The is an excerpt from
in The American Mathematical Monthly
volume 104, Number 1, January 1997, pp 48-51
Once upon a time, when I was a teaching assistant, teaching a class of the kind mockingly called "Math for Poets," an obnoxious freshman said to me, "Zero isn't a number."
I have forgotten my answer, but I remember finding her remark a shocking expression of profound ignorance.
Years later, it dawned on me - she was right!
If I say "I own a number of calculus books" or "I have a number of friends at the Courant Institute," I don't mean zero books or zero friends. I don't even mean one book or one friend. I mean two or more. That's what "number" means in plain English. I read recently that the famous phenomenologist Edmund Husserl meant by "number" something greater or equal to 2. So did Plato.
In mathematical talk, "number" has several meanings. None is the plain English meaning. The ordinary math teacher, like me back then, is so deeply embedded in math lingo that he/she doesn't notice the inconsistency. But the inconsistency can confuse students.
I say "math lingo," not language. It's a jargon, a semidialect of English (or some other natural language), not a complete language. You can't say "I have a headache" or "You bore me" in math lingo.
In math lingo, a straight line is the simplest example of a curve. In plain English, quite otherwise: a straight line isn't a curve, and a curve isn't a straight line.
In English, what we call a "line segment" is just a "line." What we call a "line" is "an infinite line." "Difference," "product," "factor," "prime" all have different meanings in plain English and in math lingo. I may ask a student, "If you subtract zero from zero, what's the difference?" While answering math-linguistically, "zero," she may be thinking, plain-Englishly, "That's right! Who cares? What's the difference?"
In English, "adding" increases what you've got. In math lingo, it may increase it, decrease it or neither, depending on whether you happen to be adding something positive, negative or zero.
Correspondingly, subtracting decreases. In math lingo, it may decrease or increase or neither.
In English, "adding" and "subtracting" are opposite. In math lingo, they're opposite, and yet they're the same! For adding a number is the same as subtracting some other number (its negative).
In English, "multiplying" means repeated adding. It makes things bigger. In math lingo, multiplying makes them bigger, smaller, or neither, depending on what you multiply with.
Correspondingly, "divide" means cut into pieces, possibly equal pieces. In math lingo, "divide" is the same as "multiply," in the sense that dividing by a number other than zero is the same as multiplying by some other number (its reciprocal).
There's a familiar conundrum about amoebas: amoebas multiply by dividing. To untangle this nonsensical but correct statement, you must see the difference between the mathematical and the plain English meanings of "multiply" and "divide."
What should you do about all this? Be aware of it and point it out to students. By appropriate examples, make them realize that what they hear in class or read in the text is technical jargon, not plain English. Otherwise, when they try to remember what you said in yesterday's lecture, they may remember it with the wrong meaning (the plain English).
Anneli Lax reminded me of one of the commonest linguistic pitfalls: the little one-letter word "a." Her example is "Show that a number divisible by 6 is even."
No seasoned math teacher is surprised to receive the wrong answer, "42 is divisible by 6. 42 is even." Why is this answer wrong? 42 is divisible by 6, and 42 is even. What's wrong is that the question has been misunderstood. By "a," the questioner meant "every"; the student misinterpreted it as "some." This is a quantification problem, which in principle could be cured by using symbolic logic instead of English. But in a case like this, something deeper is wrong. The student should realize that with the interpretation "some," the question is too trivial to be on the test. Grounding in the context saves the student from most verbal pitfalls. One goal of teaching is to ground the student in the context. Linguistic ambiguities can hurt.
In logic, the pitfalls of "or" and "implies" are familiar.
Take "or." In plain English, "Tea or coffee?" means one or the other, not both. It's called the "exclusive or."
"Are you coming or going?"
"Was that your husband or your boy friend?"
"Do it now or later?"
All are exclusive. It's hard to think of a colloquial example of the other "or," the inclusive one. A reasonable example might be, "Like a hug or a kiss?"
In logic, "or" is inclusive by convention. "A or B" is true if A or B or both is the case. I think it's customary to explain on the first day of elementary logic class that logicians have decreed "or" to be inclusive. A student can accept that logicians felt they had to pick one or the other. Perhaps they had a reason for picking the inclusive.
Peter Lax tells about the famous logician Abraham Fraenkel, of German origin and Israeli residence. Once in Jerusalem or Tel Aviv he was on a bus scheduled to leave the station at 9A.m. At 9:05 the bus was still sitting in the station. Fraenkel waved a bus schedule at the bus driver, who asked, "What are you, a German or a professor?" Fraenkel inquired in return, "Do you use the inclusive 'or' or the exclusive?"
"Implies" is worse. In plain English, "A implies B" means that if A is true, B must be true. If A is false, the "implies" statement is vacuous, neither true nor false.
