Introduction to Mathematical Philosophy

Bertrand Russell's Introduction to Mathematical Philosophy opens a strikingly clear path into the logical foundations of mathematics, treating the subject not as a sealed technical discipline but as an inquiry into what makes number, continuity, and proof possible at all. Russell contrasts the ordinary ascent from arithmetic to analysis with the philosophical descent from familiar ideas to their simplest logical components, arguing that the most obvious mathematical notions are often the least fundamental. He begins with the natural numbers and Peano's reduction of arithmetic to a few primitive ideas and axioms, including induction, showing both the power of this framework and its limits, since many different sequences can satisfy the same formal structure. From there, Russell turns to Frege's overlooked breakthrough and the problem of defining number itself. He distinguishes number from plurality, explains classes and one-one relations, and shows how equality of quantity can be established without counting. The book then develops the theory of relations, limits, and boundaries in ordered series, leading to a rigorous treatment of continuity through limiting points, perfect series, and the distinction between Dedekindian and Cantorian continuity. Throughout, Russell's aim is both explanatory and provocative: to show how mathematical logic can illuminate philosophy while demanding a deeper definition of number and structure. This edition has been fully restored with modern typesetting, custom cover design.