Transductive Reconstruction of Hippocrates? Dynamical Geometrical Diagrams

VISOKOLKIS, SANDRA; CARRIÓN, GONZALO; VARGAS, EVELYN

Diagrammatic Representation and Inference

Springer

Año: 2020; p. 10 - 25

Abstract. This paper analyzes the problem of producing diagrams of mathematicalexplanations that are not necessarily conclusive, instead of diagrammaticproofs. In order to do so, we focus on a case study, namely, the investigation thatHippocrates of Chios carried out in the fifth century B.C., concerning the squareof the circle by means of lunules. More specifically, we analyze the discussionregarding two versions that Simplicius presented about the first quadrature, onedeveloped by Alexander of Aphrodisias and the other by Eudemus of Rhodes. Ourpurpose is to address the relevance of the perspicuity of proof in diagrammaticterms. Classical historiography has regarded the Hippocratic explanation of theallegedly failed quadrature of the circle as not being axiomatic, or able to producea conclusive demonstration of his results -on the grounds of having analyzed onlysome 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. Inthis sense, taking as a starting point Jens Høyrup?s approach of Hippocrates proofas ?reasoned procedures? that are ?explanations?, we develop this perspective interms of transduction, a variant of C.S. Peirce?s concepts of abduction. Transductionis dominated by a cluster of non-deductive activities and skills such as: iconicvisual inferences, analogies, metaphors, inductive generalizations, among others,all contributing to the construction of one or more hypotheses that explain theemergence of some creative insight, in response to a problem that motivates anddrives the creative process.Keywords: Diagrammatic proof · Mathematical explanation · Transduction