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The Constancy of Mathematics

Many years ago I worked for the American Mathematical Society, contributing to the AMS’s flagship product MathSciNet. Mathematicians have a unique way of looking at the world. At one point in my life I thought I might become a mathematician and even majored in mathematics in school, but I found in the end that I am better at the application of mathematics than I was at theoretical thought.

Mathematicians are proud of the fact that mathematics is more enduring than nearly any other field of research. This pride manifests itself in a range of measurements through which mathematicians tie themselves to their predecessors.

Citation Distribution

Mathematicians, for instance, still rely on the work that Euclid did over two thousand years ago.

A project that I was involved in while at the AMS demonstrated this fact by examining the distribution of citations over time. In many other scientific fields, citation frequency to a given research paper drops dramatically after a certain amount time and eventually approaches zero as the science presented in the paper becomes obsolete. In mathematics, on the other hand, the tendency is for citation frequency to decline relatively steadily and eventually to level off. Certainly there are exceptions to this tendency, but as a general rule mathematics has more constantancy than other scientific fields.

Collaboration Distance

At the AMS I also worked on a related project involving what is known as the Erdős number. This number describes the “collaboration distance” between any publishing mathematician and Paul Erdős. Wikipedia defines the Erdős number in this way:

To be assigned an Erdős number, an author must co-write a research paper with an author with a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else’s Erdős number is k + 1 where k is the lowest Erdős number of any coauthor.

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators; these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2…

You can actually compute the Erdős number of nearly any mathematician using MathSciNet’s Collaboration Distance tool, which I developed the front end for. The results visually demonstrate the relationships between each of the coauthors in the chain of publication. This fun game caught on outside of mathematics with the Bacon Number, which measures collaboration distance from actor Kevin Bacon.

Ranking Content

Another facet of this emphasis on citations is demonstrated by the AMS’s Mathematical Citation Quotient (MCQ). The MCQ essentially provides a ranking of journals, books, and articles based on citation counts over various ranges of years. You can actually see the current year’s rankings on MathSciNet’s Top Journal MCQs page (which I also developed the front end for). This is not too dissimilar to how Google ranks pages based on link count.

The Great Conversation

In the 1950s, Encyclopedia Britannica Inc. published a set of books called Great Books of the Western World. The series was composed of fifty-four volumes containing books by authors from antiquity up to the present day. Included were books by authors such as Plato, Aristotle, Euclid, Augustine, Dante, Sir Isaac Newton, and Ernest Hemingway. Mortimer Adler, a key figure in the publication of this series, said regarding the great books,

What binds the authors together in an intellectual community is the great conversation in which they are engaged. In the works that come later in the sequence of years, we find authors listening to what their predecessors have had to say about this idea or that, this topic or that. They not only harken to the thought of their predecessors, they also respond to it by commenting on it in a variety of ways.

Paul Erdős and the field of mathematics then epitomizes this concept. Erdős brought mathematicians together in a way rarely realized. He published more collaborative mathematical papers than any other mathematician in history. As an itinerant mathematician, he would show up in the office of a colleague and stay long enough to work on a few papers, then depart, often consulting his current host on whom he might visit next.

The Great Conversation in a Digital Age

Advancements in transportation and communication technology enabled Paul Erdős to bring the mathematical community together in a new way and to create a radical explosion of collaboration and a rapid growth of the great mathematical conversation. As we move further into an age in which a community can go from 0 to Book in 3 Days, what levels of collaboration might we imagine? In what other fields might global conversations be enabled? Who will be the Paul Erdős of the digital publishing age?

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