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514 BRUCE MCEWEN: just as Kant formerly felt unable to characterise all the procedure of Mathematics as synthetic and intuitive, so now he cannot yet affirm that it is entirely concerned with objects of the sense-faculty. " Pure Mathematics," he says, " con- siders Space in geometry, and Time in pure Mechanics. To these we must add a certain conception, namely the conception of Number, dealt with by Arithmetic. This conception, in itself indeed, is of an intellectual nature, but nevertheless its realisation in concrete requires the aid of the notions, Space and Time (in adding together quantities in succession and combining them)." And so it comes about that, while Kant can announce his result : " Therefore pure Mathematics expounds the form of all our sense-knowledge and is the organon of every kind of intuitive and distinct cognition" ; he is still forced to except Arithmetical proce- dure, a most important exception from a general treatment of Mathematics. To be sure, in "Arithmetic" we have " cognitio symbolica " if we only use that phrase vaguely enough, and we might therefore suppose it was something akin to the symbolical knowledge we have of objects of the intellect in Philosophy. Now after 1770, if Kant intended to make his treatment of "Arithmetic " consistent, two courses were open to him. He might boldly deny that it was in any sense a part of Mathematics, and no doubt the consequences of this denial would, in Kant's time, have been easily accepted by many philosophers. He might affirm that the principles of Arith- metic were purely analytic, that it was concerned only with objects of the intellect and therefore proceeded not by intuition but only by analysing universal concepts, in fact, by purely symbolic cognition. But by proclaiming such a result Kant would at once have ranged himself on the side of the philo- sophic defenders of the Cartesian ' mathematical analysis,' for the central assumption of their theory of mathematics lies in some distinction between Algebraical and Geometrical method, such as that which would have been directly implied by Kant's adopting the course we have suggested. Or if Kant hesitated to regard " Arithmetic " as a branch of Philosophy and wished rather to bring it into strict accord with pure Mathematics (Mathesis Purd), which he formerly spoke of as limited to Geometry and Mechanics, it was necessary for him to prove that the procedure of "Arithmetic " was really non-philosophical. Now Kant chose this latter alternative, and accordingly, in 1781, we can easily trace the presence of a distinct improvement in his doctrine. In the paragraph of the Critique on the Axioms of Intuition