The aspects of Bertrand Russell's views on philosophy cover the changing viewpoints of philosopher and mathematician Bertrand Russell (1872–1970), from his early writings in 1896 until his death in February 1970.
Philosophical work Russell is generally credited with being one of the founders of analytic philosophy, and he also produced a body of work that covers logic, the philosophy of mathematics, metaphysics, ethics and epistemology.
Analytic philosophy Bertrand Russell helped to develop what is now called "Analytic Philosophy." Alongside G. E. Moore, Russell was shown to be partly responsible for the British revolt against idealism, a philosophy greatly influenced by G. W. F. Hegel and his British apostle, F. H. Bradley. This revolt was echoed 30 years later in Vienna by the logical positivists' "revolt against metaphysics." Russell was particularly critical of a doctrine he ascribed to idealism and coherentism, which he dubbed the doctrine of internal relations; this, Russell suggested, held that to know any particular thing, we must know all of its relations. Russell argued that this would make space, time, science and the concept of number not fully intelligible. Russell's logical work with Whitehead continued this project.
Russell and Moore were devoted to clarity in arguments by breaking down philosophical positions into their simplest components. Russell, in particular, saw formal logic and science as the principal tools of the philosopher. Russell did not think we should have separate methods for philosophy. Russell thought philosophers should strive to answer the most general of propositions about the world and this would help eliminate confusions. In particular, he wanted to end what he saw as the excesses of metaphysics. Russell adopted William of Ockham's principle against multiplying unnecessary entities, Occam's razor, as a central part of the method of analysis.
Logic and philosophy of mathematics Russell had great influence on modern mathematical logic. The American philosopher and logician Willard Van Orman Quine said Russell's work represented the greatest influence on his own work."Quine, Willard Van Orman." Encyclopædia Britannica. 2008. Encyclopædia Britannica Online. Accessed 23 February 2008.
Russell's first mathematical book, An Essay on the Foundations of Geometry, was published in 1897. This work was heavily influenced by Immanuel Kant. The book was highly praised but according to the author "far more in fact than it deserved".The Autobiography of Bertrand Russell, George Allen and Unwin Ltd., 1971, p.130. This work's academic reputation was so good at the time that "reviewers have generally said of each successive book of mine that it showed a falling-off". Russell later realised that the conception it laid out would make Albert Einstein's schema of space-time impossible. Thenceforth, he rejected the entire Kantian program as it related to mathematics and geometry, and rejected his own earliest work on the subject.
Interested in the definition of number, Russell studied the work of George Boole, Georg Cantor, and Augustus De Morgan. Materials in the Bertrand Russell Archives at McMaster University include notes of his reading in algebraic logic by Charles Sanders Peirce and Ernst Schröder.Bertrand Russell Archives at McMaster UniversityIrving H. Anellis, "Schröder Material at the Russell Archives", Modern Logic 1 (1990–1991), 237–247. In 1900 he attended the first International Congress of Philosophy in Paris, where he became familiar with the work of the Italian mathematician, Giuseppe Peano. He mastered Peano's new symbolism and his set of axioms for arithmetic. Peano defined logically all of the terms of these axioms with the exception of 0, number, successor, and the singular term, the, which were the primitives of his system. Russell took it upon himself to find logical definitions for each of these. Between 1897 and 1903 he published several articles applying Peano's notation to the classical Boole-Schröder algebra of relations, among them On the Notion of Order, Sur la logique des relations avec les applications à la théorie des séries, and On Cardinal Numbers. He became convinced that the foundations of mathematics could be derived within what has since come to be called higher-order logic which in turn he believed to include some form of unrestricted comprehension axiom.
Russell then discovered that Gottlob Frege had independently arrived at equivalent definitions for 0, successor, and number, and the definition of number is now usually referred to as the Frege-Russell definition.Griffin, Nicholas ed. The Cambridge Companion to Bertrand Russell. Cambridge. 2003. p. 412 Russell drew attention to Frege's priority in 1903, when he published The Principles of Mathematics (see below).Russell's The Principles of Mathematics The appendix to this work, however, described a paradox arising from Frege's application of second- and higher-order functions which took first-order functions as their arguments, and Russell offered his first effort to resolve what would henceforth come to be known as the Russell Paradox. Before writing Principles, Russell became aware of Cantor's proof that there was no greatest cardinal number, which Russell believed was mistaken.The Autobiography of Bertrand Russell, George Allen and Unwin Ltd., 1971, p.147. Cantor's paradox in turn was shown (for example by Crossley) to be a special case of the Russell Paradox. This caused Russell to analyse classes, for it was known that given any number of elements, the number of classes they result in is greater than their number. Russell's paradox involves the class of all classes, divided into two kinds: those classes that contain themselves, and those that do not. This class demonstrates a fatal flaw in the so-called principle of comprehension, which had been taken for granted by logicians of the time: it results in a contradiction, whereby Y is a member of Y if and only if Y is not a member of Y. Russell outlined a resolution of this paradox in an appendix to Principles, which he later developed into the theory of types. Aside from exposing a major inconsistency in naive set theory, Russell's work led directly to the creation of modern axiomatic set theory. It also crippled Frege's project of reducing arithmetic to logic. The Theory of Types and much of Russell's subsequent work have also found practical applications in computer science and information technology.
