Abstraction: How Do Linguists Think Mathematically?

As a devoted student of mathematics, I have never been fond of social sciences. Vague borderlines between different concepts and strong emphasis on seemingly temporal ideas were not exactly appealing to me, who was rather used to discussing well-defined ideas and the abstract world. Therefore when I was exposed to linguistics for my first time through this course, I was profoundly surprised by the logical structure of the subject and how its impression resonates with what I get from mathematics. Although linguistics is classified as social sciences, it essentially shares a very similar approach with mathematics, i.e. abstraction.

Abstraction is a term in the theory of meta-mathematics that describes the act of extracting the essential characteristics from a diverse collection of complex and sophisticated concepts. Abstraction is a universal approach in mathematics that is used in any sub-branch one can imagine. It is usually presented in the form of integrating individual examples into a more general setting, in which we can remove the less important features of each object and discuss their essential similarities. Consider the following example in topology, a branch of mathematics that studies things that are preserved under stretching and bending.

 It is an old joke that a topologist cannot distinguish a doughnut from a coffee mug. A doughnut and a mug is equivalent in the sense of topology, because we can deform one into the other by only enlarging or pressing some parts of it, without having to cut or glue. By establishing such an equivalence, we are removing the specific features of each object that are not relevant to our focus of study, the features that are not preserved under such continuous transformations. The remaining properties in the class of equivalent objects are the characteristics that matter to a topologist. By extracting such core characteristics, mathematicians only have to examine the properties of the generalized object (a torus in this case) and simply apply the general conclusion to the special cases (like the mug) to derive topological answers about them. This way, mathematicians shift their area of interest from the concrete to the abstract, hence organizing their theories to be more consistent, applicable, and above all, aesthetic.

Now we could notice that there is a linguistic counterpart to the same approach, within the theory of phonology and morphology that we have been learning for the past few weeks.

The concept of phoneme is a strong example of abstraction found in linguistics. When we discuss sounds in phonology, we do not place our primary attention to the individual realizations of sounds; we rather remove the phonetic variations in the sounds we perceive, and leave the remaining, preserved form of abstract sounds—phonemes—as our subject of interest. 

Revisiting the example that we have frequently used throughout our class, the aspirated bilabial stop [p^h] and the normal bilabial stop [p] are “phonemically equivalent” in the sense that although they are phonetically distinct sounds, their difference do not coincide with difference in meanings (in English). As linguists studying phonology are concerned about different sounds that result in distinct meanings, the phonetic details that distinguish [p^h] from [p] are negligible, and hence they remove such features and examine the equivalence class of the sounds instead, namely /p/. This procedure is inherently similar to how topologists reduce coffee mugs and doughnuts to “same” objects. During our transition from phonetics to phonology, we were experiencing the important process of abstraction.

The approach of abstraction that we use in our discussion of phonemes in phonology, and similarly morphemes in morphology, leads us to a remarkable resemblance between linguistics and mathematics. Linguists reason just as rigorously and abstractly as mathematicians do, making it a fascinating realm of knowledge for mathematicians to adventure.