The Burden of Proof
Henri Poincare was one of the world’s foremost mathematicians and philosophers of science at the turn of the century, and his work has substantially influenced contemporary ideas about cosmogony, relativity and topology. (He was also the first cousin of Raymond Poincare, the president of France during World War I, but that’s not part of this story.)
In the 20th Century, topology--the study of the properties of shapes that remain unchanged even when the shape is distorted--has become an important topic of mathematics, in part because of its relationship to modern geometry and physics. Understanding the nature of the Einsteinian universe requires some understanding of the properties of curved space in different dimensions, though it’s hard for all of us bound to 3-space to visualize what higher dimensions might look like.
In developing his ideas, Poincare proposed what has come to be known as the Poincare conjecture, which deals with the structure of space. The proof of the Poincare conjecture has become one of the most important outstanding problems in mathematics. Twenty years ago Stephen Smale of UC Berkeley proved it for five dimensions or more, and in 1982 Michael Freedman of UC San Diego proved it for four dimensions--an accomplishment for which he was awarded the Fields Medal earlier this year.
But the Poincare conjecture in three dimensions--the most important case, after all--remained unproved despite prodigious efforts by leading mathematicians throughout this century. Early in September, however, Colin Rourke of the University of Warwick in Britain and Eduardo Rego of the University of Oporto in Portugal announced in a cover article in the British science magazine New Scientist that they had proved the Poincare conjecture for three dimensions. They were announcing this result in a popular science magazine rather than in a mathematical journal, they said, because it was so important that they did not want to have to endure the long delay that scholarly publication requires.
Many mathematicians were skeptical, and with good reason. A few weeks ago Rourke spent a week at Berkeley defending his proof at a topology seminar. Under the criticism, he acknowledged that the proof contained an error. His announcement had been premature at best.
Recognition is the prize in science, and scientists in all fields compete for credit and fame. Increasingly in recent years this competition has led to “publication by press conference,” in which results are announced before they are subjected to the scrutiny of other scientists. The claimed proof of the Poincare conjecture is only the latest example of the danger that lies in attempting to short-circuit the procedures of scholarship and the quest for truth.
Better to wait and get it right than to rush into print with a mistake.
