Page:Mind (New Series) Volume 12.djvu/203

 RECENT WORK ON THE PHILOSOPHY OF LEIBNIZ. 189 thought that the formulas for intension and extension were the same, which is only true when addition is everywhere changed into multiplication and vice versa (p. 374). M. Couturat sums up his account by saying that Leibniz possessed almost all the principles of Boole and Schroder, and in some points was more advanced even than Boole ; but he failed to constitute symbolic logic because it cannot be based upon the vague idea of intension (pp. 386-387). There is, no doubt, a certain broad truth in this statement : the Logical Calculus undoubtedly requires a point of view more akin to that of extension than to that of intension. But it would seem that the truth lies somewhere between the two, in a theory not yet developed. This results from the consideration of infinite classes. Take e.g. the proposition " Every prime is an integer ". It is impossible to interpret such a proposition as stating the results of an enumeration, which would be the standpoint of pure extension. And yet it is essentially concerned with the terms that are primes, not, as the intensional view would have us believe, with the concept prime. There appears to be here a logical problem, as yet unsolved and almost unconsidered ; and in any case, the matter is less simple than M. Couturat represents it as being. Leibniz's Geometrical Calculus, which is discussed in chapter ix., is distinctly disappointing. He was not satisfied with analytic geometry, for it is not autonomous, but requires synthetic proofs of its foundations (p. 400). Not Algebra, he says, but a " more sublime analysis " is the true Characteristic of Geometry (p. 388). What he should have invented was Grassmann's Calculus of Extension ; he had at one time the idea of projective Geometry, i.e., of a Geometry using only straight lines, and for this he wanted a "linear analysis " (p. 404, note, and p. 409). He held the view which, in spite of Kant, is now known to be correct that Geometry does not depend upon figures for its proofs, but on intelligible relations (p. 401). He made endeavours to analyse these relations : position, he says, distinguishes objects having no intrinsic distinction, but this applies equally to magnitudes, and he failed to make a philosophic analysis of position (pp. 407-408). The fact is that the above is a mark of all asymmetrical relations whose terms are simple ; but this fact was a contradiction for Leibniz, as for most modern philosophers, owing to the subject- predicate theory of propositions. Leibniz at first endeavoured, in his geometrical calculus, to deal with the two relations of similarity and congruence ; but later, he dealt with congruence only (pp. 411, 417). From congruence alone he obtained definitions of the straight line and plane ; but he was unable to deduce that there are straight lines, or that they are determined by any two of their points (p. 420). He justly remarks: " Imagination, taken from the experience of the senses, does not permit us to imagine more than one intersection of two straight lines; but it is not on this that the science should be