  Steiner proposed his model of mathematical explanation in 1978a. In developing his own account of explanatory proof in mathematics he discusses—and rejects—a number of initially plausible criteria for explanation, e.g. the (greater degree of) abstractness or generality of a proof, its visualizability, and its genetic aspect that would give rise to the discovery of the result. In contrast, Steiner takes up the idea “that to explain the behaviour of an entity, one deduces the behavior from the essence or nature of the entity” (Steiner 1978a, 143). In order to avoid the notorious difficulties in defining the concepts of essence and essential (or necessary) property, which, moreover, do not seem to be useful in mathematical contexts anyway since all mathematical truths are regarded as necessary, Steiner introduces the concept of characterizing property. (Let me mention as an aside that Kit Fine distinguishes between essential and necessary properties and that perhaps the distinction could be exploited in this context). By characterizing property Steiner means “a property unique to a given entity or structure within a family or domain of such entities or structures”, where the notion of family is taken as undefined. Hence what distinguishes an explanatory proof from a non-explanatory one is that only the former involves such a characterizing property. In Steiner’s words: “an explanatory proof makes reference to a characterizing 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) in such a proof gives rise to an array of corresponding theorems, which are proved—and explained—by an array of “deformations” of the original proof. Thus Steiner arrives at two criteria for explanatory proofs, i.e. dependence on a characterizing property and generalizability through varying of that property (Steiner 1978a, 144, 147).