Math Lingo: a bad meme virus

It takes one to know one -- and vice versa!

Alfred E. Neuman

At the beginning of the year, the American Mathematical Monthly, unwittingly let escape from one of its virus. Our only hope is that apparently the meme has not yet reached such powerful social transmitters as PBS or the Los Angeles Times. Thus the epidemic may be prevented and the meme contained and successfully fought within the small and united community of mathematicians, math teachers and math fans.

The meme spreads the notion that, unlike natural languages (which are complete and human), and unlike programming languages (which are special languages), mathematical language is a lingo. Because of the startling novelty of the assertion, infected individuals may want to consult the commonly available dictionaries to verify a couple of entries:

Language 1.a. The use by human beings of voice sounds, and often written symbols representing these sounds, in organized combinations and patterns in order to express and communicate thoughts and feelings. b. A system of words formed from such combinations and patterns, used by the people of a particular country or by a group of people with a shared history or set of traditions. 2.a. A nonverbal method of communicating ideas, as by a system of signs, symbols, gestures, or rules: the language of algebra. b. Computer Science. A system of symbols and rules used for communication with or between computers. 3. Body language; kinesics. 4. The special vocabulary and usages of a scientific, professional, or other group: "his total mastery of screen language-camera placement, editing-and his handling of actors" (Jack Kroll). 5. A characteristic style of speech or writing: Shakespearean language. 6.a. Abusive, violent, or profane utterance: "language that would make your hair curl" (W.S. Gilbert). b. A particular manner of utterance: gentle language. 7. The manner or means of communication between living creatures other than human beings: the language of dolphins. 8. Verbal communication as a subject of study. 9. The wording of a legal document or statute as distinct from the spirit.

Lingo. 1. Language that is unintelligible or unfamiliar. 2. The specialized vocabulary of a particular field or discipline: spoke to me in the lingo of fundamentalism.

---American Heritage Dictionary

Worthy of note: the population raised in the spirit promoted by AHD would believe that there is a language of algebra, a language of the screen and a body language. Dolphins have a language. The Math-Lingo meme would have you believe that the mathematical language is, at best, unfamiliar and, at worst, unintelligible. The meme has ready arguments to aggravate the infection. The cornerstone of mathematics, the notion of number, is flawed. The best of them are not actually numbers at all. When one says, I have a number of friends, one does not mean no friends or a meager one. So 0 and 1 are out. Plato would of course agree. Never mind that in 2400 years some ideas might have changed. The meme carries a gene responsible, in grown up individuals, for the notion that, in mathematical talk, "number" has several meanings, and that none of them is the plain English meaning.

Does there exist a plain English meaning? The best way to confront the meme is to consider some everyday phrases:

  1. The crowd was small in number.
    Note that trying to substitute any specific integer into this sentence generates complete nonsense: The crowd was small in 666. Does this imply that the Number of the Beast is not a number?

  2. David Copperfield's number was the pinnacle of the show.
    Good for David, his number ought to be really large.

  3. The South had leaders, the North numbers.
    Does this mean that the federal conscripts pelted their southern counterparts with harmless figures?

  4. The suspects will do their usual number - protesting innocence - and then confess.
    So, jails are the best schools, after all?

  5. Mosquitoes without number filled the yard.
    There was nothing else but mosquitoes in the yard. Virus, go away.

So what plain English meaning mathematical "numbers" do not match? In passing, two books by the title of Strength in Numbers are currently available in bookstores:

  1. Christa Brelin (Editor), Strength in Numbers : A Lesbian, Gay and Bisexual Resource, 1996
  2. Sherman K. Stein, Strength in Numbers : Discovering the Joy and Power of Mathematics in Everyday Life, 1996

It must be noted that the research in the above mentioned laboratory had a laudable purpose: finding a way to alert teachers to the fact that their students may not be well prepared to handle math terminology. Unlike chemistry, biology, and other sciences mathematics is known to use "soft words for hard ideas," i.e., recycle common terms by imputing them a new meaning. Examples are well known: ideal, radical, field, family, group, ring, module, vector, etc. Is it any worse than using "0-anhydrosulfaminobenzoic acid" for sugar?

