Explanation, Existence and Natural Properties in Mathematics - A Case Study: Desargues’ Theorem

Abstract

Nicole and Arnauld took explanation (“divining into the true reason of things” (1717, 427)) to be as important in mathematics as it is in natural science. More recently, the mathematician William Byers (2007, 337) has characterized a “good” proof as “one that brings out clearly the reason why the result is valid”. Likewise, empirical researchers on mathematics education have recently argued that students who have proved and are convinced of a mathematical result often still want to know why the result is true (Mudaly and de Villiers 2000), that students assess alternative proofs for their “explanatory power” (Healy and Hoyles 2000, 399), and that students expect a “good” proof “to convey an insight into why the proposition is true” even though explanatory power “does not affect the validity of a proof” (Bell 1976, 24). However, none of this work investigates what it is that makes certain proofs but not others explanatory. Recently, as Mancosu (2008) and Tappenden (2008a) note, philosophers have renewed their interest in mathematical explanation, which has received