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 astronomer and mathematician, had speculated in 1611 about the most efficient way to pack equal-sized spheres into a crate of given size. Would it be the staggered layers used by grocers to stack oranges for display or some other arrangement. The problem is considered by mathematicians in terms of the density of the packing rather than the total number of spheres. While the statement of the problem is simple, Hales' proof involved hundreds of pages of text and gigabytes of data.
The Annals of Mathematics assigned a panel of 12 referees to assess the correctness of the proof. Four years later, Robert MacPherson, editor of the journal, wrote Hales that the referees had been unable to certify the correctness of the proof and had run out of energy to devote to the problem. But MacPherson wrote that chief referee Gabor Fejes Toth was 99 percent certain that the proof is correct. Hales himself launched an effort, which he calls Project Flyspeck, to create a rigorous verification of his proof in which all intermediate logical steps are supplied. As a practical matter, it has meant more reliance on computers to verify the previous computer-assisted work. The verification project could take many more years.
Some mathematicians worry about an erosion in the tradition of publishing proofs for validation and archiving. Complex proofs are being posted online for rapid dissemination but unpublished elsewhere. Steven Krantz, a professor of mathematics at Washington University in St. Louis, is worried that a new generation of mathematicians may become indifferent to publishing their work in traditional peer-reviewed journals, even as the process of review itself becomes more daunting.
He cites the example of the Poincare Conjecture, a long-standing problem in topology named after the French mathematician who first posed it in 1904. A claimed proof, posted on the Internet in 2003, has been the focus of considerable study but no one is sure yet whether it is correct or not. Grisha Perelman, a Russian mathematician, has posted several papers regarding the claimed proof, drawing on his work and that of Richard Hamilton of Columbia University. Perelman has lectured on the proof but has not submitted it for publication.
"It is not clear whose name should be attached to the proof, if indeed there is a proof," writes Krantz. "It is not clear that there is any single person who is qualified to verify the proof. It is not clear that the proof will ever be published in the usual manner. There is some danger that, in five years, nobody will know what is true, what parts of the Poincare Conjecture have been verified, and what parts require further work." While mathematicians grapple with such dilemmas, educators face challenges of their own in trying to bolster lagging achievement levels in math by U.S. students compared to students from other industrialized countries.
They have found a surprising ally in a popular television series, NUMB3RS, which airs on CBS on Fridays. The show features a young mathematician, played by David Krumholtz, who helps his brother, an FBI agent, solve crimes using mathematical methods. In the initial episode, the mathematician, Charlie Eppes, developed a formula to determine the likely location and workplace of a criminal, based on his crime patterns. He also has used math to compare DNA sequences, analyze the structural stability of a building and develop a new technique for enhancing video images. Gary Lorden of the California Institute of Technology, who serves as an adviser to the show, said that Krumholtz gives an effective portrayal of a mathematician, including the ability to write equations on a board while also talking. Krumholtz spent time in Caltech math classes while preparing for his role. "He soaked it up, like Meryl Streep playing a violinist," Lorden said.
Lorden and Krumholtz are participants in a symposium at the AAAS meeting on NUMB3RS and the challenge of changing public perceptions about math. For his part, Lorden is pleased that the show is able to reveal some of the usefulness and beauty of math within the constraints of television drama. "I am impressed that they are able to say something in 30 or 40 seconds that is not completely meaningless," Lorden said. "It's neat that they are actually able to plant a few concepts."
Devlin, who also is impressed by NUMB3RS, notes that popular shows and movies have spurred interest in science fields in the past, including an upsurge in interest in forensic science due to the success of the show CSI.
"Surely almost anything that can improve the image of mathematics in the population at large deserves the support of the mathematics community," Devlin wrote shortly after the show's premiere. While the writers and actors inevitably use some dramatic license in their portrayal of mathematicians and their world, he said, "at heart they get the mathematician and the math right."
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