210518 Viktor Blasjo

Why did Greek mathematicians spend hundreds of years trying to make an angle the third of another, or a cube twice the volume of another, in dozens of different ways? What sin could be so grave that they imposed on themselves such a Sisyphean task? In my view, constructions were a foundational program to ensure consistency, validate diagrammatic reasoning, and protect against hidden assumptions. Pushing this view to its logical conclusion leads to accepting that geometry is based in physical reality, not in abstract thought - a view that is in much better agreement with ancient sources than many commentators, both ancient and modern, have cared to admit. Furthermore, this perspective suggests new interpretations and reconstructions of operationalist aspects of solutions to the classical problems that are missing in surviving sources. Notably that: Archytas's cube duplication was originally a single-motion machine; Diocles's cissoid was originally traced by a linkage device; Greek conic section theory was based on the conic compass, and in a few cases string constructions.