Page:Mind (New Series) Volume 8.djvu/532

 518 BEUCE MCEWEN : time in the Prolegomena and was literally transferred from it to the Critique, Kant seems to repeat his former doctrine. " Some few fundamental principles," he says, " which are preposited by geometricians, are indeed really analytic and depend upon the law of contradiction, but they serve, after the fashion of identical propositions, only as links in the chain of method, and not as principles, for example a = a, the whole is equal to itself; or a + b >a, the whole is greater than its part." Here the old examples are dropped and of the two substituted the latter alone is a recognised axiom of Euclid. There certainly is an axiom which runs, Things which are equal to the same thing are equal to one another; but I know of no geometrician who ever thought it necessary to preposit the law of Identity a = a to his works. But let that pass and let us understand Kant to mean that two analytical propositions are actually used by mathematicians. Then read the next sentence, "And yet even these same propositions, although they are valid according to pure con- ceptions, are admitted in Mathematics only for the reason that they can be presented in intuition ". Now these words mean, as plainly as words can, that in Mathematics these same principles are synthetic, which in the previous sentence were called analytic. I do not believe that Kant could quote the law of contradiction in one sentence and then proceed to break it with his next; some explanation of the disagree- ment must be found, and I think a sufficient explanation has already been shadowed forth in our own discussion of axioms de qiiantitate. The general statement of any of these "axioms " may be analytic, just because when formulated most gener- ally it is intended to be applicable to all parts of experience, but in Mathematics it has its application confined to questions of mathematical magnitude, which, as we have said, always depend ultimately upon direct intuition. The logical or analytic proposition, the whole is greater than its part, may have a vast variety of different meanings under different circumstances, but in Mathematics it has a meaning it can have nowhere else. If in Geometry we wish to compare any two quantities as to their relative magnitude, they must be superposed and their equality or inequalit}' intuitively perceived; before we can say that one quantity is greater than another, this same intuition must have preceded our judgment, that is to say, we must regard the two as super- posed one upon the other. Thus it comes about that the proposition, a + b >a, is " admitted in Mathematics only for this reason that it can be presented in intuition ". Moreover the general analytical expression of this axiom