The Concept of Existence and the Role of Constructions in Euclid's Elements

Mathematics and Rhetoric. Introduction

“…19 Orna Harari has furthermore reconstructed the presuppositions which, for Aristotle, rendered classical syllogisms incompatible with mathematical demonstration. 20 Catherine Goldstein has taught us how to follow mathematical objects and texts from their constitution through to their apparent permanence in time. This permanence allows mathematicians to look at a theorem in a variety of versions, for the later historian to study.…”

Historiography of Mathematics in the 19th and 20th Centuries

“…For example, Heath seems to have been quite ready to believe that the point of certain Euclidean constructions was as existence proofs: "The use of actual construction as a method of proving the existence of figures having certain properties is one of the characteristics of the Elements" (Euclid 1908, 234). (More recent scholars have often argued that the problems are in fact to show that certain figures can be constructed under the specified restrictions (Saito 2012;Harari 2003).) Heath's position was one which had its roots in the late antique commentary of , but it is surely no accident that it tended to make (some parts of) Greek mathematics comparable to the constructivist programmes of L.E.J.…”

Rethinking geometrical exactness

“…footnote (18), above. 27 Harari has argued against Zeuthen's classical "existential interpretation" of Euclid's constructions (Harari (2003); Zeuthen (1896)). According to her, this interpretation assigns to Euclid three theses that he does not actually endorse, namely that: i ) "the correspondence between a defined term and the reality to which it refers cannot be taken for granted, but it rather should be established by means of proofs" (Harari (2003), 4); ii ) geometrical constructions are "means of justification, i. e., [.…”