112 Ãé. COUNTINGAncient Greek and Sanskrit, as well as modern English and other European
languages, share a great many root words and morphological features. In a book
on Greek mathematics (1884) the British mathematician James Gow (1854 1923)
of Trinity College, Cambridge speculated on the possibility of using comparative
philology to discover the history of mathematical terms. He noted in particular that
the words for one, two, three, and four are declinable in Greek, but not the words
for five and above. That fact suggested to him that numbers above four are an
artificial creation. (It also dovetails neatly with the observations of Karen Fuson,
discussed in Chapter 1, on the counting abilities of children.) Gow noted that
in Slavonic, which is a European language, all numerals are declined as feminine
singular nouns (those ending in 5 or above still are, in modern Russian), but he
regarded this usage as later and hence not relevant to his inquiry. He also noted
that all numerals are declined in Sanskrit, but thought it an important difference
that no gender could be assigned to them.
In a comprehensive study of numbers and counting (1969) the mathematician
Karl Menninger (1898-1963) conjectured that the words for one and two may be
connected with personal and demonstrative pronouns. In favor of Menninger's con-
jecture, we note that in formal writing in English, and sometimes also in formal
speaking, the word one is used to mean an unspecified person. This English us-
age probably derives from similar usage of the special third-person pronoun on in
French. Speaking of French, there is a suggestive similarity between this pronoun
and the word (un) for one in that language. The Russian third-person pronouns
(on, ona, ono in the three genders) are the short forms of the archaic demonstrative
pronoun onyi, onaya, onoe (meaning that one), and the word for the number one
is odin, odna, odno.
Menninger also suggested that the word two, or at least the word dual, may
be related to the archaic second-person singular pronoun thou (du, still used in
Menninger's native language, which is German). Menninger noted that the concept
of two is closely related to the concept of "other." Consider, for example, the
following sentences:
This is my favorite style of gloves. I have a second pair in my closet.
This is my favorite style of gloves. I have another pair in my closet.
Menninger suggested a connection between three and through, based on other lan-
guages, such as the Latin tres and trans. Despite these interesting connections,
Menninger emphasized that the words for cardinal numbers have left no definite
traces of their origin in the modern Indo-European languages. All the connections
mentioned above could be merely coincidental. On the other hand, the ordinal num-
ber words first and second have a more obvious connection with non-mathematical
language. The word first is an evolved form of fore-est (foremost), meaning the one
farthest forward. The word second comes from the Latin word sequor (I follow).
In cultures where mathematics and counting are developed less elaborately,
number words sometimes retain a direct relation with physical objects exemplifying
the numbers. Nearly always, the words for numbers are also used for body parts
in the corresponding number, especially fingers. In English also, we find the word
digit used to describe both a finger or toe and the special kinds of numbers that
occur in representations of the positive integers in terms of a base.
Are there languages in which body parts stand for cardinal numbers? Could
the number two be the word for eyes, for example? Gow (1884) cited a number