a historical standpoint, the origin of number is one that
has 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 exp