Page:Mind (New Series) Volume 8.djvu/534

520 BRUCE MCEWEN : may be effected either analytically or synthetically, that is, either immediately or mediately. Now when in Mathematics we ask the question, What do we really think in a conception ? we must add an intuition to that conception, and apart from such an intuition no real mathematical judgment arises. Kant's meaning in the paragraph, from which we have quoted, I take to be perfectly plain. He has already pointed out that all mathematical judgments are synthetical with the exception of " certain principles which serve as links in the chain of method ". But on a further examination of these apparently analytical principles, Kant finds that they are admitted in Mathematics only conditionally ; they are not accepted simply because Philosophy claims to guarantee their validity from pure conceptions, they must from the first have their truth made self-evident by means of an intuitive representation of a particular sensible and concrete instance, just as is necessary in geometry for the acceptance of the .axiom, that two straight lines cannot enclose a space. In- side Mathematics these axioms are synthetic and their .analytic appearance is a mere delusion brought about by the fact that the words, in which they are expressed, have another distinct and non-mathematical interpretation that really is analytical. And thus in 1783 the universal affirma- tive proposition is possible for the first time : " All mathe- matical judgments, without exception, are synthetical," for the former supposed exception is now finally removed. At this point we intend to discuss shortly a few of the criticisms that have been passed upon Kant's view of ma- thematics, not so much because we wish to defend Kant's doctrine in a way which must necessarily lead us beyond the study of Kant's own writings, but because so many of these criticisms seem to imply a different interpretation from ours of the passages which discuss axioms de quantitate. We have neither time nor inclination to treat of those argu- ments which object to Kant's doctrine of the synthetic nature of mathematical definitions, and by producing particular instances seek to cast doubt upon his whole theory. The statement, that the conception of a triangle contains the mark of having three sides, is a perfect type of them, and I confess it is difficult to regard such arguments seriously. I wonder how often in his works Kant lays down the maxim that Mathematics proceeds by construction of conceptions, always assuring itself by intuition of the possibility of the object to which they correspond. Without hoping ever to convince any one, to whom such an argument as that men- tioned above seems conclusive, I would merely suggest for