A few mathematicians do spend their professional lives in solitary contemplation of a deep mathematical problem. In general, however, mathematical research is a remarkably social process. Colleagues meet constantly to compare notes, discuss problems, look for hints, and work on proofs together.
The abundance of conferences, symposia, workshops, seminars, and other gatherings devoted to mathematical topics attests to a strong desire for interaction.
Paul Erdős (1913-1996), perhaps more than any other mathematician in modern times, epitomized the strength and breadth of mathematical collaboration. Because he had no permanent home and no particular job, Erdős simply traveled from one mathematical center to another, sometimes seeking new collaborators, sometimes continuing a work in progress. His well-being was the collective responsibility of mathematicians throughout the world.
Paul Erdős (1913-1996). MAA Convergence Portrait Gallery.
"My brain is open," Erdős typically declared on stepping into a mathematician's office, and the work would begin. For him, doing mathematics was as natural as breathing, and he did if for more than 65 years.
Driven by the notion that life represents a cosmic struggle to uncover beautiful truths hidden away by a stubborn, contrary God, Erdős applied himself to his pursuit with frenetic zeal.
"A mathematician is a device for turning coffee into theorems," Erdős wryly remarked.
To Erdős, mathematics and its elements were more real than trees and grass, transcending reality and awaiting discovery. At the same time, though he did not like having possessions, Erdős was not an ascetic. He liked to eat well in good restaurants and stay in fine hotels when he got the chance.
A compassionate, generous, gentle man, he was well informed on almost any topic of conversation and deeply aware of the tragedies of world politics.
Erdős wrote hundreds of research papers on a wide range of mathematical topics. Especially astonishing was the extent to which he also worked with other mathematicians to produce joint papers.
Collaboration on such a scale had never been seen before in mathematics, and it has entered the folklore of the mathematical community. Of course, there's a characteristically mathematical way to describe this webbiness—a quantity called the Erdős number.
Mathematicians assign Erdős the number 0. Anyone who has coauthored a paper with him has the cherished Erdős number 1. As of August 2020, there were 512 such coauthors. Another 12,600 mathematicians have the Erdős number 2, because they wrote a paper not with Erdős himself but with someone who wrote a paper with Erdős.
People belonging to these two categories already encompass a significant proportion of all mathematicians in academia worldwide.
The Erdős number 3 goes to anyone who has collaborated with someone who has collaborated with someone who coauthored a paper with Erdős, and so on. Thus, any person not yet assigned an Erdős number who has written a joint mathematical paper with a person having Erdős number n earns the Erdős number n + 1..
Anyone left out of this assignment process has the Erdős number infinity.
Keeping track of these mathematical links has become a kind of game, punctuated by published, tongue-in-cheek explorations of the properties of Erdős numbers and the extent of mathematical links to the rest of the world.
Theoretical physicist Albert Einstein (1879-1955), for instance, has an Erdős number 2. Andrew Wiles, who in 1994 proved Fermat's Last Theorem, has an Erdős number no greater than 4.
It's possible to draw a collaboration graph in which every point represents a mathematician, and lines join mathematicians who have collaborated with each other on at least one published paper. The resulting tangle is one of the largest, most elaborate graphs available to mathematicians.
Sone scholars have conjectured that this monstrous graph snares nearly every present-day mathematician and has threads into all areas of mathematics, into computer science and physics, and even into the life sciences and social sciences.
This collaboration graph's breadth is astonishing and vividly demonstrates the vital role of collaboration in math and science.
Several decades ago, mathematician Jerrold W. Grossman took on the task of maintaining a comprehensive listing of mathematicians who have earned an Erdős number 1 or 2. "It's fun," Grossman said. "But more seriously, it shows that mathematical research is very webby, with almost everybody talking to people who talk to people."
Indeed, fascinating patterns emerge from the study of Erdős collaboration lists. For example, the average number of authors per research article in the mathematical sciences increased dramatically during Paul Erdős's lifetime.
About 70 years ago, more than 90 percent of all papers were solo works, according to Grossman. In more recent years, barely half of the papers were individual efforts.
In the same period, the fraction of two-author papers rose from less than one-tenth to roughly one-third. In 1940, virtually no papers had three authors. Now, more than 10 percent of all papers have three or more authors.
Erdős himself may have played a role in this relatively recent explosion of collaboration.
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