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

5Mathematical Explanation: Whyit MattersPAOLO MANCOSUThe topic of mathematical explanation has recently been the subject of muchinterest. Although I will point out below that attention to mathematicalexplanation goes back to the Greeks, the recent revival in the analytic literatureis a welcome addition to the philosophy of mathematics. In this introductionI have set myself two goals. The first is that of giving a survey of theliterature on mathematical explanation and how the different contributions inthis area are connected. Secondly, I would like to show that mathematicalexplanation is a topic that has far-reaching ramifications for many areas ofphilosophy, including, in addition to philosophy of mathematics, epistemology,metaphysics, and philosophy of science.Let us begin by clarifying two possible meanings of mathematical explanation.In the first sense ‘mathematical explanation’ refers to explanations in the naturalor social sciences where various mathematical facts play an essential rolein the explanation provided. The second sense is that of explanation withinmathematics itself.5.1 Mathematical explanations of scientific factsThe first sense is well illustrated by the following example taken from a recentarticle by Peter Lipton:There also appear to be physical explanations that are non-causal. Suppose thata bunch of sticks are thrown into the air with a lot of spin so that they twirland tumble as they fall. We freeze the scene as the sticks are in free fall andfind that appreciably more of them are near the horizontal than near the verticalorientation. Why is this? The reason is that there are more ways for a stick to be