was appointed principal of the high school where she taught. Kun was ousted by Miklos Horthy, who founded the first fascist regime in Hungary after the war and ruled the country until 1945. Horthy violently purged Budapest of communist sympathizers. Many were tortured, murdered or expelled.

Frightened by what was happening, Anna Erdos proposed to her son that they convert to Christianity, but the seven-year-old boy replied that he would remain a Jew the way he was born. Hoffman writes that while Erdos said that his Jewishness didn't hold any special meaning for him, by an early age, he felt it was important never to compromise on his principles, even if such a stance put him in danger.

Erdos did end up waging all kinds of struggles against "fascist" regimes of all kinds, be it "Sam" (the U.S.) or "Joe" (the USSR) - in the form of the American immigration authorities, or the FBI, during the McCarthy period, or the Hungarian secret police.

When Erdos was seven, his father returned from captivity and began teaching him English, a language that he had taught himself while he was a prisoner. This is probably why Erdos always spoke English with a heavy and hard-to-decipher accent. In 1930, the 17-year-old Erdos entered university in Budapest. Within four years, he had completed a doctorate in mathematics.

He used to meet with other young Jewish mathematicians and, because of concern that the police might interrogate them because of their background, Erdos began developing his own private language. He referred to communists as people "on the long wavelength," since, on the electromagnetic spectrum, the red waves are the longest, as Hoffman explains. Fascists and Horthy supporters were "on the short wavelength." He also came up with some of his other terms at this time. "Give me an Epsilon of poison," he'd say when he wanted a sip of wine. Over time, this language spread among mathematicians. Erdos was already addicted to mathematics by then and was obsessed with prime numbers, which continued to fascinate him throughout his life.

In 1934, Erdos left Hungary because of his Jewish background. He went to England, where he was given a four-year, post-doctoral scholarship in Manchester. Hoffman quotes one of Erdos' friends who says that the mathematician's wanderlust began that same year, and that from that time on, he hardly ever slept in the same bed for seven nights in a row.

Hoffman's book provides a captivating description of Erdos's progress from his first encounter at Cambridge with J.H. Hardy, the father of modern analytic number theory, to his own entry into the pantheon of the world's greatest prime number theorists.

The woman in his life

In the late 1930s, Erdos went to the United States. He was sick with worry over his mother, who was still in Nazi-occupied Hungary. In America, Erdos worked with mathematician Stanislaw Ulam in Madison, Wisconsin. Ulam later joined the team of physicists at Los Alamos who worked on America's first atomic bomb. Erdos also wanted to work at Los Alamos, but he wasn't hired because he refused to sign a document stating that he would not talk about the bomb after the war. The FBI already had its eye on Erdos because of a 1941 arrest for loitering suspiciously near a military radio station. He was arrested along with two others, another Hungarian and a Briton.

When the war ended, Erdos learned that the Nazis had killed four of his mother's five siblings, and that his father had died of a heart attack in 1942. Erdos did not return to Budapest until 1948. Even then, he was forced to cut his visit short when Stalin started closing the borders. During the Cold War and the McCarthy period, he was interrogated by the U.S. immigration authorities. He told them that he wanted to go back and visit communist Hungary because his mother and good friends still lived there. The Americans refused to grant him a visa to re-enter the country. He subsequently lost his job and his work permit and left for Amsterdam.

He was disappointed in Europe as well. The Netherlands granted him a visa for just a short time. He was saved when Hebrew University in Jerusalem offered him a three-month position. He became an Israeli resident, but declined to become an Israeli citizen. He kept his Hungarian passport and declared himself a citizen of the world. From 1964 onward, his elderly mother started joining him on his world travels. The only place she declined to go was India, due to her fear of diseases. Hoffman writes that they ate all their meals together and that, at night, Erdos held her hand until she fell asleep. She watched over him as if he were still a small boy and tried to stop him from going out for a stroll on the boardwalk when they were staying in Manhattan Beach, California.

Erdos did not heed her advice that time and ended up getting lost.

Aside from his mother, women were never a part of Erdos' life. He explained it this way: "I have a psychological abnormality. I cannot stand sexual pleasure. It's peculiar. You know, I have a basic character that I always wanted to be different from other people." He was averse to talking about sex as well.

