Much of the unsatisfactoriness of Kant’s argument is traceable to his mode of conceiving the “construction”[525] of mathematical concepts. All concepts, he seems to hold, even those of geometry and arithmetic, are abstract class concepts--the concept of triangle representing the properties common to all triangles, and the concept of seven the properties common to all groups that are seven. Mathematical concepts differ, however, from other concepts in that they are capable of _a priori_ construction, that is, of having their objects represented in pure intuition. Now this is an extremely unfortunate mode of statement. It implies that mathematical concepts have a dual mode of existence, first as abstracted, and secondly as constructed. Such a position is not tenable. The concept of seven, in its primary form, is not abstracted from a variety of particular groups of seven; it is already involved in the apprehension of each of them as being seven. Nor is it a concept that is itself constructed. It may perhaps be described as being the representation of something constructed; but that something is not itself. It represents the process or method generative of the complex for which it stands. Thus Kant’s distinction between the intuitive nature of mathematical knowledge and the merely discursive character of conceptual knowledge is at once inspired by the very important distinction between the product of construction and the product of abstraction, and yet at the same time is also obscured by the quite inadequate manner in which that latter distinction has been formulated. Kant has again adhered to the older logic even in the very act of revising its conclusions; and in so doing he has sacrificed the Critical doctrines of the _Analytic_ to the pre-Critical teaching of the _Dissertation_ and _Aesthetic_. _Mathematical concepts are of the same general type as the categories; their primary function is not to clarify intuitions, but to make them possible._ They are derivable from intuition only in so far as they have contributed to its constitution. If intuition contains factors additional to the concepts through which it is interpreted, these factors must remain outside the realm of mathematical science, until such time as conceptual analysis has proved itself capable of further extension.