The phenomenon of context-dependence covers all the cases in natural language where the semantic content of an expression depends on the context of its utterance. The explicit context-dependence of indexicals (such as “I” or “here”) is a basic illustration of that phenomenon. Context-dependence is supposed to elude any formal representation (this diagnosis defines “contextualism”), unless it is modeled on indexical context-dependence (“indexicalism”), or reduced to it (“minimalism”). In this paper, I will argue that these logico-philosophical stances are based on too narrow an expectation of what a formal representation can be; that context-dependence, once approached as a radically dynamic process based on context-shift, is amenable to a formal representation; but that new mathematical concepts, belonging to Grothendieck’s descent theory, have to be called upon to that end. I will further argue, more generally, that linguistic meaning is better understood, not as an abstraction (from particular cases), but rather as an amalgamation (of local pieces), which fits Grothendieck’s conception of “the abstract collection of glueing data itself as the only true reality” (Carlos Simpson).