Page:Mind (New Series) Volume 8.djvu/526

 512 BRUCE MCEWEN : a priori, and with apodeictic certainty and never from general conceptions". At first sight there appears to be considerable confusion here, and many of Kant's opponents have promptly taken advantage of it to deny his theory, that mathematical judgments are one and all synthetical. Still the fact remains that in Kant's earlier view a synthetic nature is claimed only for the Definitions of Mathematics : the definitions are syn- thetic, but the remainder of mathematical procedure Kant still considers to be in great part analytic. " In Arithmetic," he says, " we treat not the things them- selves but their signs which indicate the magnitude of the things and their relations as greater and less, etc. Afterwards we work with these signs according to easy and sure rules by transposition, addition or subtraction, and different sorts of operations, so that the things indicated by the signs are in the meantime completely dropped out of consideration, until finally in the conclusion the meaning of the symbolic result is deciphered." By Arithmetic Kant means " both the general doctrine of indeterminate magnitudes," or Al- gebra, and the " particular doctrine of numbers, where the relation of the magnitudes to unity is strictly determined," Arithmetic in the ordinary narrow sense. The view ex- pressed in the foregoing quotation is that, so long as we work with signs, we proceed in abstract and analytically without being assured that any real object corresponds to our signs in concrete. But in Mathematics "the meaning of the signs must be sure " at every point, and therefore a synthetic and intuitive process must precede our using signs ; in fact when we say, Let x or let the figure 3 represent the objects, this process is always subject to the condition that it be " possible for these signs actually to represent the things " and that " one should clearly understand what meaning he assigns to them". Otherwise the symbolical result we reach can never be properly deciphered, indeed may be perfectly absurd. Every schoolboy knows how clearly he must lay down the meaning of his symbols be- fore he begins to use them, for he has already experienced, if at least he is conscientious, many an instance of the complete bewilderment which ensues, when after a long exercise in Arithmetic he obtains an answer in figures, say 19, but knows not whether it is 19 men, 19 shillings, 19 eggs, or 19 of anything else real or imaginary. Suppose an answer so obtained to be numerically correct, then from his earlier point of view (in 1763) Kant would say that the analytical process of sign-manipulation was correct, the synthetic and intuitive adoption of signs instead of things equally cor-