Bertrand Russell was a British philosopher, mathematician, and Nobel Laureate, who became the 3rd Earl Russell. His contribution to logical analysis influenced 20th century philosophy and had an decisive impact on mathematics.
He studied at Trinity College, Cambridge, where he immersed himself in the study of logical and mathematical questions. He achieved prominence with The Principles of Mathematics (1902) where he attempted to remove mathematics from the realm of abstract philosophy and give it a precise scientific framework.
But he is chiefly remembered in mathematics for his collaboration with the British mathematician Alfred North Whitehead. Together, they wrote Principia Mathematica (a title previously used by Newton), which tried to develop the whole of mathematics from a set of axioms together with some general postulates concerning class and membership of class. They proved that numbers can be defined as classes of a certain type, and developed logical concepts that established symbolic logic as a specialised area within philosophy.
During the development of the ideas that went into the book, Russell stumbled on a paradox. Here is how he describes this in his autiobiography.
“At the end of the Lent Term, Alys and I went back to Femhurst, where I set to work to write out the logical deduction of mathematics which afterwards became Principia Mathematica. I thought the work was nearly finished, but in the month of May I had an intellectual set-back almost as severe as the emotional set-back which I had had in February. Cantor had a proof that there is no greatest number, and it seemed to me that the number of all the things in the world ought to be the greatest possible. Accordingly, I examined his proof with some minuteness, and endeavoured to apply it to the class of all the things there are. This led me to consider those classes which are not members of themselves, and to ask whether the class of such classes is or is not a member of itself. I found that either answer implies it’s contradictory. At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case. Burali-Forti had already discovered a similar contradiction, and it turned out on logical analysis that there was an affinity with the ancient Greek contradiction about Epimenides the Cretan, who said that,
’ All Cretans are liars.’
“A contradiction essentially similar to that of Epimenides can be created by giving a person a piece of paper on which is written:
‘The statement on the other side of this paper is false.’
“The person turns the paper over, and finds on the other side:
‘The statement on the other side of this paper is true.’
“It seemed unworthy of a grown man to spend his time on such trivialities, but what was I to do? There was something wrong, since such contradictions were unavoidable on ordinary premises. Trivial or not, the matter was a challenge. Throughout the latter half of 1901 I supposed the solution would be easy, but by the end of that time I had concluded that it was a big job.”
Russell and Whitehead solved the paradox by further restricting the idea of class - membership of a class could not be ascribed to the class itself. They set out their ideas in their book. By page 362 of Principia Mathematica, Russell and Whitehead finally laid the groundwork that would enable them to prove that 1+1=2.
Russell was a pacifist during the first World War (1914-1918), for which he was imprisoned. It was in prison that he wrote Introduction to Mathematical Philosophy combining the two fields of study. Maintaining his firm anti-war beliefs, Russell became an opponent of nuclear weapons following the second World War (1939-1945).
He was awarded the Order of Merit (1949), by King George VI, and received his Nobel Prize for Literature (1950) being cited as “the champion of humanity and freedom of thought”. He also made many other contributions to a broad range of subjects, including education, politics, history, religion and science.