For six years, mathematicians have pored over hundreds of pages of a paper by Dr. Thomas C. Hales, a professor of mathematics at the University of Pittsburgh.



But Dr. Hales’s proof of the problem, known as the Kepler Conjecture, hinges on a complex series of computer calculations, too many and too tedious for mathematicians reviewing his paper to check by hand.


Believing it thus, at some level, requires faith that the computer performed the calculations flawlessly, without any programming bugs. For a field that trades in dispassionate logic and supposedly unambiguous truths and falsehoods, that is an uncomfortably gray in-between.



Because of the ambiguities, the journal, the prestigious Annals of Mathematics, has decided to publish only the theoretical parts of the proof, which have been checked in the traditional manner. A more specialized journal, Discrete and Computational Geometry, will publish the computer sections.



The decision represents a compromise between wholehearted acceptance and rejection of the computer techniques that are becoming more common in mathematics.



The debate over computer-assisted proofs is the high-end version of arguments over using calculators in math classes — whether technology spurs greater achievements by speeding rote calculations or deprives people of fundamentals.



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