Precision of Speech

       Among the causes of America's woeful performance in mathematics
education are often listed poverty, prejudice, alcohol, television,
parental indifference, teachers' ignorance, lack of school discipline, low
teachers' salaries, educational frills such as driver and sex education,
racism, sexism, capitalism and sloth.  Anyone who reads enough about the
low test scores will be able to add to the list; the lecture circuits and
PTA meetings are stocked with well-paid sociological seers who will
explain any one of these causes in detail, suggest a new Federal program
to fix it, and answer questions from the audience in the coffee period

     As a mathematician I, too, am a seer of sorts, though not as affluent
as the average circuit lecturer; and I have another cause to offer. I
don't guarantee it as THE cause of mathematical under-performance, but it
is surely a concomitant of our educational woes, and an accurate symptom.
My analysis is suggested by an Associated Press story printed in the
Rochester Democrat & Chronicle, February 11, 1994.

       That story was not about education, but was about children --
teenagers, actually -- children buying and smoking cigarettes; and its
last sentence read:

Cummings compiled government data estimating there are 2.7 million
smokers ages 12 to 18 in the United States who smoke up to 12
cigarettes a day, most of which are bought illegally, not borrowed or

     No, I'm not trying to say that cigarettes have caused the American
decline in mathematical performance; there is something else about that
quotation that represents the cause I am thinking of:  It is the kind of
nonsensical language used in the quoted sentence, language whose
unthinking acceptance in daily speech and writing corrupts the

understanding of our youth as surely as cocaine.

       Any mathematician or scientist, or engineer or lawyer, anyone whose
trade requires precise reasoning, will tell you that the quoted AP
sentence on teenagers has told us almost nothing.  They smoke "up to 12
cigarettes a day," it says.  How many is that?  There's an arithmetic
problem for you:  How many cigarettes is "up to 12"?  Three?  One?  One
every month?  Sure; all these answers are correct.  Is that what the
author was trying to say?  The only thing actually stated in that sentence
is that the "smokers" referred to do not smoke more than 12 per day.

     Surely this is not the point the writer of that Associated Press
story meant to convey; it is probably not even true.  It is hard to know
just what he was intending, in fact, beyond a generalized regret about
teen-age smoking.  Yes, the article was full of numbers, very statistical
sounding.  But the phrases "12 to 18", "2.7 million", and "up to 12"  were
not really intended to convey information; they were decorations to the
news story, designed to sound scientific but without actually straining
anyone's understanding with facts or reason.

     A society that tolerates this sort of thing on a daily, even hourly,
basis, that has learned to accept meaninglessness as the normal currency
of discourse, will never understand mathematics.  It is some kind of
miracle that some Americans still do.

       As a professor of mathematics I see a new bunch of 17 or 18 year
old students of calculus every year, and looking at their papers and
examinations, or even hearing them ask questions, I see the damage done by
17 or 18 years of exposure to logical nonsense presented as if it were

       Try asking an American teenager to define anything, anything at
all, a window, say, or a lake.  What is a lake, you ask.

     "Well, a lake is a like water, you know, like a lot of water, you
know, like Lake Ontario."

     If that's the answer you get, which except for the example (not asked
for) could apply to a teaspoon and the Pacific Ocean as well, think of the
answer you will get when you ask for a definition of honesty, or beauty.
Actually, if you are teaching mathematics, history, psychology or French,
and were to ask a student what a lake is, the most probable answer would
be, "But that wasn't on the assignment."

     In mathematics and all the sciences one needs definitions for objects
more subtle than a lake, though perhaps not as elusive as beauty. What is
"force," "atom," "complex number," "memory"?  Until you can say what the
words mean, any theories and observed relationships between them must be
meaningless however accurately they seem to be stated.  But then when the
relationships too are stated illogically, when the "up to 12" is
indistinguishable from "12" and "at least 12" in daily speech, not even
definitions will help.  Definitions are both a summary of prior
understanding and a guide to future exploration, and the inability to
state or use them is the very definition of intellectual poverty.

     If you go into our American graduate schools of mathematics,
engineering and the sciences, to count who is emerging with PhDs, you will
find that they are mostly imports from China, Japan, Vietnam and India.
Since in their first eighteen years they weren't taught nonsense by the
Associated Press and the self-esteem peddlers on the PTA circuits, they
were already several notches ahead of the natives here.  They therefore
had a chance to succeed that we aren't giving our homegrown youngsters,
and it shows.

      If I were writing for the Associated Press I might conclude by
saying that among the new mathematicians being graduated by American
universities there is a significant foreign component of "up to fifty
percent or more."  If I wanted to get my point across without ambiguity,
however, I would report, as the American Mathematical Society did last
year, that about half the PhDs in mathematics from United States universities

in 1993 were foreign-born foreign citizens.  It is our good luck
that some of them will turn out to be immigrants, too.

     Ralph A. Raimi
     11 February 1994