Page:Mind (New Series) Volume 12.djvu/208

 194 B. KUSSELL: Chapter iv. deals with the problems of continuity, infinity and the infinitesimal. The exposition is historically careful, and ap- pears to take note of all important passages ; but the author's own views are, on these subjects, apparently more in agreement with Leibniz's than modern mathematics will permit. He writes, however, in this chapter, with a certain reserve (e.g. p. 218), which makes it difficult to feel certain as to his opinions, or even whether they are definite. The differential, we are told, is constituted by the qualitative unity of a law, while the integral denotes a magnitude as generated by a law (p. 170). Zero as a limit has positive significance : dx r though quantitatively zero, retains the character of what vanishes, and is intelligible, not as a single quantum, but only in the process. Leibniz showed the impossibility of regarding the continuum as a single datum : only by a law of becoming can it be understood. Thus continuity requires change, but change thereby becomes the necessary presupposition of the concept of reality (p. 185). A simple substance, for Leibniz, is the law of a series, whose terms are the states of a substance (pp .187-188) : or again, it is the general term of the series (p. 538). The constancy presupposed in the conception of being is no longer the unchangeability of a thing, but the methodical constancy of the rule according to which the content varies (p. 189). In these views, which are supported by texts from Leibniz, we must, when we inquire into their truth, distinguish two elements, the mathematical and the philosophical. Leibniz's belief t;iat the Calculus had philosophical importance is now known to be erroneous : there are no infinitesimals in it, and dx and dy are not numerator and denominator of a fraction. The doctrine of limits, by careful statement, has been found alone adequate, and has shown that the Calculus is an advanced and purely technical development of the science of order. The con- tinuum is essentially a s : ngle datum, in the sense that it is the field of a given relation ; but the essential properties of continuity belong primarily to the relation, and belong to the terms composing its field not qua class of terms, but only qua field of a continuous relation. Continuous relations, so far from depending upon time or change, are not known even to occur in temporal series : the only indubitable instances of such relations are derived from Arithmetic. So far for what mathematics has to say. As regards philosophical questions, I confess that I fail wholly to understand what is meant when it is said that reality presupposes change, or that the constancy presupposed in Being is not unchangeability, but the constancy of a rule of variation. Change of what ? from what ? into what ? one must ask ; and these questions can only be answered by means of logical concepts, whose Being is free from dependence upon time, and is thus necessarily unchangeable. Change in an identical content means difference in its relations to different moments of time ; but the content must remain strictly self-identical, and this self-identity