PROOF.
Granted that the world has no beginning in time; up to every given
moment of time, an eternity must have elapsed, and therewith passed
away an infinite series of successive conditions or states of things
in the world. Now the infinity of a series consists in the fact that
it never can be completed by means of a successive
synthesis. It
follows that an infinite series already elapsed is impossible and
that, consequently, a beginning of the world is a necessary
condition of its existence. And this was the first thing to be proved.
As regards the second, let us take the opposite for granted. In this
case, the world must be an infinite given total of coexistent
things. Now we cannot cogitate the dimensions of a quantity, which
is not given within certain limits of an intuition,* in any other
way than by means of the synthesis* of its parts, and the total of such
a quantity only by means of a completed synthesis, or the repeated
addition of unity to itself. Accordingly, to cogitate the world, which
fills
all spaces, as a whole, the successive synthesis of the parts of
an infinite world must be looked upon as completed, that is to say, an
infinite time must be regarded as having elapsed in the enumeration of
all co—existing things; which is impossible. For this reason an
infinite aggregate of actual things cannot be considered as a given
whole, consequently, not as a contemporaneously given whole. The world
is consequently, as regards extension in space,
not infinite, but
enclosed in limits. And this was the second thing to be proved.
[*]
What is meant by successive synthesis must be tolerably plain. If I am
required to form some notion of a piece of land, I may assume an arbitrary standard, — a
mile, or an acre, — and by the successive addition of mile to mile or acre to acre till the
proper number is reached, construct for myself a notion of the size of the land. — Tr.
[*]
We may consider an undetermined quantity as a whole, when it is
enclosed within limits, although we cannot construct or ascertain
its totality by measurement, that is, by the successive synthesis of
its parts. For its limits of themselves determine its completeness
as a whole.
