KANT'S CRITIQUE OF TELEOLOGY IN BIOLOGICAL EXPLANATION

92 The Unconditioned and the Infinite Series
Kant's question is thus whether the regress back in time is
potentially infinite or merely indefinitely large. It is undisputed that
we can continue the regress at will, just as we can continue to count
the natural numbers at will. We can also represent the progress of
real historical events after any particular point in time reached by
the mental regress if we count backwards from the number
reached. This is precisely what we do when we assign dates in
ancient history, e.g. 390 B.C. etc. If the regress is potentially infinite
(like the progress of the natural numbers), we could represent it as
the mirror image of the denumeration of the natural numbers (e.g.
-1, -2, -3, -4, -5 ... or 1 B.C., 2 B.C., ... ). If that is true, then the real
historical course of the world out of the past must have accomplished
something like counting the natural numbers backwards.
An infinite progress does not imply that someone has already covered
the same ground in the other direction, but a regress in infinitum
back in time implies that the world has already covered the
same path going forwards (progress). Kant rejects the notion that
the temporal series of events can have the same kind of open structure
on its "past" side as it has on its future side, he rejects the
regress in infinitum and replaces it with an indefinite regress,
since the potential infinity of the regress back in time would imply
that an infinite (beginningless) progress out of the past had already
been completed, and thus that an actual infinite set, a "real infinity"
of past states, had been accumulated. To modify Strawson's description
of the argument slightly, let us replace his etherial ticks of the
clock with somewhat more substantial drops of water: Kant is
asserting that, if your tap had been dripping forever, so that you
could now count the drops of water in the tub in infinitum, your tub
would contain an actually infinite amount of water, which is absurd;
therefore your tap has not been dripping forever.
In such a case the philosopher must according to Kant take
care whether he says in infinitum or in indefinitum. The world
series must be indefinitely large.
This cosmic series can, therefore, be neither greater nor smaller than the
possible empirical regress upon which alone its concept rests. And since
this regress can yield neither a determinate infinite nor a determinate finite
(that is, anything absolutely limited), it is evident that the magnitude of the
world can be taken neither as finite nor as infinite. The regress, through
which it is represented, allows of neither alternative. (B546n)
It is worth pointing out that Bertrand Russell saw the problems
involved in this antinomy rather clearly and articulated the