The character of the modern treatment of elementary mathematics is such that mathematics might reasonably be described as the Science of Classification. The mathematician may be supposed to begin as a collector, whether of butterflies, minerals, or even of cigarette cards, does not matter. Let us consider that he has collected so many different kinds of things as to become virtually a collector of collections, each collection having its own classification, and that his next step is to compare these classifications with each •other, that is, to become a classifier of classifications. The mere comparison of classes requires the i-iea of cardinal number, since the only thing which can be said about two classes, irrespective of the nature of their individual members, is the number of individuals in each class. But the comparison of classifications which do not consist merely of single classes introduces ideas which are essential to the further development of the idea of number, and to the development of Geometry, and which should therefore precede that development. It is in this super-classification or comparative study of classifications that the foundations of mathematics are to be sought, and the question therefore arises, whether the presentation of the rudiments of mathematics should not be modified so as to recognize this fact. Of the fundamental ideas of mathematics, four especially, namely those of correspondence, order, group, and of multiple correspondence, seem to lend themselves, in their elementary treatment at any rate, to the method in which they are regarded as belonging to the Science of Classification, and will be considered in that light here, the last three being regarded as developments of the first.