# Paul Joseph Cohen

### Stanford News Obituary

Paul Cohen, winner of world's top mathematics prize, dies at 72

Paul Joseph Cohen, an emeritus professor of mathematics famed for work on set theory and 1966 winner of the world's top math prize, died March 23 at Stanford Hospital of a rare lung disease. He was 72.

Cohen won two of the most prestigious awards in mathematics--in completely different fields. He won the American Mathematical Society's Bôcher Prize in 1964 for analysis and the Fields Medal, considered the "Nobel Prize" of mathematics, in 1966 for logic.

"Paul Cohen was one of the most brilliant mathematicians of the 20th century," said Princeton Math Professor Peter Sarnak, who received his doctorate from Stanford in 1980 under Cohen's direction. "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. This breadth was highlighted in a conference held at Stanford last September celebrating Cohen's work and his 72nd birthday. The gathering consisted of leading experts in different fields who normally would not find themselves listening to the same set of lectures."

Stanford Mathematics Chair Yasha Eliashberg, the Herald L. and Caroline L. Ritch Professor in the School of Humanities and Sciences, was among many mathematicians who recalled Cohen's passion for tackling extremely difficult, longstanding mathematical problems--and solving them. "His solution of the Continuum Hypothesis will be remembered as one of the crown achievements of mathematics in the last 50 years, along with Andrew Wiles's proof of Fermat's Last Theorem and the recent proof of the Poincaré conjecture by Grigori Perelman," Eliashberg said.

"He is best known for his solution of the first of the 23 problems that the German mathematician David Hilbert posed in his very influential address to the International Mathematical Union in 1900," Sarnak said. "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 put forth a hypothesis that said 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 Kurt Gödel showed it was impossible to disprove the continuum hypothesis.

Despite having never worked in set theory, Cohen proved the extremely surprising result that both the Continuum Hypothesis and the Axiom of Choice--two of the most basic ideas in mathematics--were actually 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, caused healthy turbulence among philosophers, logicians and mathematicians concerned with the concept of truth.

"Kurt Gödel, whose work in the 1930s on the consistency of the continuum hypothesis was completed by Cohen's proof of its independence from the usual axioms of set theory, lauded Cohen's work as 'no doubt...the greatest advance in the foundations of set theory since its axiomatization,'" wrote Solomon Feferman, an emeritus professor of mathematics and philosophy famed for work in logic, in an e-mail interview.

Alexander S. Kechris, a descriptive set theorist at Caltech, said Cohen invented and used in his independence proofs a fundamental mathematical technique called forcing. "It has since become a major tool in modern mathematical logic," Kechris said. Feferman, the Patrick Suppes Family Professor of Humanities and Sciences, Emeritus, explained that forcing is used to construct unusual models of the axioms of set theory, whereby statements are made true in stages and forced to remain so in all further stages.

Angus MacIntyre, a professor of applied logic at Queen Mary, University of London, said Cohen "had done work that should long outlast our times. For mathematical logic, and the broader culture that surrounds it, his name belongs with that of Gödel. Nothing more dramatic than their work has happened in the history of the subject."

Cohen was born April 2, 1934, in Long Branch, N.J., to parents Abraham and Minnie, both Jewish immigrants from Poland. According to a 1968 article from the Trenton Evening Times, Cohen, the youngest of four children, was only nine years old when his sister Sylvia checked out a book about calculus from a New York library for him. Librarians were reluctant to let her have the book, much less for her younger brother, arguing that even some college professors didn't understand calculus.

Cohen grew up in Brooklyn and graduated from Stuyvesant High School in New York City in 1950. He attended Brooklyn College from 1950 to 1953 but left before receiving a bachelor's degree when he learned he could pursue graduate studies in Chicago with just two years of college under his belt. From the University of Chicago's Mathematics Department, he received a master's degree in 1954 and a doctorate in 1958. He wrote his thesis, "Topics in the Theory of Uniqueness of Trigonometric Series," under the supervision of Antoni Zygmund.

Before the award of his doctorate, Cohen taught math at the University of Rochester from 1957 to 1958. He also taught at the Massachusetts Institute of Technology during the 1958-59 academic year. From 1959 to 1961, he was a fellow at Princeton's Institute for Advanced Study.

Cohen joined Stanford's faculty in 1961 as an assistant professor of mathematics, becoming an associate professor in 1962, the year he received an Alfred P. Sloan research fellowship. In 1963, he won a Research Corp. Award for "his proof of the independence of the continuum hypothesis and of the axiom of choice, and initiating a whole series of advances in the field." He became a full professor in 1964.

