A giant among giants

Paul Erdős should be a household name. By the way, his Hungarian name is pronounced “Air dish” not “Err dos.”

To those who don’t know, Erdős was one of the most prolific and impactful mathematicians in history. He had broad mathematical interests, ranging from number theory, combinatorics, to set theory. There are fields of mathematics such as random graph theory that owe an essential debt to him. Erdős was incredibly productive: he authored or co-authored 1,500 research mathematics papers and posed hundreds of problems many of which remain open to this day.

Erdős was born in Budapest, Hungary in 1913. A child prodigy, he had an insatiable passion for mathematics. By the age of 21, he received his doctorate from the University of Budapest. One of the remarkable things about Erdős was his nomadic lifestyle. Especially later in life, he didn’t seek out tenured positions at universities but instead roamed the world on permanent sabbatical working with his hundreds of co-authors.

Much has been written about Erdős’s life and work, and I direct the interested reader to search his name on the web to find out more about him. My focus instead is on his impact as a collaborative mathematician.

Before and after Erdős

One of the most important things Erdős did for mathematics was to help firmly plant it as a social activity.  He had over five hundred co-authors! The stereotype of a mathematician is someone working away in solitude, but the reality is far from that for most of us. Mathematics is often a highly social affair, where we work together solving problems staring at a blackboard, over coffee in a cafe, or chatting together in our offices. Even if mathematicians choose to work alone, to be successful we must present our work at conferences and expose it to anonymous review by others.

One thing I appreciate about Erdős’s style of collaboration was that he was not a picky: he worked with the greats and be equally enthused working with a fresh graduate student (or with a child prodigy like Tao in the image above).

Your Erdős number counts the length of a shortest path in the collaboration graph from you to him. Erdős’s co-authors each have Erdős number 1. Mathematicians who wrote a paper with a co-author of Erdős but never co-authored directly with Erdős have Erdős number 2, and so on. For example, I never co-authored with Erdős, but I’ve written papers with Peter Cameron, who has Erdős number 1, so my Erdős number is 2. For more on the Erdős number and great bibliometric data on math co-authorship see the Erdős number project.

The culture now

Go to any mathematics conference and you will find cliques of researchers huddled together hashing out ideas. It’s routine now to see mathematics papers with four or more co-authors. I think in part this owes a debt to Erdős and his passion for collaboration.  I call this the Erdős effect: his style of working with many collaborators has influenced the nature and style of mathematics we do today.

As a rough bibliometric experiment, I searched the database MathSciNet for all papers with subject code 11 or “Number Theory” in randomly chosen years from 1920 to the present. Most papers earlier in the century (and even up to 1980) are single authored. In 2000, many papers had at least two authors, and several with three, four or more. In 2016, this trend continues. Similar results hold for other searches in fields like geometry, analysis, and combinatorics.  J.W. Grossman did a deeper dive into statistics on mathematical collaboration in his 2002 paper.

Unexpected synergies can occur when mathematicians work together. We each bring our own bag of tricks to the table and discussion can help foster a novel proof. Having someone real-time push back on your claims can also be critical.

Whatever one’s views on collaborative work in mathematics, I think it is here to stay. Gone are the days when mathematicians worked and published exclusively in solitude. I’d like to think Erdős had a role to play with making mathematics more social.

Anthony Bonato