abitually speaks of space
as a perception. No doubt he considers that his view that
space is only a characteristic of phenomena justifies
the identification of space and the perception of it.
Occasionally, however, he distinguishes them. Thus he
sometimes speaks of the representation of space (e. g.
B. 38-40, M. 23-4); in _Prol._, Sec. 11, he speaks of a pure
perception of space and time; and in B. 40, M. 25, he says
that our representation of space must be perception. But this
language is due to the pressure of the facts, and not to his
general theory; cf. pp. 135-6.

[34] Secs. 6-11.

It begins thus: "Mathematics carries with it thoroughly apodeictic
certainty, that is, absolute necessity, and, therefore, rests on no
empirical grounds, and consequently is a pure product of reason, and,
besides, is thoroughly synthetical. How, then, is it possible for
human reason to accomplish such knowledge entirely _a priori_?... But
we find that all mathematical knowledge has this peculiarity, that it
must represent its conception previously in _perception_, and indeed
_a priori_, consequently in a perception which is not empirical but
pure, and that otherwise it cannot take a single step. Hence its
judgements are always _intuitive_.... This observation on the nature
of mathematics at once gives us a clue to the first and highest
condition of its possibility, viz. that there must underlie it _a pure
perception_ in which it can exhibit or, as we say, _construct_ all its
conceptions in the concrete and yet _a priori_. If we can discover
this pure perception and its possibility, we may thence easily explain
how _a priori_ synthetical propositions in pure mathematics are
possible, and consequently also how the science itself is possible.
For just as empirical perception enables us without difficulty to
enlarge synthetically in experience the conception which we frame of
an object of perception through new predicates which perception itself
offers us, so pure perception also will do the same, only with the
difference that in this case the synthetical judgement will be _a
priori_ certain and apodeictic, while in the former case it will be
only _a posteriori_ and empirically certain; for the latter [i. e. the
empirical perception on which the _a posteriori_ synthetic judgement
is based] contains only that which is to be found in contingent
empirical perception, while the former [i. e. the pur