Page:Mind (New Series) Volume 8.djvu/525

 KANT'S PEOOF OF THE PROPOSITION, ETC. 511 favourite phrase is contrasted with Kant's more general statement, that Mathematical judgments are always syn- thetical, it has again and again been pointed out that this latter expresses more than was really necessary for Kant's purpose. It is admitted that the three moments of all mathematical procedure consist, as Kant has said, of Defi- nition, Axiom and Demonstration, yet the synthetic nature of Mathematics generally is sufficiently proved when it is shown that the Definitions are synthetic. But with any such partial result Kant is not satisfied : he extends his inquiries at once to every part of mathematical processes. Here, unfortunately, Kant's opinions were slow to crystallise into their final form, and the result is that there is always an appearance of wavering about his statements. At times he seems to admit analytical principles into mathematical method, again to deny them, sometimes to give expression to both views almost in the same sentence. The difficulty lies in the fact that, in Algebra and Geometry alike, the mathematician speedily passes away from an actual representation of real things to a consideration of letters, figures, or words, which, for the time being, are treated as mere signs. Take for example an ordinary mathematical syllogism x = y and w = v .'.x--w = y--v and x + w > y. The letters, x, y, w, v, here used, may be mere algebraic symbols, or on the other hand in pure geometry they may stand for lines, or plane angles, and so on, but in any case certain fundamental propositions (Grundsatze) are involved in manipulating them, namely two well-known Axioms of Euclid : If equals be added to equals the wholes are equal ; and : The whole is greater than its part. Now these " axioms " are quite obviously of a different class from such axioms as : Two straight lines cannot enclose a space ; which is Kant's best example of a purely synthetic and intuitive proposition. This latter axiom is applicable only in geometry, while the two former seem to extend to Algebra, and even to the non-mathematical procedure of human Reason. Are such axioms then synthetic ? Are they not rather analytic ? It is possible, says Kant in one passage, " to prove them in strict philosophic fashion out of mere conceptions " ; it can never be " the proper business of Mathematics to prove that the whole is greater than the part ". And yet we may quote a passage from the Transcendental ^Esthetic which runs : " Every geometrical axiom is derived from intuition