§ 15. If now with our conviction that perception is the primary source of
all evidence, and that only direct or indirect connection with it is
absolute truth; and further, that the shortest way to this is always the
surest, as every interposition of concepts means exposure to many
deceptions; if, I say, we now turn with this conviction to mathematics, as
it was established as a science by Euclid, and has remained as a whole to
our own day, we cannot help regarding the method it adopts, as strange and
indeed perverted. We ask that every logical proof shall be traced back to
an origin in perception; but mathematics, on the contrary, is at great
pains deliberately to throw away the evidence of perception which is
peculiar to it, and always at hand, that it may substitute for it a
logical demonstration. This must seem to us like the action of a man who
cuts off his legs in order to go on crutches, or like that of the prince
in the “_Triumph der Empfindsamkeit_” who flees from the beautiful reality
of nature, to delight in a stage scene that imitates it. I must here refer
to what I have said in the sixth chapter of the essay on the principle of
sufficient reason, and take for granted that it is fresh and present in
the memory of the reader; so that I may link my observations on to it
without explaining again the difference between the mere ground of
knowledge of a mathematical truth, which can be given logically, and the
ground of being, which is the immediate connection of the parts of space
and time, known only in perception. It is only insight into the ground of
being that secures satisfaction and thorough knowledge. The mere ground of
knowledge must always remain superficial; it can afford us indeed rational
knowledge _that_ a thing is as it is, but it cannot tell _why_ it is so.
Euclid chose the latter way to the obvious detriment of the science. For
just at the beginning, for example, when he ought to show once for all how
in a triangle the angles and sides reciprocally determine each other, and
stand to each other in the relation of reason and consequent, in
accordance with the form which the principle of sufficient reason has in
pure space, and which there, as in every other sphere, always affords the
necessity that a thing is as it is, because something quite different from
it, is as it is; instead of in this way giving a thorough insight into the
nature of the triangle, he sets up certain disconnected arbitrarily chosen
propositions concerning the triangle, and gives a logical ground of
knowledge of them, through a laborious logical demonstration, based upon
the principle of contradiction. Instead of an exhaustive knowledge of
these space-relations we therefore receive merely certain results of them,
imparted to us at pleasure, and in fact we are very much in the position
of a man to whom the different effects of an ingenious machine are shown,
but from whom its inner connection and construction are withheld. We are