Unit V Notes: Direct Bridge to geometor/codes & Dedekind Cuts¶
Algebraic & Computational Commentary on Book V
Resonance with geometor/codes¶
In the GEOMETOR ecosystem, geometor/codes investigates discrete and symbolic encodings of ratio, continued fractions, Ostrowski/Sturmian words, and Eudoxian anthyphairesis. Oliver Byrne's Book V notation (Pickering pp. 153–212) is the direct 1847 typographic ancestor of geometor/codes:
Where Euclid’s Greek text introduces new letters $E, F, G, H, K, L$ every time equimultiples $m A, m B, n C, n D$ are formed (creating an explosion of fresh variable bindings), Byrne writes explicit symbolic scalar products of his four colored tokens—turning Book V into an ordered $\mathbb{N}^+$-semimodule calculus!
Eudoxus's
V.def.5trichotomy (⊏,=, ⊐ over all $(m, n) \in \mathbb{N}^+ \times \mathbb{N}^+$) is mathematically isomorphic to Richard Dedekind’s 1872 definition of real numbers by rational cuts $(m/n)$.