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 EECENT WOBK ON THE PHILOSOPHY OF LEIBNIZ. 193 be accepted as final truth. A concept, Dr. Cassirer says, is not for Leibniz merely a sum of given marks, but the result of a judgment (p. 117). Yet M. Couturat's account of the attempts to construct a Symbolic Logic shows that the opposite statement is at least equally correct, and that there is in fact a contradiction at this point. Possibility Leibniz says, may be proved by experi- ence of actuality as well as a priori. This, the author remarks, shows that the decision of possibility goes beyond ordinary logic, and presupposes the foundations of scientific knowledge (pp. 112-113). The consequence, I think, is scarcely Leibnizian ; for where there is no a priori proof of possibility, this is because a complete analysis has not been effected, so that we do not know what it is whose possibility is proved by experience. Logic, the author continues, is to be transformed from a science of the forms of thought into one of objects ; this is to be effected by mathematics, which mediates between ideal logical principles and the reality of nature (p. 123). Chapter ii., on the fundamental concepts of quantity, points out that Leibniz, like Descartes, starts from quantity, but in the form of number, not of extension : the effect of having started from discreteness is visible throughout his work. He was guided, says the author, by the notion of the identity of logic and mathe- matics, where logic, to begin with, must be the logic of quantity. But Algebra is not the general logical method, and the science of quantity leads to that of quality. The next chapter, on the geo - metrical problem of space, asserts that the further development of the notion of quantity is to be derived from the Infinitesimal Calculus, whose presuppositions are not arithmetical merely, but spatial. As a statement of Leibniz's view, this is probably correct ; as a statement of the facts, it has been disproved by Weierstrass and the arithmetical theory of irrationals. The essence of space, Leibniz points out and this is an important truth is not magni- tude, for magnitude belongs also to number, time, and motion, and does not belong to the point, which is yet spatial. Leibniz's x in his Characteristic, Dr. Cassirer says, is not a true variable, but a collection : it is not obtained, as in the true notion of the variable, by varying one identical element (p. 155). This remark is not easy to understand, but if it means, as it seems to do, that a variable varies, or has some dependence upon time and change, it is certainly mistaken. The nature of the variable is the funda- mental problem of mathematical philosophy, and I do not know any satisfactory theory on the subject. But it is quite certain that the variable is a purely logical notion, introducing only such concepts as class, any, some, and logical implication ; to make it depend upon time is to make the mathematical treatment of time itself logically impossible, and to misunderstand the abstractness of Symbolic Logic, in which, though time is absent, the variable -is present throughout. The nature of the variable, in fact, is more akin to that of logical disjunction than to any notion, involv- ing variation or change. 13