Mathematics and linguistics (part 1)

Last Friday, I suggested that one of the main reasons that language isn’t mathematics is that mathematics, unlike language, is a field of study. Schematically, we have the following:

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 and mathematics will certainly interact where there is overlap between the objects they study. What this schema doesn’t reveal is the more fundamental relationship between the two fields. This relationship is, unfortunately, obscured by more than just a tabular presentation. It is obscured by the fact that both fields are generally misunderstood by non-specialists.

Misperceptions about these fields probably shouldn’t be too surprising if we consider the non-specialist’s exposure to them. Let’s start with mathematics. Here are some of society’s perceptions of mathematicians, as presented by Keith Devlin in a commencement address in 1997:

All of you graduating here today have a good head for figures.

You like adding up long columns of numbers in your head.

You have always found it easy to balance your checkbook.

You revel in solving ten simultaneous linear equations in ten unknowns. In your head.

You are all going to be math teachers or accountants.

You’re dull.

You’re boring.

You have no sense of humor.

You concentrated on mathematics because it is predictable, because there is always a right answer you can check in the back of the book, because you like following very precise rules, because it allows you to escape from everyday life into a world that has nothing to do with everyday life, and because mathematics does not require the creativity that you completely lack.

None of these, of course, is accurate. Some of them are, at least, understandable once we realize that most of the mathematics taught to non-specialists is prescriptive in nature. Here is part of Paul Lockhart’s critique of the K-12 mathematics education in the United States:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes. (p. 5)

A bit further on in his essay, he addresses what is in our mathematics classes:

The truly painful thing about the way mathematics is taught in school is not what is missing— the fact that there is no actual mathematics being done in our mathematics classes— but what is there in its place: the confused heap of destructive disinformation known as “the mathematics curriculum.” …

In place of discovery and exploration, we have rules and regulations. …

In place of meaningful problems, which might lead to a synthesis of diverse ideas, to uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in mathematics, we have instead joyless and redundant exercises, specific to the technique under discussion, and so disconnected from each other and from mathematics as a whole that neither the students nor their teacher have the foggiest idea how or why such a thing might have come up in the first place.

In place of a natural problem context in which students can make decisions about what they want their words to mean, and what notions they wish to codify, they are instead subjected to an endless sequence of unmotivated and a priori “definitions.” (pp. 14–15)

Linguistics finds itself in a slightly different situation, if only because we don’t generally have any classes in the K-12 environment called ‘linguistics’. What we do have are classes in English and in Foreign Languages—French, German, and Spanish were offered where I went to High School. Most linguists, however, will immediately identify with some of Lockhart’s comments. I wouldn’t be at all surprised to hear a linguist comment on “the confused heap of destructive disinformation known as ‘grammar’ ”. (To give some idea of how popular the prescriptivist view of grammar is, consider the fact that Amazon’s bestseller list for Writing Skills places Strunk and White’s book—as of April 30—in positions 1, 4, 7, and 16.) Prescriptivism shows up in the lexicon as well, with people claiming that something isn’t a word because “its not in the dictionary”. This argument has even been used in a 1999 opinion from the US 9th Circuit Court of Appeals (see section 32).

In both mathematics and linguistics, the non-specialist’s experience of the field is that it is made up of knowledge and rules handed down from some mysterious authority. If we’re going to improve our understanding of the relationships between these fields, we’ll need to move beyond these non-specialists misperceptions and develop an understanding of the actual scope of each of them.

Copyright © 2008 Michael L. McCliment.