The story of the Kevin Bacon of mathematics: Paul Erdős

By May 30, 2014Science heroes

This story is by Dr Nick Beaton. You can hear him tell the story here.

This is the story of a mathematician. This particular mathematician was extraordinary not only because of his remarkable intellect and his influence on mathematical thinking, but also because of his generosity, the way that he viewed mathematics as a social activity to be shared with anyone who was interested, his 30-year amphetamine habit, his eccentric behaviour, and because of his singular drive in pursuing mathematical truth.

Paul Erdős was born on March 26, 1913, in Budapest, Hungary, to two high school mathematics teachers, Anna and Lajos. While Anna was giving birth to him in hospital, their two daughters, aged three and five, contracted septic scarlet fever and died within a day. This tragedy led to Paul and his mother having an extremely close relationship for the remainder of her life.

Erdős was a mathematical prodigy. At three he could multiply three-digit numbers in his head, and at four he discovered negative numbers. He entertained himself by “computing crazy things like how long it would take a train to reach the Sun.” He amused his mother’s friends by asking them how old they were and then calculating in his head how many seconds they had lived.

While he was in high school and university, he and his friends formed a sort of “mathematics club”, where they would meet in town and discuss mathematics and politics. As expected, Paul excelled in mathematics at university, producing a number of highly original results.

After university, he had short-term positions at Manchester, The Institute for Advanced Study at Princeton (at the same time as Einstein and Gödel), Purdue and Notre Dame. Notre Dame offered to make his position permanent, but he refused, not wishing to be tied down to a permanent job.

In 1954, he was invited to a conference in the Netherlands, but the US government refused to give him a return visa. He was so offended at the thought of being stuck in one country for the rest of his life that he quit his job at Notre Dame and left for Europe, and would end up not returning to the US until the mid-60’s.

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Erd̋os is a significant figure in mathematical history for a number of reasons. To start with, if we count by the number of published scientific articles, he is easily the most prolific mathematician in history, with about 1525 articles to his name! (Counted by number of pages, he is surpassed only by the great Leonhard Euler.)

He worked on problems in many diverse areas of mathematics, including number theory, combinatorics, probability, complex analysis, topology and set theory.

His approach was very collaborative—those 1525 papers were written with some 511 different co-authors. He had no interest in fame or fortune, and was always very happy to share his methods and results with others—he had a gift for judging the right kind of problem to suggest to people, and what the “next step” should be after a problem was solved. Famously, he would judge the difficulty of problems by assigning them monetary values—ranging from a few dollars to thousands—and would mail a cheque to the first person to provide a correct solution.

His first major result, at the age of 20, was a proof of a result in number theory called Bertrand’s Postulate or Chebyshev’s Theorem, which states that for a whole number n>3, there is at least one prime between n and 2n-2. One of his most famous results, with Atle Selberg, was the first elementary proof of the Prime Number Theorem, which estimates how frequently we expect to find prime numbers.

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He is perhaps most famous, however, for something called Erdős numbers. In the 50’s and 60’s Erdős and his collaborators, notably fellow Hungarian Alfréd Rényi, started investigating random networks. These are an ideal mathematical model for modelling people and their acquaintances. (Just think about Facebook: every person is a node of the network, and we join two nodes with a line if those people are friends.)

There are lots of questions which then arise: How many connections does the average node have? If we pick two nodes at random, how far apart will they be? Are there weak points in the network, where the removal of one node could split the whole thing in two? Nowadays, with the rise of the internet, there’s a lot of research into random networks, and this informs the development of social networks, advertising and security on the web, and the physical infrastructure of the internet. There are also many applications beyond the scope of the internet, like models for the spread of diseases.

In the 60’s, as a bit of a joke, some friends of Erdős’s wondered about the network of mathematicians. Wanting to be precise, they said that two mathematicians should be linked in the network if they’ve published a paper together. They soon realised that Erdős was at the heart of this network, and so they coined the term Erdős number.

Erdős’s number is 0. Anyone who’s published with Erdős has number 1. Anyone who has published with someone with number 1, but not with Erdős himself, has number 2, and so on. If there is no connection between a given mathematician and Erdős, they have Erdős number infinity.

Most mathematicians nowadays would know their Erdős number (mine is 3). Of mathematicians with a finite Erdős number, about 80% have number 5 or less. The average is about 4.65. The largest known finite Erdős number is 13, and there’s only a few people with numbers those high.

There are variants: you may have heard of Bacon numbers, which are used to connect actors to Kevin Bacon, via movie collaborations. Others include Einstein numbers and Black Sabbath numbers. They can even be combined (just add the separate numbers together) to get such curiosities as Erdős-Bacon numbers and Erdős-Bacon-Sabbath numbers (Stephen Hawking and a mathematician/choreographer named Karl Schaffer have the lowest, at 8).

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From the time that he quit his job at Notre Dame, Paul more or less gave up having a permanent job. He travelled the world, visiting collaborators, meeting people at conferences and occasionally giving lecture series at universities. He travelled with two half-empty suitcases — one with a few pieces of clothing, the other with mathematical papers.

He would frequently stay at the houses of his collaborators, depending entirely on them for transport, food, etc, (he could barely cook for himself or clean his clothes) and often working them to the bone. At conferences, he would frequently skip the talks and gather with other mathematicians in hotel rooms to work. He once met simultaneously with six different mathematicians working on six different problems!

Erdős constantly gave away money to relatives, colleagues, students and strangers. He could not pass a homeless person without giving them money. In 1984 he received the Wolf Prize, with a $50000 award, and proceeded to donate all but $720 to a scholarship in Israel.

In 1971, his mother, who had been travelling with him since 1964, died. After this he put in nineteen-hour days, keeping himself fortified with 10 to 20 milligrams of Benzedrine or Ritalin, strong espresso, and caffeine tablets. “A mathematician,” Erdős 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.”

In 1979, Ron Graham (one of Erdős’s most frequent collaborators) bet Erdős $500 that he couldn’t stop taking amphetamines for a month. Erdős accepted the challenge, and went cold turkey for thirty days. After Graham paid up, Erdős 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 just 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.

Erdős died September 20, 1996 in Warsaw, of a heart attack, at age 83. His memorial service on October 18, 1996 in Budapest was one of the largest ever held in Hungary, with more than five hundred people in attendance.

His epitaph, which he wrote for himself, is “Végre nem butulok tovább”, which translates as “Finally, I am becoming stupider no more”.