Summary — Introduction: Our Problem Defined (§§ 1–9)

Primary Text: Read Verbatim Treatise Track (pp. 1–6) · Commentary: Critical Notes

Executive Summary

The Introduction (Our Problem Defined by Its Relations to Logic, Psychology and Mathematics, §§1–9, pp. 1–6) severs Kant’s identification of the a priori with the psychologically subjective. Russell defines a postulate as logically a priori if and only if without it experience of the science’s subject-matter would be impossible (§§3–5), and outlines his central thesis (§9): projective geometry and the general axioms of free mobility, manifold continuity, and multiple dimensions are a priori conditions of any form of externality, whereas the specific Euclidean zero-curvature value and exact triad of dimensions are empirical.

Argumentative Architecture

flowchart TD
            A["§§1–4 Severing Logical A Priori from Psychological Subjectivity"] --> B["§§5–8 Hypothetical Necessity & Transcendental Ground"]
            B --> C["§9 Program: Projective & General Metrical Axioms A Priori; Space-Constant Empirical"]
        

Key Arguments & Theses

§§1–5: Epistemology Freed from Empirical Psychology

Whether space perception is innate or learned belongs to psychology; epistemology asks purely logically what axioms are presupposed in the possibility of experiencing outer diversity.

§9: Partition of Geometric Axioms

Axioms shared by Euclidean, hyperbolic, and spherical/elliptic spaces are a priori; axioms distinguishing Euclid (zero space-constant and $n = 3$) are empirical.