Bertrand Russell was the grandson of Lord John Russell, who had twice served as Prime Minister under Queen Victoria. Educated at first privately, and later at Trinity College, Cambridge, Russell obtained first class degrees both in mathematics and in the moral sciences.

Although elected to the Royal Society in 1908, Russell's career at Trinity appeared to come to an end in 1916 when he was convicted and fined for anti-war activities. He was dismissed from the College as a result of the conviction. Two years later Russell was convicted a second time. This time he spent six months in prison. It was while in prison that he wrote his well-received Introduction to Mathematical Philosophy in 1919. He did not return to Trinity until 1944. Married four times and notorious for his many affairs, Russell also ran unsuccessfully for Parliament, in 1907, 1922, and 1923. Together with his second wife, he opened and ran an experimental school during the late 1920s and early 1930s. He became the third Earl Russell upon the death of his brother in 1931.

While teaching in the United States in the late 1930s, Russell was offered a teaching appointment at City College, New York. The appointment was revoked following a large number of public protests and a judicial decision, in 1940, which stated that he was morally unfit to teach at the College. Nine years later he was awarded the Order of Merit. He received the Nobel Prize for Literature in 1950.

During the 1950's and 1960's, Russell became something of an inspiration to large numbers of idealistic youth as a result of his continued anti-war and anti-nuclear protests. Together with Albert Einstein, he released the Russell-Einstein Manifesto in 1955, calling for the curtailment of nuclear weapons. In 1957, he was a prime organizer of the first Pugwash Conference, which brought together scientists concerned about the proliferation of nuclear weapons. He became the founding president of the Campaign for Nuclear Disarmament in 1958 and was once again imprisoned, this time in connection with anti-nuclear protests, in 1961. Upon appeal, his two-month prison sentence was reduced to one week in the prison hospital. He remained a prominent public figure until his death nine years later at the age of 97.

Russell discovered the paradox which bears his name in 1901. The paradox arose in connection with the set of all sets which are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.

Although first introduced by Russell in 1903, his theory of types finds its mature expression in an 1908 article and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica. In it, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. A fourth volume on geometry was planned but never completed.

Of equal significance during this same period was Russell's defense of logicism, the theory that mathematics was in some important sense reducible to logic. Russell's logicism consisted of two main theses. The first is that all mathematical truths can be translated into logical truths. The second is that all mathematical proofs can be recast as logical proofs.

In much the same way that Russell wanted to use logic to clarify issues in the foundations of mathematics, he also wanted to use logic to clarify issues in philosophy. He is one of the founders of "analytic philosophy". Along with Kurt Gödel, he is usually credited with being one of the two most important logicians of the twentieth century.