Saying the Unsayable: The Sophists, Geometry, and the Incommensurability of Meaning

The ancient Greek Sophists in the 5^th century BCE were as engaged in questions of geometry as in questions of language.  The point of conjunction between geometry and rhetoric appears to be that commensurable mathematical ideas are expressible in language and that incommensurable ideas are not. For example, it is impossible to express the value of π as an exact measurement. The disjunction is that incommensurable ideas, while not expressible in language, are expressible in imagistic practices such as sculpture, painting and architecture. Many appealing visual forms, such as the Parthenon and the Canon of Polycleitos, are based on √5, a quantity impossible to verify linguistically, yet a measurement capable of visual geometric evaluation. An inquiry into the relationships between commensurability in language and incommensurability in imagistic forms has heuristic implications for the field of rhetorical theory.  To begin with, Plato's arguments against the Sophists' rhetorical practices mirror his objections to the kind of geometry that fascinated them. A look at the extant fragments of the Sophists which indicate a preoccupation with inexpressibility (arhetos) runs contrary to the more popular historical record where the Sophists are depicted as teachers of expression (rhetos).  How the Sophists' interests in geometry parallel and make sense of their work with rhetoric have not been addressed adequately in the discipline.