   Examples of this general procedure can be found throughout Russell’s writings, and Russell also credits others with having achieved similar successes.  Russell’s work in mathematical logic provides perhaps the most obvious example of his utilization of such a procedure. It is also an excellent example of Russell’s contention that analysis proceeds in stages. Russell saw his own work as the next step in a series of successes beginning with the work of Cantor, Dedekind and Weierstrass. Prior to the work of these figures, mathematics employed a number of concepts, number, magnitude, series, limit, infinity, function, continuity, etc., without a full understanding of the precise definition of each concept, nor how they related to one another. By introducing precise definitions of such notions, these thinkers exposed ambiguities (e.g., such as with the word “infinite”), revealed interrelations between certain of them, and eliminated dubious notions that had previously caused confusion and paradoxes (such as those involved with the notion of an “infinitesimal”). Russell saw the next step forward in the analysis of mathematics in the work of Peano and his associates, who not only attempted to explain how many mathematical notions could be “arithmetized”, i.e., defined and proven in terms of arithmetic, but had also identified, in the case of arithmetic, three basic concepts (zero, successor, and natural number) and five basic principles (the so-called “Peano axioms”), from which the rest of arithmetic was thought to be derivable.