Bertrand Russell's Views On Philosophy - Philosophical Work - Logic and Philosophy of Mathematics

Logic and Philosophy of Mathematics

Russell had great influence on modern mathematical logic. The American philosopher and logician Willard Quine said Russell's work represented the greatest influence on his own work.

Russell's first mathematical book, An Essay on the Foundations of Geometry, was published in 1897. This work was heavily influenced by Immanuel Kant. Russell later realized 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. 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. Russell drew attention to Frege's priority in 1903, when he published The Principles of Mathematics (see below). 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 Cantor Paradox in turn was shown (for example by Crossley) to be a special case of the Russell Paradox. This caused Russell to analyze classes, for it was known that given any number of elements, the number of classes they result in is greater than their number. This in turn led to the discovery of a very interesting class, namely, the class of all classes. It contains two kinds of classes: those classes that contain themselves, and those that do not. Consideration of this class led him to find a fatal flaw in the so-called principle of comprehension, which had been taken for granted by logicians of the time. He showed that it resulted in a contradiction, whereby Y is a member of Y, if and only if, Y is not a member of Y. This has become known as Russell's paradox, the solution to which he outlined in an appendix to Principles, and which he later developed into a complete theory, 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 with 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 specialty of mathematical or symbolic logic. Two more volumes were published, but their original plan to incorporate geometry in a fourth volume was never realized, 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. 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, within that system, determine that every proposition that could be formulated within that system was decidable, i.e. could decide whether that proposition or its negation was provable within the 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.