Russell Bertrand Arthur William

<history of philosophy, biography> orphaned at the age of four, Bertrand Russell (1872-1970) studied (and later taught) both mathematics and philosophy at Cambridge. As the grandson of a British prime minister, Russell devoted much of his public effort to matters of general social concern. Jailed as a pacifist during the First World War, he later supported the battle against Fascism but deplored the development of weapons of mass destruction, as is evident in "The Bomb and Civilization" (1945), New Hopes for a Changing World (1951), and his untitled last essay. Throughout his life, Russell was an outspoken critic of organized religion, detailing its harmful social consequences in "Why I Am Not a Christian" (1927) and defending an agnostic alternative in "A Free Man's Worship" (1903). His Marriage and Morals (1929) is an attack upon the repressive character of conventional sexual morality. Russell's Autobiography (1967-69) is an excellent source of information, analysis, and self-congratulation regarding his interesting life. Its pages include his eloquent statements of "What I Have Lived For" and "A Liberal Decalogue". Russell was awarded the Nobel Prize for literature in 1950. Through an early appreciation of the philosophical work of Leibniz, published in A Critical Exposition of the Philosophy of Leibniz (1900), Russell came to regard logical analysis as the crucial method for philosophy. In Principia Mathematica (1910-13), written jointly with Alfred North Whitehead, he showed that all of arithmetic could be deduced from a restricted set of logical axioms, a thesis defended in less technical terms in Russell's Introduction to Mathematical Philosophy (1919). Applying simlarly analytical methods to philosophical problems, Russell believed, could resolve disputes and provide an adequate account of human experience. Indeed, his A History of Western Philosophy (1946) tried to show that the philosophical tradition had moved slowly but steadily toward just such a culmination. The attempt to account clearly for every constituent of ordinary assertions soon proved problematic, however. Russell proposed a ramified theory of types in order to avoid the self-referential paradoxes that might otherwise emerge from such abstract notions as "the barber who shaves all but only those who do not shave themselves" or "the class of all classes that are not members of themselves". In the theory of descriptions put forward in On Denoting (1905), Russell argued that proper analysis of denoting phrases enables us to represent all thought symbolically while avoiding philosophical difficulties about non-existent objects. As his essay on "Vagueness" (1923) shows, Russell long persisted in the belief that adequate explanations could provide a sound basis for human speech and thought. In similar fashion, the analysis of statements attributing a common predicate to different subjects in "On the Relations of Universals and Particulars" (1911) convinced Russell that both particulars and universals must really exist. He developed this realistic view further in The Problems of Philosophy (1912). Our Knowledge of the External World (1914) continues this project by showing how Russell's philosophy of logical atomism can construct a world of public physical objects using private individual experiences as the atomic facts from which one could develop a complete description of the world. Although Russell's philosophical positions were soon eclipsed by those of Wittgenstein and the logical positivists, his model of the possibilities for analytic thought remains influential. Recommended Reading: Primary sources: Bertrand Russell, A Critical Exposition of the Philosophy of Leibniz: With an Appendix of Leading Passages (Routledge, 1993); Alfred North Whitehead and Bertrand Arthur Russell, Principia Mathematica (Cambridge, 1997); Bertrand Russell, The Principles of Mathematics (Norton, 1996); Bertrand Russell, Introduction to Mathematical Philosophy (Dover, 1993); Bertrand Russell, The Philosophy of Logical Atomism, ed. by David Pears (Open Court, 1985); Bertrand Russell, The Problems of Philosophy (Oxford, 1998); Bertrand Russell, Why I Am Not a Christian, and Other Essays on Religion and Related Subjects (Simon & Schuster, 1977); Bertrand Russell, A History of Western Philosophy and Its Connection With Political and Social from the Earliest Times to the Present Day (Simon & Schuster, 1975); The Autobiography of Bertrand Russell (Routledge, 2000). Secondary sources: Ray Monk, Russell (Routledge, 1999); Essays on Bertrand Russell, ed. by E. D. Klemke (Illinois, 1971); John G. Slater, Bertrand Russell (St. Augustine, 1994); Peter Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy (Oxford, 1992); Jan Dejnozka, Bertrand Russell on Modality and Logical Relevance (Ashgate, 1999). Additional on-line information about Russell includes: McMaster University's The Bertrand Russell Archives. The Bertrand Russell Society Home Page, hosted by John Lenz. A.D. Irvine's article in The Stanford Encyclopedia of Philosophy. Mark Sainsbury's article in The Oxford Companion to Philosophy. Also see: acquaintance and description, analysis, analytic philosophy, logical atomism, Cambridge philosophy, descriptions, logical empiricism, English philosophy, impredicative definition, logic, logically proper names, logicism, philosophy of mathematics, mnemic causation, names, the persecution of philosophers, the axiom of reducibility, referential opacity, the nature of relations, skepticism about religion, Russell's paradox, set theory, 'to be', the verb, the theory of types, and vicious circles. The article in the Columbia Encyclopedia at Bartleby.com. The thorough collection of resources at EpistemeLinks.com. Eric Weisstein's discussion at Treasure Trove of Scientific Biography. Snippets from Russell in The Oxford Dictionary of Quotations. Bjoern Christensson's brief guide to Internet material on Russell. A short article in Oxford's Who's Who in the Twentieth Century. An entry in The Oxford Dictionary of Scientists. Discussion of Russell's logical treatment of mathematics from Mathematical MacTutor. A brief entry in The Macmillan Encyclopedia 2001.

[A Dictionary of Philosophical Terms and Names]

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Russell's Attic

<mathematics> An imaginary room containing countably many pairs of shoes (i.e. a pair for each natural number), and countably many pairs of socks. How many shoes are there? Answer: countably many (map the left shoes to even numbers and the right shoes to odd numbers, say). How many socks are there? Also countably many, we want to say, but we can't prove it without the Axiom of Choice, because in each pair, the socks are indistinguishable (there's no such thing as a left sock). Although for any single pair it is easy to select one, we cannot specify a general method for doing this.

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Russell's Paradox

<mathematics> A logical contradiction in set theory discovered by the British mathematician Bertrand Russell (1872-1970). If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa. This contradiction infects set theory when it is permissible to speak of "all sets" or set complements without qualification, or when a set is defined loosely as any collection of any elements, or when every predicate (intension) determines a set (extension). See complement The paradox stems from the acceptance of the following axiom: If P(x) is a property then

	{x : P}

is a set. This is the Axiom of Comprehension (actually an axiom schema). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent.

In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself:

	r = \ x . not (x x)

If we now apply r to itself,

	r r = (\ x . not (x x)) (\ x . not (x x))
	    = not ((\ x . not (x x))(\ x . not (x x)))
	    = not (r r)

So if (r r) is true then it is false and vice versa.

An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R.

Zermelo Fr"nkel set theory is one "solution" to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself.

A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway.

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Russell's theory of descriptions

<logic, philosophy of language> roughly, the view that sentences in which phrases of the form the-so-and-so appear can be reduced to more revealing logical forms in which "the" disappears and in which there is no longer any temptation to think that such phrases are like proper names. E.g. "The present king of France is bald" becomes "There exists something which is presently kind of France and there is no other individual who is such and that individual is bald". Russell's theory has been called a paradigm of philosophy.

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