Paulsen is justified in maintaining that Kant, in both editions of the _Critique_, recognises the validity of mathematics and pure natural science. The fact of their validity is less explicitly dwelt upon in the first edition, but is none the less taken for granted. The sections transferred from the _Prolegomena_ to the _Introduction_ of the second edition make no essential change, except merely in the emphasis with which Kant’s belief in the existence of valid _a priori_ synthetic judgments is insisted upon. As has already been stated, only by virtue of this initial assumption is Kant in position to maintain that there is an alternative to the strict synthetic method. The _problem_ from which he starts is common to both methods, and for that reason the formulation used in the _Prolegomena_ can also be employed in the _Introduction_ to the _Critique_. Only in their manner of solving the problem need they differ.[180] Kant’s Critical problem first begins with this presupposition of validity, and does not exist save through it.[181] He does not first seek to discover whether such judgments are valid, and then to explain them. He accepts them as valid, but develops a method of argument which suffices for proof as well as for explanation. The argument being directed to both points simultaneously, and establishing both with equal cogency, it may legitimately be interpreted in either way, merely as explanation, or also as proof. Kant does not profess or attempt to keep exclusively to any one line of statement. Against the dogmatists he insists upon the necessity of _explaining_ the validity of _a priori_ synthetic judgments, against the sceptics upon the possibility of _proving_ their validity. And constantly he uses ambiguous terms, such as ‘justification’ (_Rechtfertigung_), ‘possibility,’ that may indifferently be read in either sense. But though the fundamental demand which characterises the synthetic method in its distinction from the analytic thus falls into the background, and is only occasionally insisted upon, it is none the less fulfilled. So far as regards the main argument of the _Critique_ in either edition, the validity of synthetic _a priori_ judgments is not required as a premiss. It is itself independently proved.