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RUSSELL, Bertrand (Russell Bertrand)

( English philosopher and mathematician who made significant contributions to the development of mathematical logic.)

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Biography RUSSELL, Bertrand (Russell Bertrand)
Bertrand Arthur William Russell was born in Trelleke (Wales), May 18, 1872. Grandson of Lord John Russell, 1 st Earl Russell, Bertrand Russell inherited the title in 1931. Entered at Trinity College at Cambridge University in 1890. Subsequently, a member of the Royal Society, was elected a Fellow of Trinity College at Cambridge University, lectured on philosophy in a number of universities and colleges. The essential results were obtained by Russell in symbolic logic and its application to the philosophical and mathematical problems.
Symbolic Logic. The most important work of Russell - Beginning Mathematics (Principia Mathematica, in three volumes, published. in 1910-1913) was written in collaboration with AN Whitehead. This work contains the exact wording of logic and detailed proof that the theorem of pure mathematics follows from the principles of logic, and concepts of mathematics can be defined in terms of logic. In later studies have shown, . that the system of Principia enough three indeterminate terms; Russell called them 'minimal vocabulary' of mathematics, . in theory all mathematics and logic can be formulated with the help of some of these terms,
. The thesis of the reducibility of mathematics to logic was put forward by Russell in the Principles of Mathematics (Principles of Mathematics, 1903), a number of important provisions of the Principia expounded to them in articles that have appeared previously. Among them - following the concept.

Theory of descriptions. Expressions 'by Waverley' and 'Golden Mountain' are examples of what Russell called 'the descriptions', ie. descriptive phrases. Russell has shown that such expressions can be eliminated from the language with the help of logical reformulations of proposals to which they belong. For example, say that 'The author Waverley was a Scot' - means to say 'Someone wrote Waverley and was a Scot'. Say 'Golden Mountain does not exist' - is to say 'Nothing exists is not the same golden and a mountain'. This theory eliminates the need to assume that such proposals as the 'Golden Mountain does not exist', argue about something that it does not exist, and thereby assume the realm of essences, which includes non-existent objects. In addition, the theory of descriptions offered a new type of definition, sometimes called 'contextual definition'. Rather than offer terms, . that could be substituted in place of descriptive expressions in, . they contain, . Russell gave the definition of substitution method in place to the proposals of other proposals, . having a different structure and does not contain a descriptive expression,
. According to Russell, the possibility of such definitions indicates that the grammatical form of the original proposal did not provide a key to its true meaning.

The elimination of cardinal numbers and classes. Russell showed that all the properties of numbers can be saved, if we define the cardinal numbers in terms of classes. A cardinal number of the class was defined as the class of all the classes that are similar to this class, the classes 'similar' if their member items can be placed in one-to-one correspondence with each other. 'Bijection' was determined with the help of a dictionary of terms of logic. Hence - no need to assume that in addition to classes there are such things as the number. (A similar definition was given the number of Frege in 1884.) Russell further testified, . that there is no need to tolerate the existence of the classes themselves, with the help of contextual definitions proposals, . are apparently talking about the classes, . can be replaced by other, . more complex sentences, . talking about the properties, . instead of classes,
. These definitions indicate that the object, such as classes and the number who had previously moved out of some of the data and the existence of which was for that reason problematic, can be interpreted as logical structures, built from the data. Thus, these definitions are the use of 'Occam's Razor' - the principle that the essence should not be multiplied without necessity. Russell called this way certain objects 'logical constructions' (or 'logical fictions').


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