Summary — Chapter XLIV: Dimensions and Complex Numbers (§§ 351–360)

Primary Text: Read Verbatim Treatise Track (pp. 371–380) · Commentary: Critical Notes

Executive Summary

Opening Part VI (Space), Chapter XLIV (Dimensions and Complex Numbers, §§351–360, pp. 371–380) recasts the concept of spatial dimension into pure relation-arithmetic. Reviewing the rise of non-Euclidean geometry (§353), Russell defines geometry as the theory of multiple series of two or more dimensions (§352) and demonstrates against Cantor’s bijection paradox ($2^{\alpha_0}$) that dimensionality belongs not to the cardinal point-set, but to the nested hierarchy of generating serial relations (§§354–356). Sections 357–360 apply this definition to universal algebra and $n$-tuple complex numbers, proving Stolz’s theorem that ordinary two-term complex numbers ($x + iy$) are the maximal extension preserving the field laws.

Argumentative Architecture

flowchart LR
            A["§§351–353 Non-Euclidean Revolution"] --> B["§§354–356 Logical Definition of n Dimensions"]
            B --> C["Cantor Cardinality vs. Relational Dimension"]
            C --> D["§§357–360 Universal Algebra & Complex Numbers"]
        

Key Arguments & Theses

§§351–353: Retrospect on Non-Euclidean Geometry

Summarizing Gauss, Lobachevsky, Bolyai, Riemann, Beltrami, and Cayley–Klein, Russell shows how metageometry forced the separation of pure relational geometry from physical space.

§§354–356: Purely Logical Definition of $n$ Dimensions

An $n$-dimensional series is defined inductively through $n$ transitive asymmetrical relations mapping terms through nested classes of series, resolving Cantor’s discovery that a line and a plane have the same cardinal number of points.

§§357–360: Complex Numbers as Ordered Couples of Reals

Drawing on Whitehead’s Universal Algebra, Russell rejects formalist ‘uninterpretable symbols’ and defines complex numbers explicitly as one-dimensional series of real-number couples.