hich are most effective for
prediction and control. Such selection gives the principle that
dominates all abstractions. Progress is movement from the less abstract
to the more abstract, but it is progress only because the more abstract
is as genuinely an aspect of the concrete starting-point as anything is.
Moreover, the outcome of progress of this sort cannot be definitely
foreseen at the beginnings. The simple activities of primitive men have
to be spontaneously performed before their value becomes evident. Only
afterwards can they be cultivated for the sake of their value, and then
only can the self-conscious cultivation of a science begin. The process
remains full not only of perplexities, but of surprises; men's
activities lead to goals far other than those which appear at the start.
These goals, however, never deny the method by which the start is made.
Developed intelligence is nothing but skill in using a set of concepts
generated in this manner. In this sense the histories of all human
endeavors run parallel.

Where the empirical bases of a science are continually in the
foreground, as in physics or chemistry, the foregoing formulation of
procedure is intelligible and acceptable to most men. Mathematics seem,
however, to stand peculiarly apart. Many, with Descartes, have delighted
in them "on account of the certitude and evidence of their reasonings"
and recognized their contribution to the advancement of mechanical arts.
But since the days of Kant even this value has become a problem, and
many a young philosophic student has the question laid before him as to
why it is that mathematics, "a purely conceptual science," can tell us
anything about the character of a world which is, apparently at least,
free from the idiosyncrasies of individual mind. It may be that
mathematics began in empirical practice, such philosophers admit, but
they add that, somehow, in its later career, it has escaped its lowly
origin. Now it moves in the higher circles of postulated relations and
arbitrarily defined entities to which its humble progenitors and
relatives are denied the entree. Parvenus, however, usually bear with
them the mark of history, and in the case of this one, at least, we may
hope that the history will be sufficient to drag it from the
affectations of its newly acquired set and reinstate it in its proper
place in the workaday world. For the sake of this hope, we shall take
the risk of being tedious by citing certai