But even as mathematics in the popular culture continues to be portrayed as the province of the lone, brilliant individual, the changing nature of mathematical proof has been creating a very different landscape for many professional mathematicians.
With the appearance of computer-aided proofs, such as the Four Color Theorem in 1977 and Kepler's Sphere Packing Conjecture in 1998, and mammoth proofs assembled from the work of hundreds of mathematicians that are too long for a single person to assess, certainty has become much more difficult to achieve. As mathematician Brian Davies wrote in a recent essay, he and his colleagues face the challenge of "proofs that are too long and complex for anyone to be able to assert with total confidence that the theorems claimed are certainly true." One group theory problem--classifying all finite simple groups---can be formulated in a few sentences but it led to a solution that is more than 10,000 pages long.
The changing nature of mathematical proof, discussed in a symposium at the 2006 Annual Meeting of the American Association for the Advancement of Science (AAAS) in St. Louis, means that some mathematicians are finding themselves in the same boat as physicists and other natural scientists. For parts of mathematics, the truth may mean "true to the best of our current knowledge," according to Keith Devlin, a Stanford University mathematician.
"The shortcomings of traditional practices of peer review become evident when applied to complex computer-assisted proofs," says Thomas Hales of the University of Pittsburgh. In 1998, Hales announced a computer-assisted proof of Kepler's Sphere Packing Conjecture. Johannes Kepler, the eminent as