Frege on the applicability of mathematics

Foundations of Arithmetic, 87:

The laws of number… are not really applicable to external things; they are not laws of nature. They are, however, applicable to judgments holding good of things in the external world: they are laws of the laws of nature.

Grundgesetze der Arithmetik, 92 (trans. Geach and Black):

We know that the same ratio between quantities (the same number) can occur in connection with lengths, with temporal durations, with masses, with moments of inertia, etc. This makes it probable that the problem how we are able to make use of arithmetic is to be solved, at least in part, independently of those sciences within which the application is made.

Grundgesetze der Arithmetik, 91 (trans. Geach and Black):

Why can arithmetical equations be applied? Only because they express thoughts. How could we possibly apply an equation which expressed nothing and was nothing more than a group of figures, to be transformed into another group of figures in accordance with certain rules? Now, it is applicability alone which elevates arithmetic from a game to the rank of a science. So applicability necessarily belongs to it.