"He inspired me when I was a young mathematician," said MacIntyre, a graduate student at Stanford from 1964 to 1967. "I never heard him lecture on set theory, but rather on algebraic geometry and p-adic fields. He had a very special style, full of enthusiasm and very 'hands on.' He used as little general theory as possible and always conveyed a sense that he got to the heart of things. His techniques, even in something as abstract as set theory, were very constructive. He was dauntingly clever, and one would have had to be naïve or exceptionally altruistic to put one's 'hardest problem' to the Paul I knew in the '60s."

Sarnak agreed: "Cohen was a dynamic and enthusiastic lecturer and teacher. He made mathematics look simple and unified. He was always eager to share his many ideas and insights in diverse fields. His passion for mathematics never waned."

At a White House ceremony on Feb. 13, 1968, President Johnson presented him with the 1967 National Medal of Science for "epoch-making results in mathematical logic which have enlivened and broadened investigations in the foundation of mathematics." Calling Cohen "one of the most brilliant of mathematical logicians," Johnson said: "His work has greatly influenced the foundation and development of mathematics."

In 1972, Cohen became the first holder of the Marjorie Mhoon Fair Professorship in Quantitative Science. Author of the book Set Theory and the Continuum Hypothesis, he was a member of the National Academy of Sciences, the American Academy of Arts and Sciences, the American Mathematical Society and the American Philosophical Society. He retired in 2004 but continued to teach until this quarter.

"In recent years, Cohen continued to work on notoriously difficult problems and in particular the Riemann Hypothesis, which is problem eight on Hilbert's list," Sarnak said. "As of today this problem remains unsolved."

Cohen played piano and violin and sang in a Stanford chorus and in a Swedish folk group, according to his son Charles. He spoke Swedish, French, Spanish, German and Yiddish and traveled widely, taking sabbaticals in England, Sweden, France and Hawaii. He met his wife, Christina, from the Swedish town of Malung, during a cruise from Stockholm to Leningrad in the summer of 1962. They married on Oct. 10, 1963.

Cohen is survived by wife Christina Cohen of Stanford; sister Tobel Cosiol of San Jose, Costa Rica; brother Ruby Cassel of Brooklyn, N.Y.; twin sons Eric and Steven Cohen of Los Angeles, Calif., and son Charles Cohen and his wife, Andreea, of Boston, Mass.

By DAWN LEVY

March 28, 2007, Stanford Report,

Cohen won two of the most prestigious awards in mathematics--in completely different fields. He won the American Mathematical Society's Bôcher Prize in 1964 for analysis and the Fields Medal, considered the "Nobel Prize" of mathematics, in 1966 for logic.

"Paul Cohen was one of the most brilliant mathematicians of the 20th century," said Princeton Math Professor Peter Sarnak, who received his doctorate from Stanford in 1980 under Cohen's direction. "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. This breadth was highlighted in a conference held at Stanford last September celebrating Cohen's work and his 72nd birthday. The gathering consisted of leading experts in different fields who normally would not find themselves listening to the same set of lectures."

Stanford Mathematics Chair Yasha Eliashberg, the Herald L. and Caroline L. Ritch Professor in the School of Humanities and Sciences, was among many mathematicians who recalled Cohen's passion for tackling extremely difficult, longstanding mathematical problems--and solving them. "His solution of the Continuum Hypothesis will be remembered as one of the crown achievements of mathematics in the last 50 years, along with Andrew Wiles's proof of Fermat's Last Theorem and the recent proof of the Poincaré conjecture by Grigori Perelman," Eliashberg said.

"He is best known for his solution of the first of the 23 problems that the German mathematician David Hilbert posed in his very influential address to the International Mathematical Union in 1900," Sarnak said. "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 put forth a hypothesis that said 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 Kurt Gödel showed it was impossible to disprove the continuum hypothesis.

Despite having never worked in set theory, Cohen proved the extremely surprising result that both the Continuum Hypothesis and the Axiom of Choice--two of the most basic ideas in mathematics--were actually 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, caused healthy turbulence among philosophers, logicians and mathematicians concerned with the concept of truth.

"Kurt Gödel, whose work in the 1930s on the consistency of the continuum hypothesis was completed by Cohen's proof of its independence from the usual axioms of set theory, lauded Cohen's work as 'no doubt...the greatest advance in the foundations of set theory since its axiomatization,'" wrote Solomon Feferman, an emeritus professor of mathematics and philosophy famed for work in logic, in an e-mail interview.

