 Also in the case of mathematical explanation of mathematical facts, we need to carefully analyze more case studies in order to get a better grasp of the varieties of mathematical explanations. The philosophical pay-offs of the work on mathematical explanation are coming from the following areas. First, mathematical explanation are being used to test models of scientific explanation.  Theories of scientific explanations aim at capturing ‘scientific’ explanations in any area of knowledge, not just explanations in the natural sciences. If they cannot accommodate mathematical explanations, this will show important limitations of the theories in question. On the other hand, if no such theory will be able to encompass mathematical explanations and explanations in natural science under a single model, this might indicate important differences between science and mathematics. Secondly, accounting for mathematical explanations of scientific facts might be related in interesting ways to an account of mathematical explanations of mathematical facts. Third, ‘explanatoriness’ is only one virtue among those that an epistemology of mathematics that does not limit itself to traditional debates about justifying axioms can fruitfully investigate. Indeed, it is clear that ‘explanation’ is closely connected to other notions such as ‘generality’, ‘visualizability’, ‘mathematical understanding’, ‘purity of methods’, ‘conceptual fruitfulness’, ‘depth’ etc. The epistemological analysis of these important notions informing mathematical practice, and the connection among them, has only recently been taken up in earnest (for recent collection in this direction see the volumes Mancosu &amp; others 2005 and Mancosu 2008a). In conclusion, let me also mention that the topic of mathematical explanation has important connections to issues of concern to historians of mathematics and mathematics education (see Hanna, Jahnke, and Pulte 2010 for a recent volume).