by Jonathan Kujawa
Popular media loves nothing better than leaning on a tired trope when telling a tale. Mathematicians are always solitary geniuses who toil away in solitude on really hard problems. When they solve one nobody can quite say why anyone should have cared in the first place. But never mind that, it was really hard and it's all very impressive that they solved it all by their lonesome with their otherworldly brain. And they'd better be peculiar! Only Sheldon Coopers need apply.
The truth is much more interesting. Of course breakthroughs are sometimes made by the hard work of single individuals (such as Grigori Perelman's proof of the Poincaré conjecture or Andrew Wiles's proof of Fermat's Last Theorem). But much more often mathematics is a social, human enterprise. For me, at least, half the fun of it is in the give and take of sharing ideas with others. Ninety percent of my research papers are cowritten. Others may prefer to work alone, but even then they build on the work of others and hope others will read, appreciate, and use their work.
Heck, even the famously reclusive Perelman built his house on the foundation of using Ricci flow to study geometric objects provided by Richard Hamilton. And while Wiles worked in his attic office in complete secrecy for six years, he, too, depended on others. It was Berkeley number theorist (and oenophile) Ken Ribet who proved that Fermat's Last Theorem would follow from the Taniyama–Shimura conjecture about certain elliptic curves (those strange number systems we ran across last year). And when Wiles's proof of the Taniyama–Shimura conjecture looked to have a fatal flaw, it was in collaboration with Richard Taylor that a fix was finally found.
The collaborative and social nature of mathematics is exemplified by Paul Erdős. We crossed his path here at 3QD nearly two years ago. A mathematical vagabond, Erdős was famous for living out of a single suitcase. He traveled the world visiting math friends for a few days or weeks, doing math with anyone who was up for it.