§6. The Divisions of Mathematics

283. Having now attained to a pretty clear apprehension of what a natural order of science is — deficient in distinctness though this apprehension be — we cannot, if we have any acquaintance with mathematics, consider that class of science, without seeing that none more manifestly falls into orders than this. The hypotheses of mathematics relate to systems which are either finite collections, infinite collections, or true continua; and the modes of reasoning about these three are quite distinct. These, then, constitute three orders. The last and highest kind of mathematics, consisting of topical geometry, has hitherto made very little progress; and the methods of demonstration in this order are, as yet, little understood. The study of finite collections divides into two suborders: first, that simplest kind of mathematics which is chiefly used in its application to logic, from which I find it almost impossible to separate it¹⁾; and secondly, the general theory of finite groups. The study of infinite collections likewise divides into two suborders; first, arithmetic, or the study of the least multitudinous of infinite collections; and second, the calculus, or the study of collections of higher multitude. Hitherto, the calculus has been entirely confined to the study of collections of the lowest multitude above that of the collection of all integral numbers. This is studied either algebraically or geometrically, or, much more commonly, and perhaps more advantageously (though it is out of fashion to think so), by the two methods combined. The traditional division of mathematics, still much used, is into geometry and algebra — the division used by Jordanus Nemorarius²⁾ in the thirteenth century. It seems to me to be not only entirely artificial, but also extremely inconvenient from every point of view except the one of conforming to usage.³⁾