Contributions To Mathematics
Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis (CH), nor the axiom of choice, can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent of the ZF axioms: these statements can be neither proved nor disproved from these axioms. In this sense, the continuum hypothesis is undecidable, and it is the most widely-known example of a natural statement that is independent from the standard ZF axioms of set theory.
For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967. The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic, as of 2012.
Apart from his work in set theory, Cohen also made many valuable contributions to analysis. He was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper "On a conjecture by Littlewood and idempotent measures", and lends his name to the Cohen-Hewitt factorization theorem.
Cohen was a full professor of mathematics at Stanford University, where he supervised Peter Sarnak's graduate research, among those of other students.
Angus MacIntyre of the University of London stated about Cohen: "He was dauntingly clever, and one would have had to be naive or exceptionally altruistic to put one's "hardest problem" to the Paul I knew in the '60s." He went on to compare Cohen to Kurt Gödel, saying: "Nothing more dramatic than their work has happened in the history of the subject." Gödel himself wrote a letter to Cohen in 1963, a draft of which stated, "Let me repeat that it is really a delight to read your proof of the ind of the cont hyp. I think that in all essential respects you have given the best possible proof & this does not happen frequently. Reading your proof had a similarly pleasant effect on me as seeing a really good play."
Read more about this topic: Paul Cohen (mathematician)
Famous quotes containing the words contributions to and/or mathematics:
“The vast material displacements the machine has made in our physical environment are perhaps in the long run less important than its spiritual contributions to our culture.”
—Lewis Mumford (1895–1990)
“Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don’t happen to have all the data. In mathematics we have all the data ... and yet we don’t understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.”
—Simone Weil (1909–1943)