Of course students must be warned that using colloquial language in science is unacceptable due to the high level of ambiguity inherent in any natural language. In early 1950s R. von Mises wrote,

No proposition that presupposes complicated experiences and appeals to a necessarily vague use of colloquial language can be fit to serve as the starting point of a rigorously systematized branch of science.

It was not common knowledge 50 years ago but nowadays it's an acceptable norm. Students must be warned that mathematics is a very special language. They may remark that, to its credit, time and again mathematics demonstrates its uncanny ability to closely describe various phenomena from the physical, social, economic, biological worlds.

It could be a great time to establish rapport with students. Natural language allows for really comic constructs:

  1. I personally feel better.
  2. Why call it a building if it's already been built?
  3. Isn't it a bit unnerving that doctors call what they do practice?
  4. Do family trees form a national forest?

Students would be greatly entertained trying to generate this kind of nonsensical stream. Tell them that not all languages are the same. What is popularization in English, is vulgarization in French. Not all people are the same either. Bulgarians move their heads sideways to indicate agreement. They nod to express disagreement. In a multicultural class, more examples will be easily generated by the students themselves.

Ill-funded researchers on the East Coast made an early attempt to confront the spread of the meme virus but received the cold shoulder from the authorities. Why the meme is harmful? Because the words language and lingo have popular connotations. Language is a subject of respect and study whereas lingo is a source of irritation. A math teacher enters a class on the first day of a school year. Consider two scenarios:

  1. Says the teacher: "You must know that we embark on a study based on a very special math lingo which is bound to annoy you greatly but please pay no heed. Mathematics is a very useful tool which, nowadays, is being employed not only by aeronautic engineers but by biologists, sociologists, futures traders, and even by dentists to compute the interest on their mortgages." The teacher then proceeds to demonstrate discrepancies between the math lingo and the natural language as an example of difficulties that lurk ahead in the course of the study and will, no doubt, confuse the students. Students, beware.

  2. The teacher starts with a quote. A good example is by Oswald Spengler (but there are dozens of others):

    ... numbers are something that marks off and captures nature-impressions, and it is by means of names and numbers that the human understanding obtains power over the world.

    The teacher then adds that mathematics draws its power from a special language. Mastering this language will enhance the students' ability to function in society and to appreciate the world around them. The teacher gives examples of the ambiguities inherent in natural language. The teacher convincingly demonstrates that other sciences have developed their own special languages. Mathematics is not alone in this respect. The teacher may draw students' attention to the fact that mathematical language often constitutes an integral part of other special languages, which is evidence of its universality.

To use a recent example, according to Barry Mazur, the field of mathematics to which Wiles' proof of Fermat's Last Theorem belongs, can be described as Arithmetic Algebraic Geometry. Does not this sound funny, even confusing? A number of teachers might be apprehensive in anticipation of questions by students who watched the Nova's screening on PBS. Hopefully, a larger number of teachers will use the opportunity to point out a positive aspect of the unifying nature of mathematics.

Even a well meaning teacher would do students a great disservice following the first scenario. In a different context, R.P. Boas said:

If I had to name one trait that more than any other is characteristic of professional mathematicians, I should say that it is their willingness, even eagerness, to admit that they are wrong.

I would suggest that, in a tightly knit community, it is all right to point out the shortcomings of the mathematical language. But to denigrate it publicly to the level of a lingo may start an epidemic of uninformed misunderstanding which will benefit no one. The term for this is vulgarization (in English).

References

  1. American Heritage Talking Dictionary, Softkey, 1994
  2. R.P.Boas, Jr., If This Be Treason..., Amer Math Monthly, 64 (1957), 247-249. (Reprinted in Lion Hunting & Other Mathematical Pursuits, collected by G. L. Alexanderson, D. H. Mugler, MAA, 1995, and in A Century of Mathematics, J. H. Ewing (ed), MAA, 1994)
  3. S. J. Gould, Bully for Brontosaurus, W.W.Norton & Co, 1991
  4. J. Newman, The World of Mathematics, Simon and Schuster, 1956
  5. W. Safire, Take My Word For It, Times Books, 1986
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