If, then, all scientific judgments are synthetic, and if both rationalism and empiricism failed to account for the manner in which such judgments go beyond what is immediately given to the mind, ought we not to say that the real problem for Kant is to show not merely how synthetic a priori judgments are possible, but how any synthetic judgments are possible? This seems at first sight plausible, but the suggestion must be rejected; for, when Kant asks how a judgment is possible, he is not asking how we come to make it, but how we know that it is valid. Now, if we consider any empirical judgment about the facts of nature, we must recognise that Locke and Hume were right in denying certainty to such judgments. In all general statements about concrete facts we to a certain extent go beyond our evidence. Empirical scientific statements are not theoretically certain. They may, of course, be certain enough for all practical purposes. They are reasonable expectations of what will happen, but reasonable expectation is a very different thing from the certainty of mathematical insight. Now Kant maintained that, while such empirical judgments are not certain, they all imply the certainty of a number of general principles on which they depend. These general principles are the synthetic a priori judgments with which he is especially concerned. When we apply the principles of trigonometry to an engineering problem, we know that our measurements are only approximate, and that the result also will only be approximate; but the possibility of arriving at such approximate results depends on the absolute truth of the trigonometrical principles, and on the assumption that they express not simply the agreement or disagreement of ideas, but hold of the real. When we apply the rules of arithmetic to counting objects, there may be a certain arbitrariness in deciding on our unit. There is no such arbitrariness in the rule. All scientific judgments of causation are only approximately certain, but they all imply the certainty of the principle of causation, and are based on the assumption that such a principle is of universal application. This and the other principles assumed in our empirical judgments are, then, the synthetic judgments with which Kant is concerned. Now, it is of the nature of our empirical knowledge that it is fragmentary and not uniform, that we are concerned with an indefinite number of things whose connections we do not wholly understand, and which we cannot therefore anticipate. Yet we assume that all these objects will obey the rules of arithmetic and geometry, and will all be subject in their changes to the principle of causation. On such assumptions all the sciences of applied mathematics depend. How are they justifiable? That is Kant’s question.