Transductive Reconstruction of Hippocrates’ Dynamical Geometrical Diagrams

Abstract

This paper analyzes the problem of producing diagrams of mathematical explanations that are not necessarily conclusive, instead of diagrammatic proofs. In order to do so, we focus on a case study, namely, the investigation that Hippocrates of Chios carried out in the fifth century B.C., concerning the square of the circle by means of lunules. More specifically, we analyze the discussion regarding two versions that Simplicius presented about the first quadrature, one developed by Alexander of Aphrodisias and the other by Eudemus of Rhodes. Our purpose is to address the relevance of the perspicuity of proof in diagrammatic terms. Classical historiography has regarded the Hippocratic explanation of the allegedly failed quadrature of the circle as not being axiomatic, or able to produce a conclusive demonstration of his results -on the grounds of having analyzed only some cases of lunules and not the totality that allows giving general results-. Therefore, we propose to analyze his argumentation from an abductive point of view. In this sense, taking as a starting point Jens Høyrup’s approach of Hippocrates proof as ‘reasoned procedures’ that are ‘explanations’, we develop this perspective in terms of transduction, a variant of C.S. Peirce’s concepts of abduction. Transduction is dominated by a cluster of non-deductive activities and skills such as: iconic visual inferences, analogies, metaphors, inductive generalizations, among others, all contributing to the construction of one or more hypotheses that explain the emergence of some creative insight, in response to a problem that motivates and drives the creative process.