The Instructive Function of Mathematical Proof: A Case Study of the Analysis cum Synthesis method in Apollonius of Perga’s Conics

Abstract

This essay discusses the instructional value of mathematical proofs using different interpretations of the analysis cum synthesis method in Apollonius’ Conics as a case study. My argument is informed by Descartes’ complaint about ancient geometers and William Thurston’s discussion on how mathematical understanding is communicated. Three historical frameworks of the analysis/synthesis distinction are used to understand the instructive function of the analysis cum synthesis method: the directional interpretation, the structuralist interpretation, and the phenomenological interpretation. I apply these interpretations to the analysis cum synthesis method in order reveal how the same underlying mathematical activity occurs at different levels of scale: at the level of an individual proof, at the level of a collection of proofs, and at the level of a single line within a proof. On the basis of this investigation, I argue that the instructive value of mathematical proof lies in engendering in the reader the same mathematical activity experienced by the author themselves.