The Principles of Mathematics (1903)

§ 9

Thus pure mathematics must contain no indefinables except logical constants, and consequently no premisses, or indemonstrable propositions, but such as are concerned exclusively with logical constants and with variables. It is precisely this that distinguishes pure from applied mathematics. In applied mathematics, results which have been shown by pure mathematics to follow from some hypothesis as to the variable are actually asserted of some constant satisfying the hypothesis in question. Thus terms which were variables become constant, and a new premiss is always required, namely: this particular entity satisfies the hypothesis in question. Thus for example Euclidean Geometry, as a branch of pure mathematics, consists wholly of propositions having the hypothesis S is a Euclidean space. If we go on to: The space that exists is Euclidean, this enables us to assert of the space that exists the consequents of all the hypotheticals constituting Euclidean Geometry, where now the variable S is replaced by the constant actual space. But by this step we pass from pure to applied mathematics.(§ 9 ¶ 1)