Christopher Pincock

Depending on how it is clarified, the applicability of mathematics can lie anywhere on a spectrum from the completely trivial to the utterly mysterious. At the one extreme, it is obvious that mathematics is used outside of mathematics in cases which range from everyday calculations like the attempt to balance one s checkbook through the most demanding abstract modeling of subatomic particles. The techniques underlying these applications are perfectly clear to those who have mastered them and there seems to be little for the philosopher to say about such cases. At the same time, moving to the other extreme, scientists and philosophers have often remarked on the remarkable power that mathematics provides to the scientist, especially in the formulation of new scientific theories. Most famously, Wigner claimed that The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve (Wigner 1960, p. 14). But Wigner is far from an isolated case. According to Kant, in any special doctrine of nature there can be only as much proper science as there is mathematics therein (Kant 1786, p 6), and others seem to agree that there is some significant tie between mathematics and modern science.