Page:Mind (New Series) Volume 12.djvu/209

EECENT WORK ON THE PHILOSOPHY OF LEIBNIZ. 195 is logically prior to change, nob subsequent to it. Again, neither Leibniz nor Dr. Cassirer have realised what is meant by the con- stancy of a rule, the law of a series, etc. These notions mean that the terms whose law is constant are the field of a serial relation : there is nothing constant, so the position may be stated, except the serial relation itself. But the constancy of this relation is precisely the absolute timeless self -identity which was to have been banished ; and this will still have to belong to terms as well as to relations, if different relations are to have different fields in any significant sense. The same desire to make conceptions fluid appears in Leibniz's definition of equality as infinitesimal inequality. Following Cohen (op. cit.), Dr. Cassirer approves this definition, and adds that, in modern language (i.e. Cantor's), two magnitudes are equal when they are defined by equivalent fundamental series, i.e. by such as have between corresponding terms differences whose limit is zero (p. 194). The gloss in italics introduces a quantitative notion wholly foreign to the essence of limits. Equality, to begin with although, where irrationals are concerned, Cantor's language is ambiguous is never defined by fundamental series, but by abso- lute identity. And fundamental series may be equivalent, i.e. may have the same limit (if any), or define the same segment in any case, although the difference of corresponding terms is constant and infinite. 1 Thus when Dr. Cassirer remarks (p. 197) that the very notion of exactitude is now altered, we must reply : Yes, into inexactitude. Infinity, the author points out, is for Leibniz that of a dis- tributive, not of a collective, whole : it is not a property of a single datum, but essentially of an infinite process. It is the continuation of a law as against every single term created by the law (p. 200 ff.). This seems to mean that there are relations whose fields cannot in any way be treated as units, and which are such that no finite number of terms constitutes the whole of the field. The difficulty of the view lies in the fact that to be the field of a given relation is in itself a kind of unity, and seems to imply necessarily the existence of a collective whole. But to pursue this subject would take us into the darkest corners of logic. Infinitesimals, it is pointed out (p. 207), are stated by Leibniz to be merely useful fictions. On this point, there is the greatest difficulty in discover- ing his true opinion, for he certainly used notions derived from the Calculus in establishing force, and in many ways the infinitesimal seems to be involved in his philosophy. But Dr. Cassirer appears to be unconscious, or nearly so, of the magnitude of this inconsist- ency 1 For example, if co represent the ordinal number of the finite integers in order of magnitude, the series whose general terms are respectively < x 2n and w (2n + l) both have eo 2 for their limit, although the difference of corresponding terms is always o>.