Logic and philosophy of mathematics
Essay on the Foundations of Geometrywas Russell’s first book, it was published in 1897. Work heavily influenced by Immanuel Kant. Russell soon realized that the conception it laid out would have made Albert Einstein's schema ofspace-time impossible, which he understood to be superior to his own system. Thenceforth, he rejected the entire Kantian program as it related to mathematics and geometry, and he maintained that his ownearliest work on the subject was nearly without value. Russell studied the work of George Boole, Georg Cantor, and Augustus De Morgan because of his Interest in the definition of number, He becameconvinced that the foundations of mathematics were to be found in logic, and following Gottlob Frege took an logicist approach in which logic was in turn based upon set theory.
He attended the firstInternational Congress of Philosophy In 1900 ( 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 forarithmetic. 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 uponhimself 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 Onthe Notion of Order, Sur la logique des relations avec les applications à la théorie des séries, and On Cardinal Numbers.
He did this in 1903, when he published The Principles of Mathematics, inwhich the concept of class is inextricably tied to the definition of number. The appendix to this work detailed a paradox arising in Frege's application of second- and higher-order functions which took...