The Erdös Number Project

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 The following copyrighted article appeared in The Economist, October 5, 1996, page 83, posted here by implicit permission of the publisher.

Paul Erdös

If the Martians had made contact with earth during the lifetime of Paul Erdös he would have made a good choice as this planet's ambassador. The aliens would have appreciated his unearthly intelligence. He spoke the universe's common tongue, the theory of numbers, with fluency and wit. Importantly, Mr Erdös would never have missed the trappings of this world. He had no children, no wife, no house, no credit card, no job, no change of shoes, indeed nothing but a suitcase containing a few clothes and some notebooks. Neither was he fussy about food, as long as he had coffee. A mathematician, he said, "is a machine for converting coffee into theorems." Mr Erdös's life was streamlined for mathematics.

 Mathematics? For many it is the most baffling of the sciences. Most people with a bit of education can follow an explanation of say, the Big Bang or genetics, but limit their interest in mathematics to sorting out the intricacies of their bank statement. To mathematicians, though, their science is the purest creation of the human mind, and for many Mr Erdös was its supreme exponent this century.

Typically, he would arrive in a city where he was to lecture, ring up a fellow mathematician, and announce, "My brain is in town." He sounds like a guest from hell, but to his hosts this brain was a shared treasure and their collective responsibility. They would lodge him, feed him, launder his clothing.

Paul Erdös was a constantly wandering Jew. For much of his life Hungary, his homeland, was run by dictators. Many members of his family were murdered by the Germans during the Hitler period. In return for the comfort of friends who took care of his perfunctory needs, Mr Erdös would cut gems of elegance from numbers, graphs and logic. His problems were often simple to pose, yet offered room for creativity and surprise. Suppose an infinite number of dots are painted on an infinite canvas in such a way that the distance in inches between any two dots is a whole number. What would the painting look like? Mr Erdös's brow would furrow as he showed that the result could only be a straight row of dots. But don't ask for an explanation of his elegant proof unless you are interested in conic sections.



In his prime

Paul Erdös's parents were maths teachers, so they presumably looked at him fondly when, at four, he said he had discovered negative numbers. While in his late teens he made discoveries about prime numbers. A number is "prime" if it cannot be divided by any smaller number except one. Examples of prime numbers are 1913, the year in which Mr Erdös was born, and 83, his age when he died of a heart attack at a mathematics conference in Warsaw. Mr Erdös helped to recast a theory about prime numbers made by an earlier mathematician, using a more elegant approach. This, it was said, was analogous to creating the Panama Canal for shipping that previously went around South America.

Elegant indeed, but were Mr Erdös's l,000 or more published papers of any practical value? He made no claim for their practicality. It was enough, he said, that a proof was "very nice". Yet mathematics, however pure, has a way of turning up in useful places. "Combinatorics", a branch of maths explored by Mr Erdös, can be used to calculate the number of tiles needed to pave an irregular space. His work on graphs has been applied to the design of communications networks.

The extraordinary left side of his brain was put at the service of numerous young mathematicians at the start of their careers. To him a mathematician of promise was an epsilon, the Greek letter used by mathematicians to describe a small quantity. To an epsilon he was "Uncle Paul". He set them problems, paying them rewards of a few hundred dollars if they came up with solutions, giving away much of his modest income from lectures and prizes (among them the Wolf Prize, an Israel-based sort of Nobel prize). A colleague likened Mr Erdös to a honeybee: an industrious creature who buzzed about the world and pollinated the fields of mathematics.

He was compared to Leonhard Euler (1707-83), an awesome Swiss regarded as the most prolific mathematician who ever lived. It is hard to exaggerate Mr Erdös's passion. For 19 hours a day, seven days a week, stimulated by coffee, and later by amphetamines, he worked on mathematics. He might start a game of chess, but would probably doze off until the conversation returned to maths. To find another life this century as intensely devoted to abstraction, one must reach back to Ludwig Wittgenstein (1889-1951), who stripped his life bare for philosophy. But whereas Wittgenstein discarded his family fortune as a form of self-torture, Mr Erdös gave away most of the money he earned because he simply did not need it. "Private property is a nuisance," he would say. And where Wittgenstein was driven by near suicidal compulsions, Mr Erdös simply constructed his life to extract from his magnificent obsession the maximum amount of happiness.