concerned. In this
sense human thought is just as sovereign as not--sovereign, and its
possibility of knowledge just as unlimited as limited. It is sovereign
and unlimited as regards its nature, its significance, its
possibilities, its historical end, it is not sovereign and limited
with respect to individual expression and its actuality at any
particular time.

It is just the same with eternal truths. If mankind only operated with
eternal truths and with thought which possessed a sovereign
significance and unlimited claims to truth, mankind would have arrived
at a point where the eternity of thought becomes realised in actuality
and possibility. Thus the famous miracle of the enumerated innumerable
would be realised.

But what about those truths which are so well established that to
doubt them is to be, as it were, crazy? That twice two is four, that
the three angles of a triangle are equal to two right angles, that
Paris is in France, that a man will die of hunger if he does not
receive food, etc.? Do we not perceive then that there are eternal
truths, final truths of last instance? Quite so. We can divide the
entire field of knowledge in the old-fashioned way into three great
divisions. The first includes all the sciences which are concerned
with inanimate nature and which can be treated mathematically, more or
less--mathematics, astronomy, mechanics, physics and chemistry. If one
like to use big words to express simple things, it may be said that
certain results of these sciences are eternal truths, final truths of
last instance, whence they are called the exact sciences. But all the
results are by no means of this character. With the introduction of
variable quantities and the extension of the variability to the
infinitely small and the infinitely large, mathematics, otherwise
erect, meets with its fall, it has eaten of the apple of knowledge and
there has been opened up to it the path of limitless progress as well
as that of error. The virgin condition of absolute purity, the
undisturbable certainty of all mathematics has vanished forever, a
period of controversy has intervened, and we have now arrived at the
state of affairs in which most people carry on the operations of
multiplication and division not because they really understand what
they are engaged in, but from mere belief because the operation has so
far always given correct results. Astronomy and mechanics, physics and
chemistry are in a still more co