deny anything that could be known a priori . “A bald man is not bald” would assert and deny baldness of the same man, and would therefore contradict itself. Thus according to the philosophers before Kant, the law of contradiction, which asserts that nothing can at the same time have and not have a certain property, sufficed to establish the truth of all a priori knowledge.
Hume (1711–76), who preceded Kant, accepting the usual view as to what makes knowledge a priori , discovered that, in many cases which had previously been supposed analytic, and notably in the case of cause and effect, the connection was really synthetic. Before Hume, rationalists at least had supposed that the effect could be logically deduced from the cause, if only we had sufficient knowledge. Hume argued—correctly, as would now be generally admitted—that this could not be done. Hence he inferred the far more doubtful proposition that nothing could be known a priori about the connection of cause and effect. Kant, who had been educated in the rationalist tradition, was much perturbed by Hume’s scepticism, and endeavoured to find an answer to it. He perceived that not only the connection of cause and effect, but all the propositions of arithmetic and geometry, are “synthetic,” i.e. not analytic: in all these propositions, no analysis of the subject will reveal the predicate. His stock instance was the proposition 7 + 5 = 12. He pointed out, quite truly, that 7 and 5 have to be put together to give 12: the idea of 12 is not contained in them, nor even in the idea of adding them together. Thus he was led to the conclusion that all pure mathematics, though a priori , is synthetic; and this conclusion raised a new problem of which he endeavoured to find the solution.
The question which Kant put at the beginning of his philosophy, namely “How is pure mathematics possible?” is an interesting and difficult one, to which every philosophy which is not purely sceptical must find some answer. The answer of the pure empiricists, that our mathematical knowledge is derived by induction from particular instances, we have