Paul Cohen Biography

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World of Mathematics on Paul Cohen

Paul Cohen's reputation as a mathematician has been earned at least partly because of his ability to work successfully in a number of very different fields of mathematics. He received the highly regarded Bôcher Prize of the American Mathematical Society, for example, in 1964 for his research on the Littlewood problem. Two years later he was awarded perhaps the most prestigious prize in mathematics, the Fields Medal, for his research on one of David Hilbert's "23 most important problems" in mathematics, proving the independence of the continuum hypothesis.

Paul Joseph Cohen was born in Long Branch, New Jersey, on April 2, 1934, but his childhood and adolescence were spent in Brooklyn, New York. His parents were Abraham Cohen and the former Minnie Kaplan. Both parents had immigrated to the United States from western Russia (now part of Poland) while they were still teenagers. Cohen's father became a successful grocery jobber in Brooklyn.

Cohen appears to have had a natural and precocious interest in mathematics from an early age. To a large extent, he was self-educated, depending on books that he could find in the public library or that his elder sister Sylvia was able to borrow for him from Brooklyn College. He told interviewers Donald J. Albers and Constance Reid for their book More Mathematical People that "by the time I was in the sixth grade I understood algebra and geometry fairly well. I knew the rudiments of calculus and a smattering of number theory."

For his secondary education, Cohen attended the Stuyvesant High School in lower Manhattan, widely regarded as one of the two (along with the Bronx High School of Science) best mathematics and science high schools in the United States. In 1950, having skipped "a few grades," as he told Albers and Reid, he graduated from Stuyvesant at the age of 16. He ranked sixth in his class and received one of the 40 national Westinghouse Science Talent Search awards given that year. He then enrolled at Brooklyn College, where he remained for two years. In 1952 he was offered a scholarship at the University of Chicago, from which he received his M.S. in mathematics in 1954 and his Ph.D. in 1958.

Solves the Littlewood Problem of Harmonic Analysis

Until he reached Chicago, Cohen had a relatively unstructured and diverse background in mathematics. He was fairly knowledgeable in some areas that interested him especially and that he had been able to teach himself. But he was still naive about some important areas of mathematics, such as logic, in which he had never had a formal course or even any informal training. Partly through the influence of one of his professors at Chicago, Antoni Zygmund, Cohen became interested in a classical problem in harmonic analysis commonly known as the Littlewood problem, named for the English mathematician John Edensor Littlewood. Cohen's solution to this problem won him the American Mathematical Society's Bôcher Prize in 1964.

On receiving his degree from Chicago, Cohen accepted a position as instructor of mathematics at the Massachusetts Institute of Technology (MIT). A year later he moved to the Institute for Advanced Studies at Princeton, New Jersey, where he was a fellow from 1959 to 1961. At MIT and Princeton Cohen continued to work on problems of analysis and seemed to have found a field to which he could devote his career. That illusion soon evaporated, however. As Cohen later told Albers and Reid, he has a restless mind and is constantly looking for new fields to conquer. "I [have been] told by many people that I should stick to one thing," he said, "but I have always been too restless."

Solution of the Consistency Proof Problem Brings the Fields Medal

An occasion for shifting gears presented itself to Cohen soon after he was appointed assistant professor at Stanford in 1961. At a departmental lunch, Cohen's colleagues were discussing the problems of developing a "consistency proof" in logic, first suggested by Georg Cantor in the late 19th century. The term consistency in mathematics refers to the condition that any mathematical theorem be free from contradiction. Developing a consistency proof had been listed as number one on David Hilbert's 1900 list of the 23 most important problems in mathematics for the twentieth century. Although he had no specific background in the field of logic, in which the consistency proof is particularly relevant, Cohen was intrigued by the challenge. He saw it as a way of providing convincing evidence "that set theory is based on some kind of truth," as he told Albers and Reid.

Cohen's work on the consistency proof went forward in fits and starts over the next two years. During one period he became so discouraged that he set the work aside and concentrated on other problems. He seems to have had a glimpse of the general approach for solving his problem during a vacation with his future wife to the Grand Canyon in late 1962. Still, it was another four months before the details of that approach were worked out and a solution produced. Two years later, Cohen received his second major award in mathematics, the International Mathematics Union's Field Prize, for his work on the consistency proof.

In 1964 Cohen was promoted to the post of professor of mathematics at Stanford, a position he has held since. He continues to work on a variety of problems, including those in the fields of analysis and logic. Cohen was married to Christina Karls, a native of Sweden, in 1963. They have three sons, Steven, Charles, and Eric. In addition to the Bôcher Prize and the Fields Medal, Cohen was awarded the Research Corporation of America Award in 1964 and the National Medal of Science in 1967.