But in logic, the "law of the excluded middle" insists that every statement be either true or false. The statement "A implies B" has to be either true or false, even if A is false. Logicians chose "true." So in logic, if A is false, then A implies B, whatever B may be. This is so unintuitive, I say logicians should have used another word, even made up a word. It's too late for that. But the student is told that "implies" in logic is different from "implies" in plain English. In pre-calculus, calculus, and post-calculus, we should be equally considerate to warn of linguistic traps.
I have just carelessly used "equally." "Equal" is used freely, from kindergarten to postgraduate. It's never defined or explained.
In plain English, its meaning varies. Sometimes it's "identical, indistinguishable." Sometimes it's "worth the same number of dollars." Or "just as good" for some purpose.
Math lingo sometimes says "equal," sometimes "equivalent," the latter if an equivalence relation has been defined. Then we explain that an equivalence relation is Reflexive, Symmetric, and Transitive; it defines a partition on a set.
But what does equal mean? When we say 1/2 = 2/4, we don't mean 1/2 is indistinguishable from 2/4. They have different numerators. They have different denominators. We regard them as equivalent for good and sufficient reasons. All this may be explained in an advanced course, on the rare occasion when a detailed construction of the rationals is carried out. But already in the fourth grade the = relation is an equivalence relation between fractions, not an identity. No one ever explains this, so there's no way for the student to understand except in terms of models like slices of apple pie.
This nonunderstanding was manifested frighteningly when a calculus student was asked, "What is the minimum of the function
and answered "correctly"
Maybe this is the outcome of years in high school spent factoring, multiplying, and dividing expressions that always remained equal.
In plain English, set and group are synonyms. When we teach groups, we define set and group, then charge ahead. But some students wonder, "What's the difference? A group is the same as a set." Mention this plain English equivalence, and state explicitly that in math these words have different meanings.
The same is true of sequence and series. Their plain English meanings are the same - what in math lingo we call "a finite list." "Series" is more colloquial than sequence - for example, it's the World Series, not the World Sequence! Here the danger of confusion is more serious than with set and group. The mathematical meanings of sequence and series are so close that the distinction between them is crucial. In teaching series, we should acknowledge that we're giving a new meaning to a common word: putting + signs instead of commas between the terms.
The first day of first - semester calculus I like to talk about driving to Santa Fe. Distance from Albuquerque is a function of time. Speed is another function of time. But what is "function" in English? If you ask, "Of what is the speed a function?," you're told, "It's a function of how much gas you give" or "Of how hard you push the accelerator pedal." "Function" in English (apart from the irrelevant reference to weddings and Bar Mitzvahs) involves causal dependence. "How fast you learn is a function of how hard you study," for example. How can anything be a function of time? But the students swallow that. They understand a graph with a time axis. Then I say, "Distance is a monotonic increasing function of time, so the inverse function exists. Time is a function of distance." How can time, the independent, uncaused variable, be caused by distance? We try to teach our technical meaning of "function" without noticing the meaning the student brings into class.
We're aware that "limit" and "converge" are deep concepts. We sweat over them. But we don't acknowledge the complication caused by plain English. A "limit" in English is a barrier, a boundary beyond which one may not pass. This may partly explain why students want to approach a limit only from one side, not in alternating fashion. As for "converge." In practical computation, an algorithm converges when it settles down to one value and stays there - stays till whoever's doing the calculation is satisfied. That's the English of converge - "settle down" " close to" some "limit." In teaching our uncomputational, abstract meaning of "converge," we should talk about the colloquial meaning and explain the difference.
In advanced mathematics, there's more linguistic confusion. Surds (absurd), irrational and imaginary numbers, singular perturbations, degenerate kernels, strange attractors - all sound dangerous, undesirable, things to avoid. Yet a degenerate kernel or a singular perturbation may be more useful than a nondegenerate or regular one.
We also talk about "function spaces." The points in a function space are functions. But a function is a graph - a curve. How can a curve be a point? A point, which has no parts! We don't acknowledge the change of meaning. Just give a definition and two examples, then charge ahead.
An example of the opposite kind (due to Peter Lax) is "simple curve." Draw a confusing tangle that doesn't intersect itself. It's complicated. We say it's simple.
What about "partial?" A partial order isn't a special kind of order. A partial differential equation isn't part of an ordinary differential equation. And an ordinary differential equation may well be extraordinary.
- give the plain English meaning of prime; differentiate; integrate.
- check your answers against a standard dictionary.
- make up three slogans, one using each of these three words, that could appear on picket signs at a demonstration.
It's fortunate that some double meanings are so far apart they can be used for a joke. A manifold is part of an automobile engine (I think), and a commutator is part of a direct current electric motor.
ACKNOWLEDGMENTS. Veronka John-Steiner, Anneli Lax, and Peter Lax gave suggestions and encouragement.  is an inspiring example of frank talk about college math teaching.
 Boas, R. P. Can we make mathematics intelligible?, Amer. Math. Monthly, 88 (1981), 727-731.
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