n striking moments of
mathematical progress; and then we shall try to interpret its genuine
status in the world of working truths.

I

BEGINNINGS OF ARITHMETIC AND GEOMETRY

The most primitive mathematical activity of man is counting, but here
his first efforts are lost in the obscurity of the past. The lower
races, however, yield us evidence that is not without value. Although
the savage mind is not identical with the mind of primitive man, there
is much in the activities of undeveloped races that can throw light upon
the behavior of peoples more advanced. We must be careful in our
inferences, however. Among the Australians and South Americans there are
peoples whose numerical systems go little, or not at all, beyond the
first two or three numbers. "It has been inferred from this," writes
Professor Boas (_Mind of Primitive Man_, pp. 152-53), "that the people
speaking these languages are not capable of forming the concept of
higher numbers.... People like the South American Indians, ... or like
the Esquimo ... are presumably not in need of higher numerical
expressions, because there are not many objects that they have to count.
On the other hand, just as soon as these same people find themselves in
contact with civilization, and when they acquire standards of value that
have to be counted, they adopt with perfect ease higher numerals from
other languages, and develop a more or less perfect system of
counting.... It must be borne in mind that counting does not become
necessary until objects are considered in such generalized form that
their individualities are entirely lost sight of. For this reason it is
possible that even a person who owns a herd of domesticated animals may
know them by name and by their characteristics, without even desiring to
count them."

And there is one other false interpretation to be avoided. Man does not
feel the need of counting and then develop a system of numerals to meet
the need. Such an assumption is as ridiculous as to assume prehistoric
man thinking to himself: "I must speak," and then inventing voice
culture and grammar to make speaking pleasant and possible. Rather, when
powers of communication are once attained, presumably in their
beginnings also without forethought, man being still more animal than
man, there were gradually dissociated communications of a kind
approaching what numbers mean to us. But the number is not yet a symbol
apart from that of the things numbered. Pic