not be done. Hence he
inferred the far more doubtful proposition that nothing could be known
_a priori_ about the connexion 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 connexion 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
already seen to be inadequate, for two reasons: first, that the validity
of the inductive principle itself cannot be proved by induction;
secondly, that the general propositions of mathematics, such as 'two
and two always make four', can obviously be known with certainty by
consideration of a single instance, and gain nothing by enumeration of
other cases in which they have been found to be true. Thus our knowledge
of the general propositions of mathematics (and the same applies to
logic) must be accounted for otherwise than our (merely probable)
knowledge of empirical generalizations such as 'all men are mortal'.

The problem arises through the fact that such knowledge is general,
whereas all experience is particular. It seems strange that we should
apparently be able to know some truths in advance about particular
things of which we have as yet no experience; but it cannot easily be
doubted that logic and arithmetic will apply to such things. We do not
know who will be the inhabitants of London a hundred years hence; but
we know that any two of them and any other two of them will make four of
them. This apparent power of anticipating facts about things of which
we have no exp