The varieties of mathematical explanation
Résumé
In this paper we will begin with some general methodological remarks about mathematical explanations. We will then point out that attention to mathematical practice reveals the presence of a great variety of mathematical explanations. This realization affects two important aspects of the discussion of the nature of mathematical explanation. First of all, most of the traditional debates (see Mancosu 1999, 2000, 2001) have focused on the opposition between explanatory and non-explanatory proofs. However, there are mathematical explanations that do not come in the form of proofs and this has in fact been recognized by several scholars. Second, the variety of mathematical explanations challenges the current philosophical accounts of mathematical explanation, i.e. those of Kitcher and Steiner. As detailed discussion of case studies is necessary to see the limitations of such accounts, in the second part of the paper we restrict our focus to Steiner's theory and to the discussion of an example of an explanatory proof which, we claim, Steiner's theory cannot account for.