KANT'S THEORY OF MATHEMATICS
NOTES AND DISCUSSIONS. 577 MR. MONCK'S argument is clear and to the point. But he will see, I think, on further consideration, that, if it is once granted to me that the equation between a number and the number next below it + 1 is the definition of the higher number and therefore an analytical proposition, I can deduce 12 = 7 + 5, without any more "counting" than is involved in the substitution of the right-hand term of such an equation for its left-hand term. This deduction, as I conceive it,
... , as I conceive it, diverges from that which Mr. Monck has given when the step 12 = 10 + 1 + 1 is reached: at this point I take the definition of 2 = 1 + 1, and substituting 2 for 1 + 1 in the equation first given, I get 12 = 10 + 2 : then substituting 9 + 1 for 101 get 12 = 9 + 1+2, and using the definition of 3=1 + 2 asl used the definition of 2, I get 12 = 9 + 3: and so on till by precisely similar steps I arrive at 12 = 7 + 5. Now I do not suppose that a Kantian will contend that the conversion of such a proposition as 12=11 + 1 isa procedure which requires us to go beyond our concepts to intuitions : and Kant has expressly declared the analytical character of the principle that the " sums of equals are equal". Hence I do not see how my conclusion-that 12 = 7 + o is deducible from propositions which Kant must admit to be analytical-can be disputed except by disputing the analytical character of such propositions as 12 = 11 + 1.