The functionalism vs. formalism debate in language research involves a basic opposition between two incompatible views on the nature of linguistic research (methodologically), and on the nature of language (theoretically). The theoretical and methodological dimensions are strictly speaking separable, but in actual practice they are closely intertwined (that is, one will rarely find a theoretical formalist who accepts a functionalist methodology, and vice versa). It will thus not always be possible to separate them in the following.
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