Mathematical Digest

Short Summaries of Articles about Mathematics
in the Popular Press

"Mathematics With a Moral," by Robert Osserman. The Chronicle of Higher Education, 23 April 2004, page B10.

Two major mathematical announcements mark the beginning and the end of the last 10 years: Andrew Wiles' proof of Fermat's Last Theorem in 1993 and Grigori Perelman's assertion last year that he had solved the Poincaré Conjecture. But, as Osserman points out, achievements such as Perelman's do not exist in a vacuum: besides Henri Poincaré, major players included Bernhard Riemann "whose radical rethinking of the foundations of geometry in 1854 would eventually lead to the solution of the Poincaré Conjecture" and William Thurston, "whose work provided the essential ingredients for connecting the ideas of Riemann with the conjecture of Poincaré." And this doesn't take into account other mathematicians who contributed---and are still contributing---to this endeavor.

What moral can be drawn from this story? Osserman sees two. First, if attempts to solve a problem fail, formulating a more difficult problem may lead to new insights and ultimately a solution to the original problem. Second, contrary to being a lonely enterprise, mathematics is more typically a social endeavor involving fruitful collaboration. And even when working alone, the individual cannot help but depend upon the results generated by other mathematicians.

--- Claudia Clark