Opening the text
A Commentary to Kant's 'Critique of Pure Reason'

Kant · en
“The proposition--the angles of a triangle are together equal to
     two right angles--Kant regards as synthetic. It is indeed deduced
     from the axiom of parallels (with the aid of auxiliary lines), and
     to that extent is understood in accordance with the principle of
     contradiction.... The angles in the triangle constitute a special
     case of the angles in the parallel lines which are intersected by
     other lines. The principle of contradiction thus serves as vehicle
     in the deduction, because once the identity of A and A´ is
     recognised, the predicate _b_, which belongs to A, must also be
     ascribed to A´. But the proposition is not for that reason itself
     analytic in the Kantian sense. In the analytic proposition the
     predicate is derived from the analysis of the subject concept. But
     that does not happen in this case. The synthetic proposition can
     never be derived _in and by itself_ from the principle of
     contradiction; ... but only with the aid of that principle _from
     other propositions_. Besides, in this deduction intuition must
     always be resorted to; and that makes an essential difference.
     Without it the identity of A and A´ cannot become known.”

=Pure mathematics.=[268]--“Pure,” as thus currently used, is opposed only
to applied, not to empirical. Kant here arbitrarily reads the latter
opposition into it. Under this guise he begs the point in dispute.

7 + 5 = 12.[269]--Though 7 + 5 = 12 expresses an identity or equality,
it is an equality of the _objects_ or _magnitudes_, 7 + 5 and 12, not of
the concepts through which we think them.[270] Analysis of the concepts
can never reveal this equality. Only by constructing the concepts in
intuition can it be recognised by the mind. This example has been
already cited in the first edition.[271] It is further elaborated in the
_Prolegomena_, § 2 _c_, and is here transcribed. Kant’s mode of stating
his position is somewhat uncertain. He alternates between “the
representation of 7 and 5,” “the representation of the combination of 7
and 5,”[272] and “the concepts 7 and 5.”[273] His view would seem to be
that there are _three_ concepts involved. For the concept of 7 we must
substitute the intuition of 7 points, for the concept of 5 the intuition
of 5 points, and for the concept of their sum the intuitive operation of
addition.