Kenneth Appel, 80; first to use a computer to prove a major math theorem

DOVER, N.H. — Kenneth Appel, a mathematician who was the first to use a computer to prove a major mathematical theorem, died April 19 in Dover, N.H. He was 80.

Dr. Appel, a longtime educator who chaired the University of New Hampshire mathematics department, had been diagnosed with esophageal cancer.


In 1976, he and Wolfgang Haken used 1,200 hours of calculations from an IBM computer to prove that a flat map can be colored with just four colors, so that contiguous countries have different colors.

Proving the 100-year-old ‘‘Four-Color Conjecture’’ was considered a major achievement at the time, though highly unpopular with some mathematicians who did not trust the performance of computers.

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‘‘The proof of the four-color conjecture is unlikely to be of applied significance,’’ a New York Times article said at the time. ‘‘Nevertheless, what has been accomplished is a major intellectual feat. It gives us an important new insight into the nature of two-dimensional space and of the ways in which such space can be broken into discrete portions.’’

Since the time of Euclid and Pythagoras, proofs of mathematical theorems had consisted of long strings of equations or geometric notations that any mathematician could read and quibble with, all marching logically, step by step, toward a conclusion.

Dr. Appel’s and Haken’s technique and conclusion depended on 1,200 hours of computer time — the equivalent of 50 days — and 10 billion logical decisions all made automatically and out of sight by the innards of an IBM computer at the University of Illinois in Urbana.


The revelation shepherded computers toward a greater role in higher math, but it also made many mathematicians uneasy. They worried about computer bugs and wondered how they could check or understand a ‘‘proof’’ they could not see. And it ignited a long-running debate about what constitutes a mathematical proof.

‘‘Like a landmark Supreme Court case, the proof’s legacy is still felt and hotly debated,’’ said Edward Frenkel, a mathematician at the University of California, Berkeley.

Kevin Short, a mathematician at the University of New Hampshire, called the feat ‘‘a watershed for modern mathematics.’’

‘'It has spawned whole fields of study,’’ he said.

While Dr. Appel’s achievements were well known at UNH, and probably most math departments around the country, Short said Dr. Appel himself was ‘‘an incredibly humble man.’’

‘‘He felt that it was really something that came out of a great collaboration with Professor Wolfgang Haken and that their interests, skills and background problems they had worked on dovetailed very well,’’ said Short, who was hired by UNH in the early 1990s when Dr. Appel was chairman of the math department.

He considered Dr. Appel a mentor.

‘‘One of the things he prided himself on was trying to help other faculty members in general, but in particular, young faculty members, to get them started in a research career,’’ Short said.

Kenneth Ira Appel was born in Brooklyn, N.Y., and attended the University of Michigan, from which he earned his doctorate in 1959.

Dr. Appel and Haken received the American Mathematical Society and the Mathematical Programming Society’s Delbert Ray Fulkerson prize in 1979.

Since retirement, Dr. Appel counseled students at Dover High School, helping to set up an Internet-based homework system developed at the University of Rochester. He also served on the Dover Board of Education.

Before their revolutionary work was published, Dr. Appel and Haken enlisted their entire families to check hundreds of pages of calculations, making sure that diagrams of map configurations matched the computer printouts and did not have typos. Dr. Appel’s son, Andrew, said his sister, Laurel, found some 800 mistakes, most of which she could fix herself.

Laurel F. Appel, a biology professor at Wesleyan University, died this year. Besides his son Andrew, a computer science professor at Princeton, Dr. Appel leaves his wife, Carole S. Stein; another son, Peter; a sister, Lois Green; and five grandchildren.

Despite the criticism in more traditionalist quarters, Dr. Appel never agonized about his reliance on a computer to arrive at the four-color theorem, his son Andrew said. The mathematician Alan Turing, he noted, had shown long ago that even very short theorems could have very long proofs, running hundreds of pages.

As his son recalled, Appel used to say, ‘‘Without computers, we would be stuck only proving theorems that have short proofs.’’

Material from the Associated Press and New York Times was used in this obituary.
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