Of Syntax and Grammar from English to Math and Science

Adapted from J. Engr Education, 2008, 97(3), 301 - 302.

Consider this:

It is very important that you learn about traxoline. Traxoline is a new form of zionter. It is montilled in Ceristanna. The Ceristannians gristeriate large amounts of fevon and then bracter it to quasel traxoline. Traxoline may well be one of our must lukized sneziaus in the future because of our zionter lescelidge.

Directions: Answer the following questions in complete sentences. Be sure to use your best handwriting.

1. What is traxoline?

2. Where is traxoline montilled?

3. How is traxoline quaselled?

4. Why is it important to know about traxoline?

- Judith Lanier

Now, anyone who is decently competent in English can answer the questions above without having any idea what the terms mean. But this judgement neglects the important point that a significant amount of grammatical and syntatic knowledge is being tested here. One could not solve these problems without understanding the differences between subjects and objects, how to identify verb tenses and endings, the role and implication of helping verbs etc.

There is a strong analogy between (the above) example and the use of mathematics in science. The mathematics in itself is about grammatical and syntactic relationships, and permits the drawing of complex conclusions about the placeholders (variables) without having any idea what those placeholders stand for.

...We not only expect students in science and engineering to be able to understand mathematics (syntax); we expect them to combine this knowledge with knowledge of what the math is talking about in a tightly integrated way to see the meaning of the symbols. This is different from straight math and can even lead to differences in the way equations are interpreted.

The Differences Between Meanings in Physics and Math: A Shibboleth

One of your colleagues is measuring the temperature of a plate of metal placed above an outlet pipe that emits cool air. The result can be well described in Cartesian coordinates by the equation

T(x,y)=k((x²+y²)

where k is a constant.

If you were asked to give the following function, what would you write?

T(r,q)=?

The above is a problem whose answer tends to distinguish engineers and physicists from mathematicians. An engineer or physicist who works regularly with polar coordinates is likely to give the response T(r,q) = kr², a result obtained by assuming the variables are related by the familiar polar to Cartesian relation r² = x² + y². A more mathematical colleague is likely to respond, T(r,q) = k(r² + q²). The function is defined mathematically by saying, “Add the squares of the two variables and multiply by k.”

The engineer will object. “You can’t add r² and q²! They have different units.” The mathematician might reply, “No problem. I see what you mean. You just have to change the name so that each symbol represents a unique functional dependence. For example, you could write T(x, y) = S(r,q) = kr².” Unfortunately, your more concrete friend (imagine a chemical engineer at this point) is still unlikely to be satisfied. “You can’t write the temperature equals the entropy! That will confuse everything.”

Many engineers, physicists, and mathematicians are surprised by this story. Each side believes that its way of using equations is the “obvious way” and that surely even a mathematician (or an engineer) would agree. Unfortunately, that is not the case. Each group strongly prefers its own interpretation of how to write an equation.
These two examples dramatically illustrate that in science and engineering, we tend to look at mathematics in a different way from the way mathematicians do. The mental resources that are associated (and even compiled) by the two groups are dramatically different...using math in science requires the blending of distinct local coherences: our understanding of the rules of mathematics and our sense and intuitions of the physical world.

Dedicated to those of you readers who did not get this joke.