After his mother's death, it was Erdos' friend Ronald Graham, one of the world's leading mathematicians, who looked after him. Unlike Erdos, Graham was an athlete and physical fitness buff. He also served as president of the International Jugglers' Organization and was an expert trampoline jumper who liked to solve mathematical problems while bouncing on the pogo stick he kept in his office. Erdos and Graham first met in 1963, at a conference on number theory that was held in Boulder, Colorado. They went on to write 27 papers and one book together - although, at their first meeting, Graham got the impression that Erdos was a sickly and elderly man (who surprised him by beating him at ping-pong).

After the death of Erdos' mother, Graham functioned as his guardian and looked after his bank accounts. In Graham's office, there was a room in which Erdos's papers were stored. It was also the address for Erdos' incoming mail. Erdos did some of his work with other mathematicians via written correspondence. It all had to be cataloged and maintained.

This week, Schonheim displayed the large stack of letters he had received from Erdos between the 1960s and Erdos' death in 1996. Many are brief notes written in childish script. They mostly contain mundane information about his whereabouts and often end with some mathematical theorem. On December 7, 1960, Erdos went on for half a page telling Schonheim how sick he was and that he would therefore have to cancel a meeting at the Technion. "My old body is falling apart," he wrote.

In December 1977, Erdos wrote: "It looks like I'll be in Israel in a few months. I hope there will be peace, but I'm not optimistic."

In the book, Hoffman describes at length how Erdos was able to get those around him, and especially his hosts, such as Graham and his wife, to treat him like a baby. Erdos did not learn to tie his shoes until he was 11 - his mother always did it for him before then - and he didn't know how to spread butter on bread or slice a grapefruit or to leave a bathroom without having it look like a hurricane had just swept through it. He didn't drive, of course, and had a terrible sense of direction. Thus, he was almost totally dependent on the people around him.

`The Odd Couple'

Schonheim first met Erdos in Budapest in 1954, at an international conference of the World Organization of Mathematicians. "Erdos had friends all over the world, families of friends who affectionately took care of him. My wife Leah and I became his family in Israel," he says. He recalls some of Erdos's eccentricities: Because of his allergies, he used to wear silk clothes. He was obsessive about personal hygiene and was also a hypochondriac. He ate very little and showed no interest in anything besides mathematics. Music, art, literature and movies all bored him. He loved to wear sandals, even at official events, and found it hard to be around people for any great length of time. He did not like physical contact - even handshakes - and was tied to his mother until the day she died.

Schonheim: "I remember sitting beside her once when he gave a lecture in Israel, and she said to me: `Look, my son can't write.' Erdos had a very childish handwriting and it annoyed her. Until the end, she guided him the way you would a child, and handled his correspondence. She wasn't a negative figure, but they were dependent on each other. He felt obligated to take care of her and he worried about her. He even bought her a fur and put her up in nice hotels.

"There was something a little pathological about it - they were like `The Odd Couple.' She died in Calgary, Canada, a place they often traveled to. He was very depressed after her death."

But, says Schonheim, Erdos' eccentricities don't really help explain who the man was: "He was a good man who gave a lot to the needy. I took him with me to some friends on a kibbutz and he talked with them about kibbutz life and about Israel. He was very curious about Israel. As a humanist, he hoped there would be peace and not killing. He never talked in `isms' or used words like `occupation.' He didn't think or speak in cliches. The man was a liberal and an individualist and he was sometimes critical of Israel."

In the last decade of his life, Erdos lived in a dormitory belonging to the Academy of Budapest. After his mother's death, he gave her apartment and all its contents to a pair of young mathematicians. Schonheim kept in touch with him. When Erdos died from heart disease, he was given an official funeral and eulogized by senior members of the Academy and the government. His body was cremated and his ashes were buried in the Jewish cemetery, at a ceremony with no rabbis present.

"The last time we met was in Budapest," recalls Schonheim, "about a year before his death. We sat in a restaurant - my wife was there, too - and when we got up to leave, this shy and modest man who was such a towering genius got up and escorted us to the car to say good-bye, something he didn't usually do. He was very warm and friendly and attentive, which was unusual.

"I miss him because now I don't have anyone to turn to when problems come up. I don't have anyone to take advice from. When I go to a conference now and he's not there, it feels like there's nothing to grab onto anymore."

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