Kant's philosophy started from the question: How are synthetic a priori judgments possible? Now we must first of all make a distinction. Kant is concerned with knowledge, not with mere belief. There is no philosophical problem in the fact that a man can have a belief which is synthetic and not based on experience—e.g. that this time the horse on which he has put his money will win. The philosophical problem arises only if there is a class of synthetic a priori beliefs which is always true. Kant considered the propositions of pure mathematics to be of this kind; but in this he was misled by the common opinion of his time, to the effect that geometry, though a branch of pure mathematics, gave information about actual space. Owing to non-Euclidean geometry, particularly as applied in the theory of relativity, we must now distinguish sharply between the geometry applicable to actual space, which is an empirical study forming part of physics, and the[Pg 175] geometry of pure mathematics, which gives no information as to actual space. Consequently this instance of synthetic a priori knowledge, upon which Kant relied, is no longer available. Other kinds have been supposed to exist—for example, ethical knowledge, and the law of causality; but it is not necessary for our purposes to decide whether these kinds really exist or not. So far as physics is concerned, we may assume that all real knowledge is either dependent (at least in part) upon perception, or analytic in the sense in which pure mathematics is analytic. The Kantian synthetic a priori knowledge, whether it exists or not, seems not to be found in physics—unless, indeed, the principle of induction were to count as such. But the principle of induction, as we have already seen, has its origin in physiology, and this suggests a quite different treatment of a priori beliefs from that of Kant. Whether there is a priori knowledge or not, there undoubtedly are, in a certain sense, a priori beliefs. We have reflexes which we intellectualize into beliefs; we blink, and this leads us to the belief that an object touching the eye will hurt it. We may have this belief before we have experience of its truth; if so, it is, in a sense, synthetic a priori knowledge—i.e. it is a belief, not based upon experience, in a true synthetic proposition. Our belief in induction is essentially analogous. But such beliefs, even when true, hardly deserve to be called knowledge, since they are not all true, and therefore all require verification before they ought to be regarded as certain. These beliefs have been useful in generating science, since they supplied hypotheses which were largely true; but they need not survive untested in modern science.