Mathematics and linguistics (part 4)

In part 1, I discussed the non-specialist’s experience with mathematics and with linguistics, and suggested that their experience is, in both cases, essentially prescriptivist in nature. Before discussing the relationship between these fields, we needed to move beyond the non-specialist’s perceptions and to understand more about the actual scope of each of these fields. I offered some observations about mathematics in part 2, and addressed linguistics in part 3. In this final part, we’ll look at the relationship between the two fields in light of the preceding discussion.

Now that we have a better understanding of what mathematics and linguistics are, let’s reconsider the question of how they are related to one another. Our initial view was that the two fields were predominantly parallel, and would interact where their respective objects of study happened to coincide (if anywhere). We represented this aspect schematically:

Fields of inquiry and their objects of study
Field of study Practitioners Objects studied
linguistics linguists language
mathematics mathematicians space, number, quantity, and arrangement

Linguistics, as we have now seen, isn’t just an arbitrary study of language. In many cases, philosophers may be said to study language. I’m not convinced that literary criticism can ever avoid studying language. But neither of these are inherently linguistic in nature. (They may be approached from a linguistic perspective; for example, one can apply pragmatics to the study of literature—but one can also study literature without using pragmatics, or any other part of linguistics.) Linguistics is the scientific study of language as a principal phenomenon.

Linguistics naturally studies the patterns that arise in languages. Even linguists who strongly reject any notion of an underlying rule-governed cognitive system propose that there are patterns in the languages that they study. Whether we adopt a rule-governed framework or not, the role of a linguistic theory is to propose that there are certain patterns—ranging from fully systematic “laws” through to weak “tendencies”—that arise in the use of language. The process of drawing conclusions from these basic propositions, and hence the entire act of moving from philosophical opinion to empirical science, is inherently an act of mathematics. Language is not mathematics. Linguistic theorization is not mathematics, but uses mathematics as its tool of reasoning. The evaluation of linguistic theories, however, is intrinsically mathematical. Has the theorist constructed a consistent theory? Do the theorist’s predictions follow from the patterns abstracted from their observations? Are there other predictions that follow from the proposed theory that also need to be evaluated empirically? These questions cannot be answered by the linguistic theory, which takes some aspect of language as its object of study, since these questions naturally take the linguistic theory itself as the object of study.

When physicists propose specific theories, these theories are evaluated not only for agreement with the empirical data and for compatibility with generally known physical properties, but they also evaluate and validate the mathematical properties of these theories. If the theory proposes a set of relationships among the observed patterns that is inconsistent, or if there is no way to construct any object satisfying the properties proposed by the theory, then that theory is rejected. The situation in linguistics is analogous. Just as the study of physical theories is part of physics, the study of linguistic theories is just as much a part of linguistics as the study of languages is.

This relationship, in which mathematics is the instrument by which we analyze linguistic theories, is the more fundamental relationship that I hinted at from the outset. It comes with an immediate corollary: mathematical modeling is necessarily a valid research methodology in linguistics. It does not replace empirical studies or the myriad research methodologies associated with them; it complements such studies. Both aspects are important for the health of linguistics as a science. For the formal evaluation of linguistic theory, however, mathematical modeling may well be the only valid methodology.

Copyright © 2008 Michael L. McCliment.


One Response to Mathematics and linguistics (part 4)

  1. […] posts the past few weeks have dealt with linguistics in very general terms. The purpose of the Mathematics and linguistics posts has been to outline a specific mode of inquiry within theoretical linguistics: the […]

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