Power and placeholders
We point out a difference between mathematics and magic.
Long ago, in secondary school, one of our consultants participated in an in-class exercise. Suppose a person has lost all his or her memory entirely, but will recall the answer immediately to any question you ask. What ten (or it may have been twenty) questions will you employ?
Each student made up a list and of course the point of the exercise was in the discussion afterward, defending one’s choices or maybe changing some. The interesting point our consultant remembers is that the class split firmly into two opposed groups: those who would ask, “What is your name?” and those who wouldn’t. It was a very clear division. The Nominalists (to use a word which actually has another meaning) placed it as their first question; the Anti-Nominalists wouldn’t ask it at all.
The idea that your name has a power over and above simple identification is ancient and pervasive. Medieval magic (as in Faust) controlled demons by invoking their names. Perhaps the most complete expression of the power of special words is in Ursula K. LeGuin’s Earthsea books, in which to cast a spell correctly the magician has to know the true name for everything involved (in the special Old Tongue), and you told your own True Name to almost no one.
So then, of course one might expect mathematicians and scientists to be Nominalists (in this sense). They are very keen on giving precise definitions to words and being careful about exactly how they’re used. There are many students at the moment who are being tripped up by the differences between speed and velocity and between exothermic and endothermic.
But in science and math the power actually goes the other way. It’s the definition that lends the word its significance; given the qualities we want, we can label them as we will. The word, the name, can be changed whenever it’s convenient.
This is sometimes hard for students to grasp, but vital. Algebra is all about giving an arbitrary label to something, normally using a letter of the alphabet. The letter is not important; what we can say about it is. Solving x3 – x2 + 1 = 0 we get the same numbers as if we solve y3 – y2 + 1 = 0. This is sometimes unsettling to students. Some are uncomfortable with variables in the first place, and plug in numbers whenever possible. Many get used to common conventions, like y = mx + b, and have trouble when the variables are given different names. (Our tutor remarks to his students, “I can call this anything. Instead of x, Harry; instead of y, George.” This probably doesn’t help.) But to make progress the students have to understand that the letters are placeholders, labels that can be changed as convenient. They are not Names with their own power.
Even Chemistry, where every element has its own fixed symbol, uses labels. Our tutor has a Chemistry student who was given a generic reaction: molecule A reacts with molecule B to form molecules C and D. A page or two later, a different problem appeared in which A decomposed to form B and C. The student was confused because he did not realize that the B in the first reaction was a label, a placeholder, and had no necessary connection with the B in the second.
Students in mathematics and the sciences do need to learn the technical terms of their field. But the power lies in the definitions, not specific words. They must be comfortable with changing labels as necessary.
