Paul Erdős

Paul Erdős: The Mathematician Without an Address

A half-empty suitcase, socks worn with sandals, no home, no job and no country he stayed in for long: for most of six decades Paul Erdős crossed the world with almost nothing, arriving unannounced on colleagues' doorsteps to work. "Property is nuisance," he said. He had not buttered his own toast until he was twenty-one, and depended on other mathematicians for housing, meals, clothes and tax returns. When he died he left roughly $25,000 and more than 1,500 papers.

Born Into Loss

Erdős was born on 26 March 1913 in Budapest to Lajos and Anna, both mathematics teachers from a Jewish family that had once been called Engländer. His arrival coincided with catastrophe. "I had two sisters, 3 and 5 years old," he later recalled, "and while Mother was in the hospital, during my birth, within 24 hours, both of my sisters died from scarlet fever." His parents became, understandably, extremely protective of the son who remained.

The First World War took his father to Siberia as a prisoner of war for six years. His mother taught, and Erdős was left largely in the care of an Austrian governess: "I spent a lot of time alone with an Austrian lady, so I learnt German at the age of 3." Mathematics arrived as conversation rather than curriculum. At around four he worked out for himself that "if we subtract 250 from 100 we get 150 under 0," which prompted his mother to explain negative numbers. At ten his father showed him Euclid's proof that the primes are infinite. "My friendship with primes started very early," he wrote.

A Friendship With Primes

Hungary in the 1930s restricted Jewish enrolment at its universities, but Erdős won a place in 1930 by topping a national examination. Before he was twenty he had produced the result that made his name among professionals: an elegant elementary proof of Bertrand's postulate, the statement that a prime number always lies between any n and 2n. He took his doctorate at the University Pázmány Péter in Budapest in 1934.

That same year a fellowship took him to Manchester, and the wandering began. He met G. H. Hardy at Cambridge in 1934 and Stanisław Ulam in 1935. Asked by his mother, during a period of antisemitic violence, whether the family should convert to Christianity, the young Erdős answered with the flat independence that would define him: "You can do whatever you want, but I'll stay what I was born to be."

Exile, and What Was Lost

He reached Princeton in 1938 and then Madison at Ulam's invitation. Europe closed behind him. His father died of a heart attack in 1942; his mother survived the war, but four of his uncles and aunts were murdered, and his cousin Magda Fredro came back from Auschwitz. He returned to Budapest in 1948, after the liberation.

America then turned on him too. A trivial 1941 trespassing incident recorded by the FBI, combined with his correspondence with a Chinese mathematician, was enough for McCarthy-era officials to refuse him re-entry to the United States in 1954. His visa was not restored until November 1963. For nearly a decade the most cosmopolitan mathematician alive was locked out of the country where most of his collaborators worked.

The Elementary Proof, and the Quarrel

In 1949 Erdős and Atle Selberg achieved something the profession had thought impossible: an elementary proof of the prime number theorem, one that avoided the complex analysis every previous proof had required. What followed was one of the most bitter priority disputes of the century. Selberg raced ahead, published first, and took the Fields Medal the following year. Erdős, characteristically, "was not much concerned with the competitive aspect." He received the American Mathematical Society's Cole Prize in Number Theory in 1951 and, much later, the Wolf Prize in 1983, which came with $50,000. He gave nearly all of it away, to students and as bounties for problems he had posed.

A Life Without an Address

Stanisław Ulam, writing in 1976, described him as "somewhat below medium height, an extremely nervous and agitated person" whose eyes "indicated he was always thinking about mathematics." He attended conferences everywhere and stayed with whoever would have him. Ronald Graham gave him a room, handled his money and his correspondence, and kept what became known as the Erdős room for the purpose. Erdős posted cash rewards for unsolved problems, calibrated to their difficulty, and paid out when someone won.

His mathematics ran to number theory, combinatorics, graph theory and the field the New York Times credited him with founding outright, discrete mathematics, now the substrate of computer science. Britannica calls him "arguably the most prolific mathematician of the 20th century, in terms of both the number of problems he solved and the number of problems he convinced" other people to attack. The Times called him "the Euler of our time." He believed mathematical truths were discovered rather than invented, and imagined a divine Book in which the most elegant proof of every theorem was already written; the highest praise he could give an argument was that it came straight from The Book.

The Erdős Number

His collaborative habit produced its own folk metric. An Erdős number of 1 means you co-authored a paper with him; 2 means you co-authored with one of those people, and so on outward. He had 458 direct collaborators, and roughly 4,500 mathematicians hold an Erdős number of 2. No comparable index exists for anyone else, because no one else worked that way.

Why Paul Is Called a Genius

The quality at issue with Erdős is unusual, and it is not the standard one. He built no cathedral of theory. He held no permanent chair, ran no school, and never won a Fields Medal, losing that race to Selberg over the very result they had reached together. Measured by the ordinary yardstick of great mathematicians, foundational frameworks that reorganise a field, his record is thinner than his reputation.

What he had instead was an extraordinary nose for the right problem: the ability to look at a question that was, in his own description, "incredibly simple, and seemingly almost unsolvable," and see the one combinatorial angle that cracked it. He did this across number theory, graph theory and set theory, thousands of times, for sixty years, and the elementary proof of Bertrand's postulate before the age of twenty and the elementary proof of the prime number theorem at thirty-six are the demonstrations. Britannica's "legendary eccentric" and the Times's "Euler of our time" are both defensible.

The honest complication is that his genius was social as much as individual. Erdős made other mathematicians better by arriving, posing exactly the question they were equipped to answer, and working alongside them until it fell. Those 458 collaborators are not a footnote to the achievement; they are the shape of it. He was a distributed intelligence with a suitcase, and the Erdős number exists because there was no other way to describe what he did.

Legacy

He died of a heart attack in Warsaw on 20 September 1996, aged eighty-three, while attending a mathematics meeting. Ronald Graham's epitaph was exact: he "died with his boots on, in hand-to-hand combat with one more problem." The problems he posed, many still carrying his cash bounties, are still being solved. The field he helped found underwrites modern computing. And the metric named after him remains the only measure in science of how close you stand to a single human being.

Achievements

• Authored ~1,500 papers, the most of any mathematician in history
• Inspired the 'Erdős number' measuring collaborative distance
• Co-developed the probabilistic method in combinatorics
• Elementary proof of the prime number theorem (with Atle Selberg)
• Founding contributions to Ramsey theory and extremal graph theory