Summary — Preface (1903)

Primary Text: Read Verbatim Treatise Track (pp. v–ix) · Commentary: Critical Notes

Executive Summary

In the December 1902 Preface to The Principles of Mathematics, Bertrand Russell announces his decisive break from the neo-Kantian and neo-Hegelian framework of his 1897 Essay on the Foundations of Geometry. Guided by G. E. Moore’s analytic realism, Giuseppe Peano’s symbolic logic, Georg Cantor’s transfinite arithmetic, and Gottlob Frege’s foundational works, Russell formulates the twin pillars of logicism: (1) that all pure mathematics—including Euclidean and non-Euclidean geometry—derives solely from a small set of logical constants without appeal to spatial intuition, and (2) that mathematical existence-theorems are secured constructively through the principle of abstraction.

Argumentative Architecture

flowchart LR
            A["1903 Preface"] --> B["Realism over Idealism (G. E. Moore)"]
            A --> C["Symbolic Logic (Peano & Frege)"]
            A --> D["Existence-Theorems via Principle of Abstraction"]
            A --> E["Candor on Russell's Class Contradiction (Ch. X)"]
        

Key Arguments & Theses

Philosophical Realism of Propositions

Against psychologism and Kantian idealism, Russell holds that what is true or false—a proposition—is non-mental, objective, and composed of mind-independent terms and relations.

Intellectual Debt to Peano, Frege, and Whitehead

Russell credits Peano’s symbolic notation for separating distinct logical relations previously conflated by ordinary grammar (membership vs. inclusion), acknowledges independent convergence with Frege, and highlights ongoing collaboration with A. N. Whitehead toward Volume II (realized in 1910–1913 as Principia Mathematica).

Constructive Existence-Theorems

Defining cardinals, ordinals, complex numbers, and geometric ideal elements as logical classes via equivalence relations eliminates unproved postulates and guarantees non-emptiness whenever the underlying relation is instantiated.