In the mathematical theory of Galois representations, a choice of algebraic closure of the rationals and an embedding of this algebraic closure in the complex numbers (e.g. section 5) is usually necessary to frame the background setting, but I never hear "the algebraic closure" or "the embedding," instead "an algebraic closure" and "an embedding." Thus I never forget that a choice has to be made and that this choice is not necessarily obvious. This is an example from mathematics where careful language is helpful in tracking background assumptions.