Construction as Existence Proof in Ancient Geometry

Abstract

The title of this essay is borrowed from a modern mathematical historian; its tag line is taken from an ancient philosopher. Their shared interest in questions dealing with existence has given rise to a familiar thesis about ancient geometry: that its constructions were intended to serve as proofs of the existence of the constructed figures. I propose here to examine that thesis, to argue its weakness as a historical account of ancient geometry, and to offer an alternative view of the role of problems of construction: 1 that constructions, far from being assigned a specifically existential role, were not even the commonly adopted format for treating of existential issues when these arose: that some central questions relating to existence were handled through postulates or tacit assumptions, rather than through explicit constructions; that, by contrast, when constructions were given, the motive lay in their intrinsic interest for the ancient geometers. On this basis I will maintain that preconceptions based on modern theories have interfered in the modern effort to interpret ancient mathematics, thus attaching to the existential view of constructions a greater credence than the ancient evidence could justify.