# "Math Lingo": a follow-up article

## by Reuben Hersh, Santa Fe, N. M.

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:

I "is an element of" H
and

You "is an element of" B.
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:

If the chairman does not scold Tadashi, then Tadashi does not
work.
Its alleged logical equivalent is:

If Tadashi works, then the chairman scolds him.
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.

## Anthology of inclusive OR, in response to "Math Lingo..."

- "Will you go or will you go?" and

- "I'll get an A in math or English," both contributed by
F. Khosraviyani.

- "I may write or telephone,"
contributed by Webster's Dictionary and J. L. Kuhns.

- "In order to graduate from some high schools, one must take a year
of chemistry or a year of physics," and

- "In order to legally drive in most countries, you must have an
international license or a license for the country you are driving
in." Both by Kenneth Ross, Oregon.

- "I was about to have a couple of hernias repaired. The anesthetist
asked whether I wanted a spinal or a general anesthetic. I replied
with no hesitation. "BOTH." He told me that wasn't an option. I
guess he wasn't a mathematician." From Larry Wallen, University of
Hawaii.

- "Admission limited to people over 18 years old or accompanied by a
parent," contributed by Kenneth Ross and by Aaron Meyerowitz of
Florida Atlantic University (independently).

- "Do you want cream or sugar in your coffee?" and

- "Do you know how to tango or waltz?" contributed independently by
Leon Harkelroad of Cornell and Eric Karnowski.

**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.

## References (most suggested by readers)

- Klein, J. Greek Mathematical Thought and the Origin of Algebra.
Cambridge: M.I.T. Press, 1968.

- MacNeal, E. Mathsemantics: Making Numbers Talk Sense. New York:
Viking, 1994.

- Mueller, I. Philosophy of Mathematics and Deductive Structure in
Euclid's Elements. (p. 58, notes.)

- Poe, E. A. "The Purloined Letter", in Collected Works. Cambridge:
Belknap Press of Harvard University Press, 1969.

- Reichenbach, H. "Analysis of Conversational Language," in Elements
of Symbolic Logic. New York: MacMillan, 1947.

- Russell, B. Principles of Mathematics. New York: Norton, 1964.

- Whitehead, A.N. An Introduction to Mathematics. New York: Holt,
1911.

**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 [email protected].

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