;
but, despite the complete similarity, the one hand cannot be exactly
super-imposed on the other; the glove of the one cannot be worn on the
other. This difference in direction, which has significance only when
viewed from a definite point, and the impossibility mentioned of a
congruence between an object (right hand) and its reflected image (left
hand) can be understood only by intuition; they must be seen and felt, and
cannot be made clear through concepts, and, consequently, can never be
explained to a being which lacks the intuition of space.

In the "transcendental" exposition of space and time Kant follows this
"metaphysical" exposition, which had to prove their non-empirical, and
non-discursive, hence their _a priori_ and intuitive, character, with
the proof that only such an explanation of space and time could make it
conceivable how synthetic cognitions _a priori_ can arise from them. The
principles of mathematics are of this kind. The synthetic character of
geometrical truths is explained by the intuitive nature of space, their
apodictic character by its apriority, and their objective reality or
applicability to empirical objects by the fact that space is the condition
of (external) perception. The like is true of arithmetic and time.

If space were a mere concept, no proposition could be derived from it which
should go beyond the concept and extend our knowledge of its properties.
The possibility of such extension or synthesis in mathematics depends on
the fact that spatial concepts can always be presented or "constructed" in
intuition. The geometrical axiom that in the triangle the sum of two sides
is greater than the third is derived from intuition, by describing the
triangle in imagination or, actually, on the board. Here the object is
given through the cognition and not before it.--If space and time were
empirical representations the knowledge obtained from them would lack
necessity, which, as a matter of fact, it possesses in a marked degree.
While experience teaches us only that something is thus or so, and not that
it could not be otherwise, the axioms, (space has only three dimensions,
time only one; only one straight line is possible between two points),
nay, all the propositions of mathematics are strictly universal and
apodictically certain: we are entirely relieved from the necessity of
measuring all triangles in the world in order to find out whether the sum
of their angles is equal to two right