The mathematician Paul Erdős had an Erdős number of 0. Anyone who has written a paper with Erdős has an Erdős number of 1. Anyone who writes a paper with someone having an Erdős number of 1 has an Erdős number of 2. And so on.

In his biography My Brain Is Open, Bruce Schechter writes about Erdős numbers and the insights Jerry Grossman's Erdős Number Project gives into the sociology of mathematical collaboration. When Erdős started publishing, very few maths papers had more than one author: in 1940, 90% of papers were one-author efforts; in 2000, fewer than half were. Discussing this growth in collaboration and the publish-or-perish mentality, Schechter presents the following verse written by Erdős:A theorem a day

Means promotion and pay!

A theorem a year

And you're out on your ear!

That's how research is now; but things used to be different. Here's an anecdote I found in The Gatekeeper, the autobiography of literary critic and English don Terry Eagleton:

I heard later of an antique Cambridge don who worked in a small department which had just acquired a new, zealous head who insisted that his colleagues produce some tangible evidence of research. In the Cambridge of those days, this would no doubt have been as stunning a demand as requiring the dons to have sexual intercourse in public with sheep. Publication was in general regarded as a mildly vulgar, publicity-seeking affair, as opposed to more enduring achievements like providing some robust chairmanship of the college wine committee. The new head of department, fed up of having to chivvy his laggard colleagues, eventually set them a deadline for producing their research, and as the hour of reckoning drew nearer, the ancient don grew more and more visibly agitated. Finally, at ten minutes to midnight on the deadline day, the window of his house in a leafy Cambridge suburb was heard to open, and his quavering voice rang out across the street: 'Stop thief! He's got my research!'