Page:Mind (New Series) Volume 8.djvu/535

 KANT'S PROOF OF THE PROPOSITION, ETC. 521 his consideration a true example of a proposition, which is analytical in Mathematics, namely, " A Duangle is a plane figure bounded by two straight lines " ; and I would ask what part of Mathematics this definition occurs in ? But in those criticisms of Kant's doctrine which discuss the nature of axioms de quant itate we have to deal with objections of quite another class. These arguments have their weight considerably increased by the fact that Kant's own words can so often be quoted in such a way as apparently to support them. Indeed we might safely say that it is universally considered to be Kant's final opinion of axioms de quantitate that they are analytic, and of course it is then easy to deny the statement that mathematical judgments are always syn- thetical. But such a view can only be held by those who are prepared either to ignore or explain away the glaring contradiction we have pointed out. One well-known com- mentator 1 has published a theory that the sentences in which Kant in our view of the matter declares the mathe- matically synthetic nature of axioms de quantitate have no immediate reference to those axioms. Therefore there must be a displacement of the order of this paragraph due either to a printer's error or to the author's negligence. It is supposed that Kant agrees to consider axioms de quantitate as analytic, and that the sentences which seem to contradict this view were originally penned in some quite different connexion from that in which they now stand. We have already tried to point out that such an artificial explanation is unnecessary and would indeed leave the opening statement of the general nature of mathematical judgments ill-founded. Moreover a displacement of a paragraph, so serious as that suggested here, might perhaps have been possible on one occasion, but that an author should have copied the same section literally into a work four years later, without correct- ing its error, seems to me an untenable hypothesis. The argument must be taken as it stands and we venture to claim that our explanation of it is the only one possible. Of the many particular instances with which critics seek to confirm or illustrate their interpretation of Kant's views on those axioms we shall take only one example and endea- vour to show, on mathematical grounds, that it is a fallacy. It is said that the Law of Identity, a thing is equal to itself, is largely used in Geometrical proofs after the fashion of an 1 The objection here is not to Prof. Vaihinger's theory of " a leaf- displacement in the Prolegomena " but to the use made of the theory in explaining this passage in his " Commentar," voL i.