The explanatory and heuristic power of mathematics

Mathematics is, and has been for a very long time, one of the most successful autonomous fields of research. However, the last five centuries have seen it become so deeply interwoven in virtually every area of scientific inquiry to convince Kant that “in any special doctrine of nature there can be only as much proper science as there is mathematics therein” (Kant 1786/2004, 6; emphases in original). While the distinction between pure and applied mathematics remains somewhat elusive, philosophers have been interested in understanding the nature of each. Moreover, the idea that there are actually two uses of mathematics, an explanatory and a heuristic one, has begun to feature more and more prominently in recent philosophical debates.