Fields Medalist Paul Cohen Dies at 72

April 5, 2007

Paul Joseph Cohen, 72, emeritus professor of mathematics at Stanford, who won fame for work on set theory and was a 1966 winner of the Fields Medal, was one of the most brilliant mathematicians of the 20th century. Kurt Gödel called Cohen's set theory work as the greatest advance in the foundations of set theory since its axiomatization. Peter Sarnak of Princeton University, who received his doctorate from Stanford in 1980 under Cohen's direction, said of Cohen, "Like many great mathematicians, his mathematical interests and contributions were very broad, ranging from mathematical analysis and differential equations to mathematical logic and number theory."

Cohen is perhaps best known for his solution of the first of the 23 problems that the German mathematician David Hilbert posed at the International Mathematical Union in 1900. By the 1950s, after the work of Gödel, this problem, known as the "Continuum Hypothesis," had become the central one in the set theory.

In the late 1870s, German mathematician Georg Cantor hypothesized that any infinite subset of the set of all real numbers can be put into one-to-one correspondence either with the set of integers or with the set of all real numbers. All attempts to prove or disprove this conjecture failed until 1938, when Gödel showed it was impossible to disprove the Continuum Hypothesis.

Despite having never worked in set theory, Cohen proved that both the Continuum Hypothesis and the Axiom of Choice—two basic ideas in mathematics—were undecidable using the axioms of set theory. This result, which meant that conventional mathematics could neither prove nor disprove concrete and well-known mathematical assertions, is said to have unsettled philosophers, logicians, and mathematicians concerned with the concept of truth.

Source: Stanford News Service.