Wednesday, September 2, 2009
Wednesday Math, Vol. 86: Paul Erdős
Not all mathematicians are eccentric, but when they are, they can be spectacularly so. That would certainly apply to Paul Erdős, part genius and part hobo, a man whose worldly possessions could fit in a suitcase for his entire adult life.
Erdős was born in Hungary in 1913. His father fought in the war and was taken prisoner by the Russians. He was not returned to his life until 1920, so Paul was seven years old when first met his father. Not surprisingly, he doted on his mother, though later in his childhood it is his father who first introduces him to higher mathematics. Both Erdős and John Von Neumann are Hungarian Jews, but Erdős's life is much more influenced by the anti-Semitism of Europe in the first half of the 20th Century, possibly because of his eccentricities.
Von Neumann is put at the top of the list of mathematicians in history, but Erdős is not. The difference between them is that Von Neumann is not just a solver of problems but a creator of theorems and even entire fields of study, like game theory and certain parts of computer science. Erdős, on the other hand, is a prodigious problem solver, and is famous for the number of articles he published, usually with co-authors.
Erdős spent his entire life staying with friends and sleeping on couches and in guest bedrooms. He would show up, sometimes unannounced, and say "My brain is open." He would work on whatever unsolved math problems his friends might have, and he always had a few problems in number theory he had formulated, just in case his friends were free of unsolved problems. He slept fitfully his entire adult life and was a habitual user of amphetamines as well as a drinker of prodigious amounts of coffee. Many people tell stories about working with Erdős on a problem and noticing him dozing off while they were talking, sometimes even snoring. But if the person he was working with stopped talking, Erdős would awaken abruptly and complete his colloborator's last sentence.
He was not completely broke. He won math prizes and was given stipends by universities from time to time, but he would often quickly give the money away to charities or as scholarships for promising students. Late in his life, he had amassed more money than he could quickly give away, and his finances were handled by the mathematician Ronald Graham.
The details of his life can be found in the book The Man Who Loved Only Numbers by Paul Hoffman and in the documentary N Is A Number by George Paul Csicsery. Some of the highlights can be quickly gleaned from his page on Wikipedia. One of the things he is best remember for is the Erdős number, which works like the game known as Six Degrees of Separation or Six Degrees of Kevin Nacon, if you prefer. If a person published a paper with Erdős, that person has an Erdős number of 1. If a mathematician never published a paper with Erdős, but published with on of his co-authors, that person has an Erdős number of 2. Erdős himself has the Erdős number of zero, the only person ever in the world with so low a number.
Technically, I've never published a paper in a peer reviewed journal, but I do have my website I worked on with Tom Roby and I'm given credit for improving the proof of a theorem in a book by Russ Merris. Erdős worked directly with Richard Pollack (Erdős number = 1), who in turn worked with Frank Sottile (Erdős number = 2), who worked with Tom Roby (Erdős number = 3). If I'm given credit for working with Tom, I would have an Erdős number of 4, and since Tom and Russ worked together and I'm only given credit for things that actually appeared in print, that would give me an Erdős number of 5 if I'm given credit for working with Russ.
I think it would be fairer to split the difference and say I have an Erdős number of 4.5. Some might say my Erdős number is only imaginary, and I can live with that as well.