My article "Math Lingo vs. Plain English: Double Entendre" appeared in the American Mathematical Monthly in January. It argued that some words have different meanings in math lingo and in plain English, and suggested we beware this possible source of misunderstanding. I mentioned "group" and "set", the inclusive and exclusive "or", and whether 0 and 1 are considered "numbers."
I received two dozen responses, some from friends I hadn't heard from in a while. Many were kind. Some called the article "delightful." Many agreed about the danger of double entendre in math teaching, and sent in examples of ambiguity between math lingo and plain English. I offer this follow-up in case some readers find the conversation interesting, and because it's easier to answer en masse than one by one.
Lee Lorch of York University tells of the class of "slowly oscillating" functions, which needn't oscillate, and "more vividly," the class of "slowly increasing" functions, which includes all decreasing functions. (Similarly, "slowly decreasing functions" include all increasing functions.) "I've seen eminent analysts confused by this expression," says Lee. "These concepts, introduced by J. Karamata, are of considerable importance in connection with Tauberian theorems."
In the same vein, Jerry Folland of Washington proposes "simple complex Lie algebra."
On a less lofty mathematical plane, Thomas L. Bartlow of Villanova writes, "Frustration with integration techniques led one student to say to me, `If Ross Barnett doesn't want to integrate, why should I?'"
I have learned I was mistaken to think that the inclusive "or" is hard to find in colloquial English. Several examples were contributed. I collect them in a list below.
John Larry Kuhns of Woodland Hills, California, caught my mistake about "set" and "group." I said that in plain English, they mean the same thing, but they don't. In plain English a group is open-ended, but a set can be either incomplete or complete, like a chess set, a set of dishes, or a set of cards. This makes the inconsistency between plain English and math lingo more flagrant. In math, a "set" is in general unstructured, and a "group" is a kind of structured set. (Kuhns also contributed an extensive logical critique of my article.)
Two readers disagreed with my statement that math lingo isn't a complete language. I wrote, "You can't say `I have a headache' or `You bore me'" in math lingo." Dr. Alexander Bogomolny of Cut The Knot Software, Inc., East Brunswick, New Jersey, disagreed. "I can say this in math "lingo" (just let me define "I" and "a headache" appropriately, which I will postpone to a more fitting occasion.)"
Firooz Khosraviyani of Texas A. & M. wrote: "Why can't we say "I have a headache" or "You bore me" in math lingo? Let H = (people with headache); B = (people who bore others). Then the above sentences are:
This is an amusing example of a narrow view of language found among some logicians and theoretical-computer people. "I have a headache" is used in actual communication between actual live human beings. It's beside the point that logic would let them use "#%@*" to mean "I have a headache."
What about Fortran, Lisp, Pascal, C++, et cetera? Aren't they "languages"? Sure, but in a different sense than English is a language. Hint: English is a human language.
Instead of looking for a sentence that can't be expressed in math lingo, I should have pointed to a life situation where plain English serves and math lingo doesn't. For instance, this article, and all my letters here quoted, are in plain English, even though (one presumes) all the writers are also fluent in math lingo. It would be fatuous to claim that "in principle" they could have been written in math lingo. In practice, they couldn't.
If Dr. Bogomolny claims to be unaware of any life situation where plain English serves and math lingo doesn't, I'll admit that for him math lingo is a complete language.
Another issue was, Are 0 and 1 numbers? Les Lange of Cal State requested references. My source was Jacob Klein. David Fowler of the University of Warwick informed me that not only Plato, but also Euclid distinguished "unity" from "numbers". Book VII, Definition i, says: "A number is a multitude of units." Euclid needs two separate proofs, VII 9 & 15, because of this distinction. See also VII 12 in the proof of VII 15. Fowler learned of this distinction in Euclid by reading Mueller.
For Plato and for Euclid, "numnber" means "numerosity" or "multiplicity". "Unity" is not numerosity. (Zero wasn't thought about in those days.) Bob McGuigan of Worcester, Massachusetts says "I have repeatedly had students tell me that "some" means "more than one."
Was my student right or wrong, long ago, when she said "Zero isn't a number"? Dr. Bogomolny says, "The obnoxious student of Professor Hersh was unequivocally wrong. Zero is a number. There is nothing to be apologetic about." Certainly she was wrong with regard to her participation in my math class. But I was wrong to think her an idiot. If she was wrong as a student, I was wrong as a teacher. Her opinion did make sense, since I had failed to explain that in mathematics we have a different usage of the word "number."
When a student's usage is different, it's natural just to label it "wrong" without asking whether it's right in another usage. On the other hand, if you think it's worth the trouble, you can listen to the student, hear how he or she uses the "wrong" usage, and explain the difference.
On "material implication," Tadashi Tokieda of McGill University contributed the following "if, then" statement:
Its alleged logical equivalent is:
Tadashi explains that we ought to realize that in plain English "if...then involves a temporal order of events."
More about plain English logic: Bob Mcguigan reports that "Common English usage accepts as equivalent the two sentences "All meats are not fattening" and "Not all meats are fattening."
Here is a problem abut the meaning of "product": "I had been teaching elementary abstract algebra, and a little while ago proved that `every positive integer greater than 2 is the product of primes, uniquely up to order.' A student asked how to represent the integer 2 in this manner, so I wrote 2 = 2. `Where is the product?' she asked." (William Singer, Fordham University.)
"I've reached the point where I start each semester with Lewis Carroll: "When I use a word," Humpty Dumpty said, in a rather scornful tone, "It means just what I choose it to mean--neither more nor less." (Joe Rotman of Chicago.)
Hy Pitt of Milwaukee belongs to "SPELL, the Society for the Preservation of English Language and Literature...When the language is abused badly, we usually send a friendly Goof Card to the violator."
Three correspondents misunderstand my intention. I didn't attack math lingo. Nor did I undertake to explain why we need it. I just pointed out possible misunderstandings. Nevertheless, I annoyed a few readers.
Eric E. Karnowski of Jamaica Plain, Mass., thought that "This article illustrates [language] problems, but not in the way Dr. Hersh intended. The author seems to be under just as many misconceptions as his students...I expected more careful thought from the Monthly." Saunders MacLane of Chicago was even angrier: "Hersh's article was the one rotten egg in the bunch." If Hersh wanted to write about mathematical language, said MacLane, then he had to write about precision. "Instead, Hersh just sounded off."
Tom Reuterdahl reminded me of a good old piece of math lingo: "vanish." "I happened to be talking about the Wronskian and the condition under which it would vanish...ah, the word VANISH!! To my complete surprise the entire class was nonplussed by my usage of the term." Yes Tom, I remember being nonplussed, at NYU, 40 years ago. Maybe I thought that in order to "vanish," the Wronskian or Hessian or Jacobian or whatever should just fade away, leaving a blank spot on the page.
Don Myers of Arizona reminds us that random variables are functions, not (independent) variables. And why don't we "derive" when we want to get the derivative?
And finally, Edward MacNeal told me about his interesting book.
Acknowledgments: In addition to the correspondents quoted above, I also thank Jo-Ann Cohen of Raleigh, Ladnor Geissinger of Chapel Hill, Arnold Lapidus of Englewood, Eugene McGovern of Ossining, N.Y., Bruce Resnick of Chicago, and Dale A. Wood of Huntsville, Alabama.
Reuben Hersh is retired and lives in Santa Fe, New Mexico. With Philip J. Davis he was the co-author of The Mathematical Experience. His book What is Mathematics, Really? has just been published by Oxford in the U.S. and Johathan Cape in the U.K. His email address is firstname.lastname@example.org.