
math and language arts
Posted:
Jul 16, 1996 12:13 AM


One poster has mentioned his interest in language arts and has apparently taken an interest in the crossword called the Sator Magic Square in this connection. In a separate discussion, the difficulties of translating mathematics written in English into mathematics written in Spanish has come up.
I would like to elaborate on my comment, in the latter discussion, about mathematics being a language and see where it leads from the standpoint of language arts.
Topic 1: Mathematical idiom
If you show someone an open bag of potato chips which still has some potato chips in it, there are various circumlocutions to describe it. "You want some chips? There's some left."
A mathematician will simply remark that there is a nonempty bag of potato chips. The word "nonempty" is typical of mathematical idiom. It doesn't normally arise in the parlance of nonmathematicians.
There are other key phrases and usages which, even if not unique to mathematics, are standard mathematical usage which one finds less often in common parlance. One is "if and only if". Someone might say, "I'll give you a quarter if you let me copy your homework." From a logical point of view, it is conceivable that the quarter will be given even if permission is not granted to copy the homework, but most likely the student intended the word if to mean "if and only if".
Another is the tendency of some people to write "We have" before displaying a formula or a conclusion. E.g., "Combining these two identities and clearing denominators, we have x=7." or "After a similar line of reasoning, which we omit here, we have that the line opposite the largest angle is the largest side."
Another example involves words that mean one thing in English and something entirely different in mathematics. An advanced example is "limit" or "compact". But more elementary notions such as "group" or even "relation" are already different.
One activity is to try to come up with amusing equivocations that play on the different usages. E.g., "A group is a collection of individuals who associate and there is usually an individual, called the identity element, with the property that any individual who associates with the identity element is no longer a member of the group."
I have not even scratched the surface. Anyway, exploring this aspect of mathematics also involves the language arts.
One source to look at for some informal and formal discussions of this, look at the discussion of the Dog Walking Ordinance, reprinted in the Word Of Mathematics from a work of Robert Graves and Alan Hodge.
Topic 2: Language Decipherment
Take an English speaking student and show him/her some mathematics written in another language, preferably one with a lot of cognates with English. There is a tremendous amount of information in whatever tables or formulas the writing contains. The notation is likely to be common to speakers of either language and that fact serves to give the reader some idea of the context of the discussion even if he/she doesn't know the language in question. Under such conditions, one can often decipher enough of the language to make sense of what is going on, even if one can't make a detailed translation.
One of the pleasant features of decipherment is that one can form hunches as to what the text actually says, but then one has to check whether the proposed translation leads to correct mathematics.
Another is realizing that one is staring at a familiar usage (e.g. if and only if) in a language one has never studied and now knowing how to say that. In some cases, one might never know how to look it up.
Topic 3: wff and proof
There is a game which was available when I was in high school and which I still see advertised sometimes. It involves tossing dice which have logical symbols on the faces and trying to arrange them so as to arrive at the longest well formed formula one can make with those symbols. It was called wff and proof. This game can be used to illustrate another aspect of mathematics and language arts.
One can also go further using predicate calculus. (BTW, calculus is another good example of a word that has different meanings in mathematics than in common parlance. Sometimes when people ask me what calculus is, I like to tell them it is a substance found on the teeth of mathematicians.)
Topic 4: Productions and natural language
I'm sure some very simple examples can be given to students to illustrate some of the concepts of formal languages. Then one can explain that similar ideas are being applied to study natural languages.
One way to introduce the examples might be as puzzles in which the students, after seeing that the teacher accepts some strings of symbols and not others, have to guess the pattern in the acceptable strings. Or a computer, with infinite time and patience, can do the accepting and rejecting.
It can be pointed out that a computer program is a string of symbols which the machine will accept or reject and that learning to program in that language in part involves learning to distinguish acceptable programs from unacceptable ones.
Similarly, learning a natural language, such as one's native language, involves learning to recognize which strings of sounds or symbols are acceptable in that language. Rules of grammar and usage amount to trying to approach a description of the acceptable strings, just as in the exercises one did at the beginning.
Maybe another variant would be to introduce "pig latin" or various other modifications of ordinary speech and to specify or guess the rules of formation.
Topic 5: Secret codes
Students can make up secret codes and pass notes using them. Other students can try to decode them. The easiest would be substitution codes.
This is to some extent a language art. It also involves some statistical knowledge of the language. It wouldn't hurt to bring to their attention the existence of such puzzles in the newspapers.
Incidentally, one can also use magic squares to encode secret messages, although the messages might take up a lot of space.
Allan Adler adler@pulsar.cs.wku.edu

