In the work of Russell from September 1900 on, the metaphysics of Platonic Atomism became interwoven with mathematics and mathematical logic. 'This interweaving is one of the points of origin of a conception of philosophy which was to have a decisive influence on the development of analytic philosophy, a conception according to which the technical is not separable from the philosophical, and mathematical logic provides the crucial method of philosophy. Russell had a long-standing interest in the philosophy of mathematics, especially of geometry, but it was not until after he attended the Congress of Philosophy in Paris, in July 1900, that he began to take a serious interest in mathematical logic. He later described his attendance at this Conference as 'the most important event' in 'the most important year' in his intellectual life. 1 By his own account, Russell was so impressed with Peano's performance at the Congress that he began to study his work. Within a month he had mastered the logic of Peano and his school, and had begun to extend the technique to new areas—most notably to the logic of relations. 2 This phase of Russell's work includes his discovery of the class paradox which bears his name, in June 1901, and culminates in The Principles of Mathematics, 3 which was completed in the summer of 1902.
Principles has, as Russell says in its Preface, two main objects. The first is to present and discuss the basic concepts and principles of logic, including the theory of classes. The second is to argue for logicism, i.e. the claim that mathematics is logic. (More fully: that all concepts of mathematics can be defined in terms of the basic concepts of logic, and