Chapter 5. Counting
Counting could conceivably occur without number words. What is required is
merely a matching of the objects in two sets. The legendary American gunslinger,
putting a notch in the handle of his gun for every person he has killed, is an example
of such counting. A vivid example is cited by Closs (1986, p. 16) as a folk tale of the
Copper Eskimo. In this story, a hunter who has killed a wolf argues with another
hunter who has killed a caribou as to which animal has more hair. To decide the
question, they pull out the hairs one at a time and pair them off. This mode of
thought has become very familiar to mathematics students over the past century
because of the rise of set theory in the undergraduate curriculum.
1. Number words
Every human language that we know about has words for numbers. In the case of
languages whose long history is known—English, for example—the number words
seem to be of such ancient origin that they have no obvious relation to the non-
numerical words in the language. Some attempts have been made to find clues as
to the origin of number words and number concepts in the grammar of various lan-
guages, but very few reliable conclusions have been reached. The guesses involved
are interesting, however, and we shall look at a few of them below, taken mostly
from the books of Menninger (1969) and Gow (1884)-
To begin with modern English, when a person says, "I know a number of
ways to prove the Pythagorean theorem," the listener will interpret the phrase "a
number of" to mean "three or more." The word number here is a synonym for the
fancier word multitude and is used to refer to any set of things having three or more
members. If the speaker knew only two ways to prove the Pythagorean theorem,
the word number would almost certainly be replaced by couple. This last word is
one of the few collective words in English with a definite numerical meaning. It
is used as a synonym for two when the two objects mentioned are related to each
other in some way, in phrases like "I have a couple of errands to run downtown."
The connection with the number two is not quite exact here, since in very informal
speech the word couple is often stretched to mean simply a small number. From
such considerations we might form the hypothesis that one and two are instinctive
concepts, and that numbers as a deliberate, conscious creation of the human mind
begin with three. Support for this idea comes from the reflection that English has
special words for ordinal (second) and partitive (half) concepts connected with the
number two and a special word (both) to apply to the whole of a set of two objects,
while the ordinal and partitive concepts merge for numbers three and higher (third,
one-third, and so on) and the same word (all) is used to denote the whole of any
set having more than two members. Of course, what is true of English does not
always apply to other languages.
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