therefore allows, as at least possible, that upon clarification of our concepts space may be discovered to be radically different from what it at first sight appears to be. In any case, the perfecting of the concepts must have some effect upon their object. But even--as the modern geometer further maintains--should our space be definitely proved, upon analytic and empirical investigation, to be Euclidean in character, other possibilities will still remain open for speculative thought. For though the nature of our intuitional data may constrain us to interpret them through one set of concepts rather than through another, the competing sets of alternative concepts will represent genuine possibilities beyond what the actual is found to embody. Thus the defect of Kant’s teaching, in regard to space, as judged in the light of the later teaching of geometrical science, is closely bound up with his untenable isolation of the _a priori_ of sensibility from the _a priori_ of understanding.[484] Space, being thus viewed as independent of thought, has to be regarded as limiting and restricting thought by the unalterable nature of its initial presentation. And unfortunately this is a position which Kant continued to hold, despite his increasing recognition of the part which concepts must play in the various mathematical sciences. In the deduction of the first edition we find him stating that synthesis of apprehension is necessary to all representation of space and time.[485] He further recognises that all arithmetical processes are syntheses _according to concepts_.[486] And in the _Prolegomena_[487] there occurs the following significant passage. “Do these laws of nature lie in space, and does the understanding learn them by merely endeavouring to find out the fruitful meaning that lies in space; or do they inhere in the understanding and in the way in which it determines space according to the conditions of the synthetical unity towards which its concepts are all directed? Space is something so uniform and as to all particular properties so indeterminate, that we should certainly not seek a store of laws of nature in it. That which determines space to the form of a circle or to the figures of a cone or a sphere, is, on the contrary, the understanding, so far as it contains the ground of the unity of these constructions. The mere universal form of intuition, called space, must therefore be the substratum of all intuitions determinable to particular objects, and in it, of course, the condition of the possibility and of the variety of these intuitions lies. But the unity of the objects is solely determined by the understanding, and indeed in accordance with conditions which are proper to the nature of the understanding....”