Paul Erdős. (26 March 1913 – 20 September 1996) was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. He is also known for his “legendarily eccentric” personality.
Paul Erdős was born in Budapest, Hungary on March 26, 1913. He was the only surviving child of Anna and Lajos Erdős ; his siblings died before he was born, aged 3 and 5. His parents were both Jewish mathematicians from a vibrant intellectual community. His fascination with mathematics developed early—at the age of three, he could calculate how many seconds a person had lived.
Both of Erdős’s parents were high school mathematics teachers, and Erdős received much of his early education from them. Erdős always remembered his parents with great affection. At 16, his father introduced him to two of his lifetime favorite subjects—infinite series and set theory. During high school, Erdős became an ardent solver of the problems proposed each month in KöMaL, the Mathematical and Physical Monthly for Secondary Schools. Erdős later published several articles in it about problems in elementary plane geometry.
In 1934, at the age of 21, he was awarded a doctorate in mathematics. Because anti-Semitism was increasing, he moved that same year to Manchester, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at Princeton University. At this time, he began to develop the habit of traveling from campus to campus. He would not stay long in one place and traveled back and forth among mathematical institutions until his death.
Possessions meant little to Erdős; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were generally donated to people in need and various worthy causes. He spent most of his life as a vagabond, traveling between scientific conferences and the homes of colleagues all over the world. He would typically show up at a colleague’s doorstep and announce “my brain is open,” staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom he should visit next. His working style has been humorously compared to traversing a linked list.
His colleague Alfréd Rényi said, “a mathematician is a machine for turning coffee into theorems”, and Erdős drank copious quantities. After 1971 he also took amphetamines, despite the concern of his friends, one of whom (Ron Graham) bet him $500 that he could not stop taking the drug for a month. Erdős won the bet, but complained that during his abstinence mathematics had been set back by a month: “Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.” After he won the bet, he promptly resumed his amphetamine use.
He had his own idiosyncratic vocabulary: he spoke of “The Book”, an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, “You don’t have to believe in God, but you should believe in The Book.” He himself doubted the existence of God, whom he called the “Supreme Fascist” (SF). He accused the SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, “This one’s from The Book!”. This later inspired a book entitled Proofs from THE BOOK.
In 1952, during the McCarthy anti-communist investigations, the U.S. government denied Erdős, a Hungarian citizen, a re-entry visa into the United States, for reasons that have never been fully explained. Teaching at Notre Dame at the time, Erdős could have chosen to remain in the country. Instead, he packed up and left, albeit requesting reconsideration from the Immigration Service at periodic intervals. The government changed its mind in 1963 and Erdős resumed including American universities in his teaching and travels.
During the last decades of his life, Erdős received at least fifteen honorary doctorates. He became a member of the scientific academies of eight countries, including the U.S. National Academy of Sciences and the UK Royal Society. Shortly before his death, he renounced his honorary degree from the University of Waterloo over what he considered to be unfair treatment of colleague Adrian Bondy. He died “in action” of a heart attack on September 20, 1996, at the age of 83, while attending a conference in Warsaw, Poland. Erdős never married and had no children.
Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler; Erdős published more papers, while Euler published more pages. He wrote around 1,525 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, having 511 different collaborators in his lifetime.
In terms of mathematical style, Erdős was much more of a “problem solver” than a “theory developer”. (See “The Two Cultures of Mathematics” by Timothy Gowers for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated.) Joel Spencer states that “his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular theorems and conjectures throughout his illustrious career.” Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the Wolf Prize, where his contribution is described as “for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over”. In contrast, the works of the three winners after were recognized as “outstanding”, “classic”, and “profound”, and the three before as “fundamental” or “seminal”.
Of his contributions, the development of Ramsey theory and the application of the probabilistic method especially stand out. Extremal combinatorics owes to him a whole approach, derived in part from the tradition of analytic number theory. Erdős found a proof for Bertrand’s postulate which proved to be far neater than Chebyshev’s original one. He also discovered an elementary proof for the prime number theorem, along with Atle Selberg, which showed how combinatorics was an efficient method of counting collections. Erdős also contributed to fields in which he had little real interest, such as topology, where he is credited as the first person to give an example of a totally disconnected topological space that is not zero-dimensional.