Mathematician Profile# Paul Erdös: Collaborative Mathematics

December 6, 2021

*By Lillie Therieau *

Paul Erdös is one of the most idiosyncratic mathematicians of all time. His long career produced over 1,500 mathematical papers, making him one of the most prolific mathematical thinkers of all time. His strange vocabulary, nomadic lifestyle, and socially-focused process of solving complex problems make him an incredibly unique thinker to study.

Paul died at the old age of 83, making him the longest living mathematician in our series so far. This is notable because he was able to refine his work throughout his life, break into new subjects, and publish many more findings than other mathematicians whose lives were cut tragically short. He also collaborated with the most mathematicians of any person in history, making his legacy rich and interwoven with many of the most famous 20th-century mathematical thinkers.

*Paul Erdos (Image Source: **George Csicsery**)*

Paul Erdös was born in Budapest, Hungary in 1913. His parents were both high school mathematics teachers and he was the only surviving child in his family, as both of his sisters died of scarlet fever during an epidemic. From 1914-20, Paul’s father was held captive in Siberia as a WWI prisoner of war. His mother worked long hours to support the family and Paul was left alone at home with his parents’ mathematics books. He taught himself to read through these books, learning complex math concepts at an extremely young age.

At the age of 16, Paul’s father introduced him to infinite series and set theory, two of the main concentrations in his later professional work. At 17, Paul entered college at the University of Budapest. He completed his undergraduate degree and was awarded his doctorate in mathematics by the time he was just 21. After he left university, Paul realized that Hungary was becoming extremely unsafe for Jewish people. It was the early 1930s and Hitler’s Nazi Party was beginning its meteoric rise. Paul left Hungary and moved to the US, reluctantly leaving his family behind.

After leaving Hungary, Paul led a strange lifestyle. He lived out of a suitcase for his entire life, owning few personal possessions and never having a permanent place to live. Instead, he bounced from conference to meeting to seminar across the world. His fame in math circles meant that he always had a willing host, whom he expected to feed him, do his laundry, and transport him to wherever he needed to go.

*Paul Erdos playing Ping-Pong (via **anthonybonato.com**)*

Most of Paul’s family perished in the Holocaust, though his mother was able to escape in hiding. Paul was tormented by his inability to communicate with his family or help them escape. There was no way to call or send mail from America to Hungary, so Paul was totally in the dark about what was happening to his family and friends. Eventually, he was able to reconnect with his mother and some of his close childhood friends. Later in his life, his mother traveled everywhere with him and they were very close until her death.

Though Paul made substantial money as a speaker and visiting professor, he donated almost all of it to the needy and charities. He cared for little but math, regularly working over 19 hours a day on his latest proofs. Besides work, Paul’s two vices were coffee and classical music, which he lovingly referred to as “noise”.

“Noise” was just one of the words in Paul’s own strange, self-formulated language of slang Erdös-isms. He called the USA “samland,” after Uncle Sam and called mathematicians who stopped doing math “dead,” while saying the dead had simply “left”. He was an atheist but referred to his own symbolic set of beliefs as the Book. He believed that the Book contained every theorem, known or unknown. When he found the solution to a particularly complex or elegant proof, he was known to say “This one is from the Book!”

Paul’s best-known mathematical contributions are collaborations with other mathematicians. He believed that the practice of mathematics should be a fundamentally social one, collaborating with the most mathematicians of any person in history. He had over 500 separate collaborators in his lifetime, often working with the same people dozens of times. The Erdös number was coined by his friends as a way to quantify the degree of separation between any given mathematician and Paul. Paul Erdös himself was 0, while his collaborators had a number of 1, and their collaborators had a number of 2, and so on.

Paul was also known for giving out prizes for “Erdös Problems,” which were math’s most notorious unsolved problems. The prizes ranged from $20 to thousands of dollars, depending on how important Paul considered the problem.

Paul’s mathematical work was broad, covering the fields of infinite series, number theory, set theory, and probability. He’s well-known for his work in the probabilistic method, a way to prove the existence of a certain kind of mathematical object. This method allows mathematicians to solve for a type of object that shares properties and behaviors with others in the prescribed set. It’s an important way to understand how certain operations will turn out, even if one cannot solve for the exact identity of the object, and is used in several fields of mathematics.

Paul Erdös died at the age of 83 after suffering a heart attack during a conference that he was attending. His death was remarkably close to the way he had long proclaimed he wanted to die: on his feet and teaching students.

Paul had been an important mentor for young mathematicians across the world. When gifted high school students would write to him, he would always respond, sometimes even coming out to visit them. He touched the lives of thousands of mathematicians, either through collaborations, correspondence, or his Erdös problems.

Many of his friends, mentees, and admirers are still working today, shaping mathematics through collaboration and cooperation, inspired by their friend, Paul Erdös.

This article is the sixth in our series exploring the lives and achievements of famous mathematicians throughout history. (Our last article was about the Russian mathematician Sofya Kovalevskaya!)

Through the lives of these brilliant folks, we hope you’ll find connections, inspiration, and empowerment.