The Number Concept | Page 2

provoked much lively discussion, and has led to a great
amount of learned research among the primitive and savage languages

of the human race. A few simple considerations will, however, show
that such research must necessarily leave this question entirely
unsettled, and will indicate clearly that it is, from the very nature of
things, a question to which no definite and final answer can be given.
Among the barbarous tribes whose languages have been studied, even
in a most cursory manner, none have ever been discovered which did
not show some familiarity with the number concept. The knowledge
thus indicated has often proved to be most limited; not extending
beyond the numbers 1 and 2, or 1, 2, and 3. Examples of this poverty of
number knowledge are found among the forest tribes of Brazil, the
native races of Australia and elsewhere, and they are considered in
some detail in the next chapter. At first thought it seems quite
inconceivable that any human being should be destitute of the power of
counting beyond 2. But such is the case; and in a few instances
languages have been found to be absolutely destitute of pure numeral
words. The Chiquitos of Bolivia had no real numerals whatever,[1] but
expressed their idea for "one" by the word etama, meaning alone. The
Tacanas of the same country have no numerals except those borrowed
from Spanish, or from Aymara or Peno, languages with which they
have long been in contact.[2] A few other South American languages
are almost equally destitute of numeral words. But even here,
rudimentary as the number sense undoubtedly is, it is not wholly
lacking; and some indirect expression, or some form of circumlocution,
shows a conception of the difference between one and two, or at least,
between one and many.
These facts must of necessity deter the mathematician from seeking to
push his investigation too far back toward the very origin of number.
Philosophers have endeavoured to establish certain propositions
concerning this subject, but, as might have been expected, have failed
to reach any common ground of agreement. Whewell has maintained
that "such propositions as that two and three make five are necessary
truths, containing in them an element of certainty beyond that which
mere experience can give." Mill, on the other hand, argues that any
such statement merely expresses a truth derived from early and
constant experience; and in this view he is heartily supported by

Tylor.[3] But why this question should provoke controversy, it is
difficult for the mathematician to understand. Either view would seem
to be correct, according to the standpoint from which the question is
approached. We know of no language in which the suggestion of
number does not appear, and we must admit that the words which give
expression to the number sense would be among the early words to be
formed in any language. They express ideas which are, at first, wholly
concrete, which are of the greatest possible simplicity, and which seem
in many ways to be clearly understood, even by the higher orders of the
brute creation. The origin of number would in itself, then, appear to lie
beyond the proper limits of inquiry; and the primitive conception of
number to be fundamental with human thought.
In connection with the assertion that the idea of number seems to be
understood by the higher orders of animals, the following brief
quotation from a paper by Sir John Lubbock may not be out of place:
"Leroy ... mentions a case in which a man was anxious to shoot a crow.
'To deceive this suspicious bird, the plan was hit upon of sending two
men to the watch house, one of whom passed on, while the other
remained; but the crow counted and kept her distance. The next day
three went, and again she perceived that only two retired. In fine, it was
found necessary to send five or six men to the watch house to put her
out in her calculation. The crow, thinking that this number of men had
passed by, lost no time in returning.' From this he inferred that crows
could count up to four. Lichtenberg mentions a nightingale which was
said to count up to three. Every day he gave it three mealworms, one at
a time. When it had finished one it returned for another, but after the
third it knew that the feast was over.... There is an amusing and
suggestive remark in Mr. Galton's interesting Narrative of an Explorer
in Tropical South Africa. After describing the Demara's weakness in
calculations, he says: 'Once while I watched a Demara floundering
hopelessly in a calculation on one side of me, I observed, "Dinah," my
spaniel, equally embarrassed on the other; she was overlooking half a
dozen of her new-born puppies, which had