Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

mathematical explanation: why it matters 143For instance, Sandborg (1997, 1998) tests van Fraassen’s account of explanationas answers to why-questions by using cases of mathematical explanation.Steiner proposed his model of mathematical explanation in 1978. In developinghis own account of explanatory proof in mathematics he discusses—andrejects—a number of initially plausible criteria for explanation, i.e. the (greaterdegree of) abstractness or generality of a proof, its visualizability, and its geneticaspect which would give rise to the discovery of the result. In contrast, Steinertakes up the idea ‘that to explain the behaviour of an entity, one deduces thebehaviour from the essence or nature of the entity’ (Steiner, 1978a, p.143). Inorder to avoid the notorious difficulties in defining the concepts of essence andessential (or necessary) property, which, moreover, do not seem to be usefulin mathematical contexts anyway since all mathematical truths are regardedas necessary, Steiner introduces the concept of characterizing property. Bythis he means ‘a property unique to a given entity or structure within afamily or domain of such entities or structures’, where the notion of familyis taken as undefined. Hence what distinguishes an explanatory proof froma non-explanatory one is that only the former involves such a characterizingproperty. In Steiner’s words: ‘an explanatory proof makes reference to acharacterizing property of an entity or structure mentioned in the theorem,such that from the proof it is evident that the result depends on the property’.Furthermore, an explanatory proof is generalizable in the following sense.Varying the relevant feature (and hence a certain characterizing property) insuch a proof gives rise to an array of corresponding theorems, which areproved—and explained—by an array of ‘deformations’ of the original proof.Thus Steiner arrives at two criteria for explanatory proofs, i.e. dependence ona characterizing property and generalizability through varying of that property(Steiner, 1978a, pp. 144, 147).Steiner’s model was criticized by Resnik and Kushner (1987) who questionedthe absolute distinction between explanatory and non-explanatory proofs andargued that such a distinction can only be context-dependent. They alsoprovided counterexamples to the criteria defended by Steiner. In Hafner andMancosu (2005) it is argued that Resnik and Kushner’s criticisms are insufficientas a challenge to Steiner for they rely on ascribing explanatoriness to specificproofs based not on evaluations given by practicing mathematicians but ratherrelying on the intuitions of the authors. By contrast, Hafner and Mancosubuild their case against Steiner using a case of explanation from real analysis,recognized as such in mathematical practice, which concerns the proof of Kummer’scriterion for convergency. They argue that the explanatoriness of the proof ofthe result in question cannot be accounted for in Steiner’s model and, moreimportantly, this is instrumental in giving a careful and detailed scrutiny of