For them linguistics does not and cannot aim at explanation. Linguistics can only be scientific in this approach if it is taken to be a formal science on a par with formal logic, rather than an empirical science. The aim of a formal science is deriving theorems from a set of axioms rather than explaining empirical observations. In linguistics this means that languages are abstract objects and sentences axioms of a formal system. From the proposed analysis it follows that we should at least assume a tripartition in the approaches to linguistics.
Functionalist approaches aim at explanations of the form in terms of the function of language.
Formalist approaches in the narrow sense study the form of language with the goal of deriving theorems from sentential axioms. Whether explanation vs. deriving theorems or emphasis on the function vs. on the form of language should be taken as the principal dividing line is not immediately obvious. Much harm has been done, however, by entirely ignoring the former opposition. Only in this way can it be explained that in many critical presentations of allegedly Chomskyan linguistics the basic assumptions of the narrowly formalist approach are attributed to Chomsky. Chicago Linguistic Society, p. 323-353.