facts, but even start from them, while in a synthetic procedure
they must strictly be derived in abstracts from concepts.

     But, in order to rise from these actual and at the same time
well-grounded pure cognitions a priori to such a possible
cognition of the same as we are seeking, viz., to metaphysics as
a science, we must comprehend that which occasions it, I mean the
mere natural, though in spite of its truth not unsuspected,
cognition a priori which lies at the bottom of that science, the
elaboration of which without any critical investigation of its
possibility is commonly called metaphysics. In a word, we must
comprehend the natural conditions of such a science as a part of
our inquiry, and thus the transcendental problem will be
gradually answered by a division into four questions:

     1. How is pure mathematics possible?
     2. How is pure natural science possible?
     3. How is metaphysics in general possible?
     4. How is metaphysics as a science possible?
   
     It may be seen that the solution of these problems, though
chiefly designed to exhibit the essential matter of the Critique,
has yet something peculiar, which for itself alone deserves
attention. This is the search for the sources of given sciences
in reason itself, so that its faculty of knowing something a
priori may by its own deeds be investigated and measured. By this
procedure these sciences gain, if not with regard to their
contents, yet as to their proper use, and while they throw light
on the higher question concerning their common origin, they give,
at the same time, an occasion better to explain their own nature.

                             * * * *

            FIRST PART OF THE TRANSCENDENTAL PROBLEM:
                HOW IS PURE MATHEMATICS POSSIBLE?

     Here is a great and established branch of knowledge,
encompassing even now a wonderfully large domain and promising an
unlimited extension in the future. Yet it carries with it
thoroughly apodictical certainty, i.e., absolute necessity, which
therefore rests upon no empirical grounds. Consequently it is a
pure product of reason, and moreover is thoroughly synthetical.
[Here the question arises:] "How then is it possible for human
reason to produce a cognition of this nature entirely a priori?"

     Does not this faculty [which produces mathematics], as it
neither is nor can be based upon experience, presuppose some
ground of cognition a priori, which lies deeply hidden, b,.--,
which might reveal itself by these its effects, if their first
beginnings were but diligently ferreted out?

     Sect. 7. But we find that all mathematical cognition has
this peculiarity: it must first exhibit its concept in a visual
form [Anschauung] and indeed a priori, therefore in a visual form
which is not empirical, but pure. Without this mathematics cannot
take a single step; hence its judgments are always visual, viz.,
"Intuitive"; whereas philosophy must be satisfied with discursive
judgments from mere concepts, and though it may illustrate its
doctrines through a visual figure, can never derive them from it.
This observation on the nature of mathematics gives us a clue to
the first and highest condition of its possibility, which is,
that some non-sensuous visualization [called pure intuition, or
reine Anschauung] must form its basis, in which all its concepts
can be exhibited or constructed, in concrete and yet a priori. If
we can find out this pure intuition and its possibility, we may
thence easily explain how synthetical propositions a priori are
possible in pure mathematics, and consequently how this science
itself is possible. Empirical intuition [viz., sense-perception]
enables us without difficulty to enlarge the concept which we
frame of an object of intuition [or sense-perception], by new
predicates, which intuition [i.e., sense-perception] itself
presents synthetically in experience. Pure intuition [viz., the
visualization of forms in our imagination, from which every thing
sensual, i.e., every thought of material qualities, is excluded]
does so likewise, only with this difference, that in the latter
case the synthetical judgment is a priori certain and
apodictical, in the former, only a posteriori and empirically
certain; because this latter contains only that which occurs in
contingent empirical intuition, but the former, that which must
necessarily be discovered in pure intuition. He.-e intuition,
being an intuition a priori, is before all experience, viz.,
before any perception of particular objects, inseparably
conjoined with its concept.

     Sect. 8. But with this step our perplexity seems rather to
increase than to lessen. For the question now is, "How is it
possible to intuit [in a visual form] anything a priori" An
intuition [viz., a visual sense perception] is such a
representation as immediately depends upon the presence of the
object. Hence it seems impossible to intuit from the outset a
priori, because intuition would in that event take place without
either a former or a present object to refer to, and by
consequence could not be intuition. Concepts indeed are such,
that we can easily form some of them a priori, viz., such as
contain nothing but the thought of an object in general; and we
need not find ourselves in an immediate relation to the object.
Take, for instance, the concepts of Quantity, of Cause, etc. But
even these require, in order to make them understood, a certain
concrete use-that is, an application to some sense-experience
[Anschauung], by which an object of them is given us. But how can