Russell continued to defend logicism, the view that mathematics is in some important sense reducible to logic, and along with his former teacher, Alfred North Whitehead, wrote the monumental Principia Mathematica, an axiomatic system on which all of mathematics can be built. The first volume of the Principia was published in 1910, and is largely ascribed to Russell. More than any other single work, it established the speciality of mathematical or symbolic logic. Two more volumes were published, but their original plan to incorporate geometry in a fourth volume was never realised, and Russell never felt up to improving the original works, though he referenced new developments and problems in his preface to the second edition. Upon completing the Principia, three volumes of extraordinarily abstract and complex reasoning, Russell was exhausted, and he felt his intellectual faculties never fully recovered from the effort.The Autobiography of Bertrand Russell, the Early Years, p. 202. Although the Principia did not fall prey to the paradoxes in Frege's approach, it was later proven by Kurt Gödel that neither Principia Mathematica, nor any other consistent system of primitive recursive arithmetic, could decide the truth of every proposition by a proof or disproof within that system (see: Gödel's incompleteness theorem).
Russell's last significant work in mathematics and logic, Introduction to Mathematical Philosophy, was written while he was in jail for his anti-war activities during World War I. This was largely an explication of his previous work and its philosophical significance.
Philosophy of language Russell made language, or more specifically, how we use language, a central part of philosophy, and this influenced Ludwig Wittgenstein, Gilbert Ryle, J. L. Austin, and P. F. Strawson, among others, who used many of the techniques that Russell originally developed. Russell, and G. E. Moore, argued that clarity of expression is a virtue.
A significant contribution to philosophy of language is Russell's theory of descriptions, set out in On Denoting (Mind, 1905). Frank Ramsey described this paper as "a paradigm of philosophy." The theory considers the sentence "The present King of France is bald" and whether the proposition is false or meaningless. Frege had argued, employing his distinction between sense and reference, that such sentences were meaningful but neither true nor false. Russell argues that the grammatical form of the sentence disguises its underlying logical form. Russell's Theory of Definite Descriptions enables the sentence to be construed as meaningful but false, without commitment to the existence of any present King of France. This addresses a paradox of great antiquity (e.g. "That which is not must in some sense be. Otherwise, how could we say of it that it is not?" etc.), going back at least as far as Parmenides. In Russell's own time, Meinong held the view of that-which-is-not being in some sense real; and Russell held this view prior to On Denoting. Principles of Mathematics (1903) pg. 449.
The problem is general to what are called "definite descriptions." Normally this includes all terms beginning with "the," and sometimes includes names, like "Walter Scott." (This point is quite contentious: Russell sometimes thought that the latter terms shouldn't be called names at all, but only "disguised definite descriptions," but much subsequent work has treated them as altogether different things.) What is the "logical form" of definite descriptions: how, in Frege's terms, could we paraphrase them to show how the truth of the whole depends on the truths of the parts? Definite descriptions appear to be like names that by their very nature denote exactly one thing, neither more nor less. What, then, are we to say about the proposition as a whole if one of its parts apparently isn't functioning correctly?
Russell's solution was, first of all, to analyse not the term alone but the entire proposition that contained a definite description. "The present king of France is bald," he then suggested, can be reworded to "There is an x such that x is a present king of France, nothing other than x is a present king of France, and x is bald." Russell claimed that each definite description in fact contains a claim of existence and a claim of uniqueness which give this appearance, but these can be broken apart and treated separately from the predication that is the obvious content of the proposition. The proposition as a whole then says three things about some object: the definite description contains two of them, and the rest of the sentence contains the other. If the object does not exist, or if it is not unique, then the whole sentence turns out to be false, not meaningless.
One of the major complaints against Russell's theory, due originally to Strawson, is that definite descriptions do not claim that their object exists, they merely presuppose that it does.