5 = 12, the
thought of certain units as a group of twelve is no mere repetition of
the thought of them as a group of five added to a group of seven.
Though the same units are referred to, they are regarded differently.
Thus the thought of them as twelve means either that we think of them
as formed by adding one unit to a group of eleven, or that we think of
them as formed by adding two units to a group of ten, and so on. And
the assertion is that the same units, which can be grouped in one way,
can also be grouped in another. Similarly, Kant is right in pointing
out that the geometrical judgement, 'A straight line between two
points is the shortest,' is synthetic, on the ground that the
conception of straightness is purely qualitative,[10] while the
conception of shortest distance implies the thought of quantity.

[10] Straightness means identity of direction.

It should now be an easy matter to understand the problem expressed
by the question, 'How are _a priori_ synthetic judgements possible?'
Its substance may be stated thus. The existence of _a posteriori_
synthetic judgements presents no difficulty. For experience is
equivalent to perception, and, as we suppose, in perception we are
confronted with reality, and apprehend it as it is. If I am asked,
'How do I know that my pen is black or my chair hard?' I answer that
it is because I see or feel it to be so. In such cases, then, when my
assertion is challenged, I appeal to my experience or perception of
the reality to which the assertion relates. My appeal raises no
difficulty because it conforms to the universal belief that if
judgements are to rank as knowledge, they must be made to conform to
the nature of things, and that the conformity is established by appeal
to actual experience of the things. But do _a priori_ synthetic
judgements satisfy this condition? Apparently not. For when I assert
that every straight line is the shortest way between its extremities,
I have not had, and never can have, experience of all possible
straight lines. How then can I be sure that all cases will conform to
my judgement? In fact, how can I anticipate my experience at all? How
can I make an assertion about any individual until I have had actual
experience of it? In an _a priori_ synthetic judgement the mind in
some way, in virtue of its own powers and independently of experience,
makes an assertion to which it claims that reality must conform. Yet
why should reality conform