In a long and varied career **Russell** published a vast number of books on logic, theory of knowledge, and many other topics. His best known work was *Principia Mathematica *.

- His contributions relating to mathematics include his discovery of Russell's paradox, his defence of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus.
- Russell discovered the paradox which bears his name in May 1901, while working on his Principles of Mathematics (1903).
- Russell's paradox arises as a result of naive set theory's so-called unrestricted comprehension (or abstraction) axiom.
- Most attempts at resolving Russell's paradox have therefore concentrated on various ways of restricting or abandoning this axiom.
- Russell's own response to the paradox came with the introduction of his theory of types.
- Using the vicious circle principle also adopted by Henri Poincaré, together with his so-called "no class" theory of classes, Russell was then able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set", should not be applied to themselves since self-application would involve a vicious circle.
- Although first introduced by Russell in 1903 in the Principles, his theory of types finds its mature expression in his 1908 article Mathematical Logic as Based on the Theory of Types and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913).
- Russell's response to the second of these objections was to introduce, within the ramified theory, the axiom of reducibility.
- Of equal significance during this same period was Russell's defence of logicism, the theory that mathematics was in some important sense reducible to logic.
- First defended in his Principles, and later in more detail in Principia Mathematica, Russell's logicism consisted of two main theses.
- Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity.
- In Principia Mathematica, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory.
- 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.
- As one of the founders of "analytic philosophy", Russell is remembered for his work using first-order logic to show how a broad range of denoting phrases could be recast in terms of predicates and quantified variables.
- Here, as in mathematics, it was Russell's hope that by applying logical machinery and insights one would be able to resolve otherwise intractable difficulties.
- 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.
- (The details of the dismissal are recounted in Bertrand Russell and Trinity (1942) by G H Hardy.) Two years later Russell was convicted a second time.
- Married four times and notorious for his many affairs, Russell also ran unsuccessfully for Parliament, in 1907, 1922, and 1923.
- While teaching in the United States in the late 1930s, Russell was offered a teaching appointment at City College, New York.
- During the 1950s and 1960s, 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.

Born 18 May 1872, Ravenscroft, Trelleck, Monmouthshire, Wales. Died 2 February 1970, Penrhyndeudraeth, Merioneth, Wales.

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Analysis, Astronomy, Origin Wales, Prize Nobel, Set Theory

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive