We count apples and divide a cake so that each guest gets an equal piece; we weigh galaxies and use Hilbert spaces to make correct predictions about spectral lines. It would seem we have no difficulty in applying mathematics to the world; yet the precise role of mathematics in its various applications is surprisingly elusive. How does it work? Eugene Wigner has gone so far as to say that 'the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it' (1960: 223). Could he be right?