"2003: Mathematicians Face Uncertainty," by Keith Devlin. Discover, January 2004, page 36.
The question "What is a proof?" can be answered in two ways, according to Keith Devlin. On the one hand, it is "a logically correct argument that establishes the truth of a given statement." On the other hand, it is "an argument that convinces a typical mathematician." Devlin argues that the former definition is generally an "unattainable ideal" and points to three events of the past year as evidence. The first was the apparent proof of a result closely related to the Twin Prime Conjecture; a few weeks after its announcement, the proof was found to be in error. The second was an outline of the proof of the Poincaré conjecture that has yet to be accepted as correct after many months. Finally, the Annals of Mathematics has agreed to publish a 1998 proof of a Kepler conjecture regarding the most efficient way to arrange spheres, but with a disclaimer regarding its veracity. These complicated proofs demonstrate to Devlin that, except for fairly simple cases, an assertion really becomes a proof when the mathematical community accepts it as such.
See also the Letter to the Editor in response to this article. The Letter, from Steven Goldberg, appeared in the March 2004 issue of Discover.
--- Claudia Clark