by the empirical philosophers, who maintained that experience was
as much the source of our knowledge of arithmetic as of our knowledge of
geography. They maintained that by the repeated experience of seeing two
things and two other things, and finding that altogether they made four
things, we were led by induction to the conclusion that two things
and two other things would _always_ make four things altogether. If,
however, this were the source of our knowledge that two and two are
four, we should proceed differently, in persuading ourselves of its
truth, from the way in which we do actually proceed. In fact, a certain
number of instances are needed to make us think of two abstractly,
rather than of two coins or two books or two people, or two of any other
specified kind. But as soon as we are able to divest our thoughts of
irrelevant particularity, we become able to see the general principle
that two and two are four; any one instance is seen to be _typical_, and
the examination of other instances becomes unnecessary.(1)

(1) Cf. A. N. Whitehead, _Introduction to Mathematics_ (Home University
Library).

The same thing is exemplified in geometry. If we want to prove some
property of _all_ triangles, we draw some one triangle and reason about
it; but we can avoid making use of any property which it does not share
with all other triangles, and thus, from our particular case, we obtain
a general result. We do not, in fact, feel our certainty that two and
two are four increased by fresh instances, because, as soon as we have
seen the truth of this proposition, our certainty becomes so great as
to be incapable of growing greater. Moreover, we feel some quality of
necessity about the proposition 'two and two are four', which is
absent from even the best attested empirical generalizations. Such
generalizations always remain mere facts: we feel that there might be a
world in which they were false, though in the actual world they happen
to be true. In any possible world, on the contrary, we feel that two
and two would be four: this is not a mere fact, but a necessity to which
everything actual and possible must conform.

The case may be made clearer by considering a genuinely-empirical
generalization, such as 'All men are mortal.' It is plain that we
believe this proposition, in the first place, because there is no known
instance of men living beyond a certain age, and in the second place
because there seem to be physiologic