KANT'S CRITIQUE OF TELEOLOGY IN BIOLOGICAL EXPLANATION

94 The Unconditioned and the Infinite Series
maintains that it is logically false to assert that someone has, for
instance, already counted the natural numbers, for an endless task
cannot be completed. But he allows that a beginningless task can be
completed. For instance, the assertion that someone has just finished
counting the natural numbers backwards is supposedly not
only logically impeccable but also possibly even empirically true.
Thus the impossible becomes possible if it is done in a manner that
is inconceivable.
Kant, in any case believes that it is absurd to maintain that
world history has done something similar to counting the natural
numbers backwards. This is the position that Russell and the others
ought to have attacked with philosophical arguments, instead of
aiming psychological speculations at mental habits. The philosophical
presupposition that underlies this sort of misinterpretation
was articulated by Russell when he maintained that: 45
The notion of infinity ... is primarily a property of classes, and only derivatively
applicable to series; classes which are infinite are given all at once by
the defining property of their members, so that there is no question of 'completion'
or of 'successive synthesis'.
Thus the reason that Kant and Russell cannot agree on
whether the world is infinitely or only indefinitely old lies neither in
Kant's mental habits nor in Russell's, but rather in philosophical
differences about the concept of the infinite: whether infinity can be
given actually as a class or set, or whether it can only be set as a
task, as potential progress in a series. Whether or not Kant's arguments
against the impossibility of an infinite past are convincing,
thus depends on whether or not one is prepared to accept the existence
of actually infinite classes, not merely in mathematics but also
in the realm of physics. 46
3) The Regress in Space
In the second part of the First Antinomy — the extent of the
world in space — the refutation of the thesis on the finite extent of
the world and of the antithesis on its infinite extent have the same
structure as those in the first part concerning time. Before we turn
45 Russell, Knowledge, p. 160.
46 Paul Lorenzen, Methodisches Denken pp. 94ff, argues against the actual
infinite in mathematics as well.