I may now summarise this general discussion. Though Kant in the first edition of the _Critique_ had spoken of the mathematical sciences as based upon the intuition of space and time, he had not, despite his constant tendency to conceive space and time as parallel forms of existence, based any separate mathematical discipline upon time. His definition of number, in the chapter on _Schematism_, had recognised the essentially conceptual character of arithmetic, and had connected it with time only in a quite indirect manner. A passage in the _Prolegomena_ is the one place in all Kant’s writings in which he would seem to assert, though in brief and quite indefinite terms, that arithmetic is related to time as geometry is related to space. No such view of arithmetic is to be found in the second edition of the _Critique_. In the transcendental exposition of time, added in the second edition, only pure mechanics is mentioned. This would seem to indicate that Kant had made the above statement carelessly, without due thought, and that on further reflection he found himself unable to stand by it. The omission is the more significant in that Kant refers to arithmetic in the passages added in the second edition _Introduction_. The teaching of these passages, apart from the asserted necessity of appealing to fingers or points,[526] harmonises with the view so briefly outlined in the _Analytic_. Arithmetic is a conceptual science; though it finds in ordered sequence its intuitional material, it cannot be adequately defined as being the science of time. CONCLUSIONS FROM THE PRECEDING CONCEPTS[527] These _Conclusions_ do not run parallel with the corresponding _Conclusions_ in regard to space. In the first paragraph there are two differences. (_a_) Kant takes account of a view not considered under space, viz. that time is a self-existing substance. He rejects it on a ground which is difficult to reconcile with his recognition of a manifold of intuition as well as a manifold of sense, namely that it would then be something real without being a real object. In A 39 = B 57 and B 70 Kant describes space and time, so conceived, as _unendliche Undinge_. (_b_) Kant introduces into his first _Conclusion_ the argument[528] that only by conceiving time as the form of inner intuition can we justify _a priori_ synthetic judgments in regard to objects.