Page:Mind (New Series) Volume 8.djvu/530

516 BRUCE MCEWEN Kant has proved the synthetic nature of these " numerical formulae," it would be quite as superfluous to point out that they are involved in all Arithmetical procedure, as it formerly was that the Definitions, Axioms and Principles of Euclid are involved in all Geometry. In both cases Kant has pene- trated to what is obviously fundamental in Mathematics ; it is immediately clear that all mathematical arguments, whether in Geometry or in Arithmetic, do ultimately depend upon an intuitive and synthetic groundwork. And if we admit that the most elementary processes in each branch of Mathematics are strictly synthetic, it is certain that this synthetic character will cling to all judgments that follow them. " For although a synthetic principle can certainly ^e discerned by means of the principle of Contradiction, yet this is possible only when another synthetical proposition precedes, from which it can be deduced." When the syn- thetical proposition that ' two straight lines do not enclose a space ' is given, we can deduce directly from it, that two particular straight lines AC and AB, meeting at A, do not meet in any other point, but this conclusion, though analyti- cally derived from our general principle, is none the less itself purely synthetical. But before Kant could honestly formulate his compre- hensive statement that ' mathematical judgments are one and all synthetic,' it was necessary that his view of the fundamental principles of Mathematics should be absolutely uniform, recognising no exceptions, and in 1781 it was just at this point that Kant failed. In the first edition of the Critique (p. 163) he deliberately styles a large number of Euclid's "Axioms " analytical, and inasmuch as it would be folly to exclude them from the category of mathematical judgments, he was fully justified in postponing the announce- ment of his final opinion. " As regards the quantity of a thing (quantitas) ," he says in the passage referred to, " namely in answering the question, How great? we have at hand various propositions synthetical and immediately certain (indemonstrabilia), but we have in the proper sense of the term no Axioms. For example, the propositions, ' If equals. be added to equals the wholes are equal,' ' If equals be taken from equals the remainders are equal,' are analytical, for I am immediately conscious of the identity of the pro- duction of the one quantity with the production of the other, whereas Axioms must be synthetic propositions a priori." Now this remark applies to no fewer than nine of Euclid's twelve axioms, and were we to judge of all of the axioms de quantitate by the two examples Kant gives, we might infer 0.1