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In its applications to empirical science, mathematics allows us to represent and reason about the physical world. A similar phenomenon occurs internally to mathematics when one branch of mathematics allows us to represent and reason about the subject matter of another. I argue that taking seriously the close analogies between these two kinds of case will help us resolve two philosophical puzzles concerning the applicability of mathematics: 1. If (as some platonists contend) mathematics studies abstract objects, but the empirical sciences study concrete objects, then how is mathematics relevant to the empirical sciences? 2. How can one branch of mathematics be applied to another if (as some empiricists contend) mathematics has no factual content?