An Introduction to the Philosophy of Mathematics

Mathematical explanation is a hot topic in current work in the philosophy of mathematics. We have already seen one reason for this: the close connection between the indispensability argument for mathematical realism and the scientific realist's reliance on inference to the best explanation. This connection is even tighter if it can be established that there are mathematical explanations of empirical phenomena. As a result, a great deal of recent work on realism-anti-realism issues in mathematics has focused on mathematical explanations in science. Irrespective of such issues, the question of mathematical explanation is important in its own right and deserves closer attention.

We start by making a distinction between two different senses of mathematical explanation. The first we call intra-mathematical explanations. These are mathematical explanations of mathematical facts. Such explanations can take the form of an explanatory proof – a proof that tells us why the theorem in question is true – or perhaps a recasting of the mathematical fact in question in terms of another area of mathematics. There is also the issue of whether mathematics can explain empirical facts. Call this extra-mathematical explanation. A full account of mathematical explanation will provide both a philosophically satisfying account of intra-mathematical explanation and an account that coheres with our account of explanation elsewhere in science.