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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 believe 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.
cc:
rhersh@math.unm.edu
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