It is the essence of science to make prediction possible. The value of prediction lies in the fact that through this function man can control his environment, or, at worst, fortify himself to meet its vagaries. To attain such predictions, however, the world need not be grasped in its full concreteness. Hence arise processes of abstraction. While all other symptoms remain unnoticed, the temperature and pulse may mark a disease, or a barometer-reading the weather. The physicist may work only in terms of quantity in a world which is equally truly qualitative. All that is necessary is to select the elements which 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 entrée. 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 certain striking moments of mathematical progress; and then we shall try to interpret its genuine status in the world of working truths.