The Category Mistake: Aristotle, Theophrastus, and the Dispute over the Early Academy's Discrete Architecture

Contrary to the assumption that Greek mathematics relied purely on continuous geometry, the early Lyceum's polemic against "indivisible lines" (atomoi grammai) proves otherwise. Driven by institutional rivalry, the Peripatetics committed a fundamental category mistake: they critiqued the Academy's strictly discrete, relational architecture using the incompatible, continuous standards of Euclidean space.

Evaluating the "diagonal of a square" continuously inevitably yields irrational magnitudes and incommensurability. However, within the Academy’s discrete operational logic, this purported paradox dissolves completely. By strictly separating specific operational logic from physical lengths, the underlying system reveals a perfectly commensurable coherence. Crucially, this systemic reconstruction directly corroborates the pioneering thesis of David Fowler (The Mathematics of Plato's Academy), demonstrating exactly how anthyphairetic principles formed the generative core of early Platonic mathematics.