Alexander S. Kechris, a descriptive set theorist at Caltech, said Cohen invented and used in his independence proofs a fundamental mathematical technique called forcing. "It has since become a major tool in modern mathematical logic," Kechris said. Feferman, the Patrick Suppes Family Professor of Humanities and Sciences, Emeritus, explained that forcing is used to construct unusual models of the axioms of set theory, whereby statements are made true in stages and forced to remain so in all further stages.

Angus MacIntyre, a professor of applied logic at Queen Mary, University of London, said Cohen "had done work that should long outlast our times. For mathematical logic, and the broader culture that surrounds it, his name belongs with that of Gödel. Nothing more dramatic than their work has happened in the history of the subject."

**Life path**Cohen was born April 2, 1934, in Long Branch, N.J., to parents Abraham and Minnie, both Jewish immigrants from Poland. According to a 1968 article from the Trenton Evening Times, Cohen, the youngest of four children, was only nine years old when his sister Sylvia checked out a book about calculus from a New York library for him. Librarians were reluctant to let her have the book, much less for her younger brother, arguing that even some college professors didn't understand calculus.

Cohen grew up in Brooklyn and graduated from Stuyvesant High School in New York City in 1950. He attended Brooklyn College from 1950 to 1953 but left before receiving a bachelor's degree when he learned he could pursue graduate studies in Chicago with just two years of college under his belt. From the University of Chicago's Mathematics Department, he received a master's degree in 1954 and a doctorate in 1958. He wrote his thesis, "Topics in the Theory of Uniqueness of Trigonometric Series," under the supervision of Antoni Zygmund.

Before the award of his doctorate, Cohen taught math at the University of Rochester from 1957 to 1958. He also taught at the Massachusetts Institute of Technology during the 1958-59 academic year. From 1959 to 1961, he was a fellow at Princeton's Institute for Advanced Study.

Cohen joined Stanford's faculty in 1961 as an assistant professor of mathematics, becoming an associate professor in 1962, the year he received an Alfred P. Sloan research fellowship. In 1963, he won a Research Corp. Award for "his proof of the independence of the continuum hypothesis and of the axiom of choice, and initiating a whole series of advances in the field." He became a full professor in 1964.

"He inspired me when I was a young mathematician," said MacIntyre, a graduate student at Stanford from 1964 to 1967. "I never heard him lecture on set theory, but rather on algebraic geometry and p-adic fields. He had a very special style, full of enthusiasm and very 'hands on.' He used as little general theory as possible and always conveyed a sense that he got to the heart of things. His techniques, even in something as abstract as set theory, were very constructive. He was dauntingly clever, and one would have had to be naïve or exceptionally altruistic to put one's 'hardest problem' to the Paul I knew in the '60s."

Sarnak agreed: "Cohen was a dynamic and enthusiastic lecturer and teacher. He made mathematics look simple and unified. He was always eager to share his many ideas and insights in diverse fields. His passion for mathematics never waned."

At a White House ceremony on Feb. 13, 1968, President Johnson presented him with the 1967 National Medal of Science for "epoch-making results in mathematical logic which have enlivened and broadened investigations in the foundation of mathematics." Calling Cohen "one of the most brilliant of mathematical logicians," Johnson said: "His work has greatly influenced the foundation and development of mathematics."

In 1972, Cohen became the first holder of the Marjorie Mhoon Fair Professorship in Quantitative Science. Author of the book Set Theory and the Continuum Hypothesis, he was a member of the National Academy of Sciences, the American Academy of Arts and Sciences, the American Mathematical Society and the American Philosophical Society. He retired in 2004 but continued to teach until this quarter.

"In recent years, Cohen continued to work on notoriously difficult problems and in particular the Riemann Hypothesis, which is problem eight on Hilbert's list," Sarnak said. "As of today this problem remains unsolved."

Cohen played piano and violin and sang in a Stanford chorus and in a Swedish folk group, according to his son Charles. He spoke Swedish, French, Spanish, German and Yiddish and traveled widely, taking sabbaticals in England, Sweden, France and Hawaii. He met his wife, Christina, from the Swedish town of Malung, during a cruise from Stockholm to Leningrad in the summer of 1962. They married on Oct. 10, 1963.

Cohen is survived by wife Christina Cohen of Stanford; sister Tobel Cosiol of San Jose, Costa Rica; brother Ruby Cassel of Brooklyn, N.Y.; twin sons Eric and Steven Cohen of Los Angeles, Calif., and son Charles Cohen and his wife, Andreea, of Boston, Mass.

By DAWN LEVY

March 28, 2007, Stanford Report,