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Roger A. Horn
1515 Mineral Square, Room 142
University of Utah
Salt Lake City, UT 84112
February 22, 1997
Dear Professor Horn:
An article by Professor R. Hersh (Math Lingo…, pp 48-51) in the recent issue (No 1, v 104)
of the Monthly has left me bewildered. As I see it, the author set out to caution his
colleagues about frequent confusion of words’ math meanings and customary plain English
usages. The attempt is to be applauded. I could not disagree with the author less.
It is quite important from the outset to tell students that Mathematics employs its own
language wherein many well known words acquire a meaning different from what one might
expect. The advice is good and applies to all other sciences and fields of human endeavor.
However, a few points in the article have left me confused.
- The number story. After many years of teaching Mathematics the author finally realized that an obnoxious student of his who claimed that 0 is not a number was actually right. For, when one says "I own a number of Calculus books", one does not mean 0 books, nor even 1
book. I cite:
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In mathematical talk, "number" has several meanings. None is the plain English meaning.
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At the risk of being dubbed humorless, I have looked up the word "number" in the American
Heritage Dictionary. In the computerized version, there are 10 (most with several subarticles)
distinct meanings for the noun and 4 additional meanings for the verb. Mathematics has been
mentioned only once. It would be a relief to learn that the intention of the article was to sum
up a part of the math folklore that has to do with math as a language. I read and reread the
article several times. Some juxtapositions are indeed funny, but on the whole the article’s tone
is quite apologetic.
- The author writes:
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I say "math lingo", not language. It’s 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.
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For one, if in "You can’t say", the author meant me, he was patently wrong as I belief many
other mathematicians who also took the remark personally would justly claim. I can say this in
math "lingo" (just let me define "I" and "a headache" appropriately which I will postpone until
a more fitting occasion.)
Secondly, I wonder what the author may think of programming languages. Should these misfits
who did not make it into math departments and make their living by writing software boast of
mastering the FORTRAN lingo, or the C++ semidialect, or, more recently, the Java jargon?
Thirdly, on reading "… a complete language" one is reminded of the Godel’s famous
Incompleteness Theorem (it’s a math article in a math magazine, after all.) This said, the
whole argument of comparing contradictory meanings of math terms becomes vacuous. On the other
hand, I am pretty sure the author did not mean to imply that, in order to be complete, a
language should incorporate the two phrases "I have a headache" and "You bore me".
- Some of the examples quite plainly have nothing to do with mathematics. E.g., "amoebas
multiply by dividing" reflects more on the difficulties experienced by biology professors in an
undergraduate biology class than on a possibility of confusing math students. As an alternative,
I would suggest the following:
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The matrix’s other meaning is a womb, so a biology student taking a Linear
Algebra class may well consider it indecency to discuss multiplication of a matrix by
a vector whose other meaning is another agent that transfers genetic material from
one location to another (The American Heritage Dictionary.)
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Or, turning again to the number story, consider the plain English
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The crowd was small in number.
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Following the author, neither 0, nor 1, nor 2, nor … what should I call the rest? - is a
number, as one cannot say "The crowd was small in zero", or "The crowd was small in
one", etc.
- Some terminological problems transcend barriers between sciences. Take, for example, the
term "simple". The author asserts (and quite rightly) that mathematically simple curves may not
appear at all simple even to an experienced math eye. A "simple leaf" is a valid biological term
which is usually juxtaposed with a "compound leaf". However, one does not need a microscope to
convince oneself on inspection that neither of the two has a really simple structure. The
American Heritage Dictionary defines simple as "Having or composed of only one thing, element,
or part." This definition suits better my perception of "indivisible". Now recollect that the
word "atom" originates from the Greek "atomos" which means just that - indivisible. But who
nowadays would dare calling an atom simple?
In conclusion, "plain English" as, for that matter, any "plain" language, is inherently
ambiguous. By the end of the last century the fact has been recognized in Mathematics.
Currently, even a Biology text that would never include a deduction, develops a peculiar
vocabulary often by redefining "plain language" words. Thus, one may assume that the need for
conscientiously accurate word usage has been recognized long ago by both scientist and pedagogs
in Mathematics as much as in other sciences. Being the first to recognize the necessity of
specifying the language, mathematicians may be expected to lead the effort of instilling this
idea in the rest of the populace.
Josiah Willard Gibbs is known to have insisted that "Mathematics is a language". And, as
Richard Feynman wrote, "To those who do not know Mathematics it is difficult to get across a
real feeling as to the beauty, the deepest beauty of nature. ... If you want to learn about
nature, to appreciate nature, it is necessary to understand the language that she speaks in."
The obnoxious student of Professor Hersh was unequivocally wrong. Zero is a number. There is
nothing to be apologetic about.
Sincerely,
Alexander Bogomolny, Ph.D.
President, Cut The Knot Software, Inc.
http://www.cut-the-knot.com
cc:
rhersh@math.unm.edu
Copyright © 1996-2008 Alexander Bogomolny
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