KANT'S CRITIQUE OF TELEOLOGY IN BIOLOGICAL EXPLANATION

Concept of Infinity 87
be given, involves a contradiction according to Kant. 39 Even in passages
where Kant describes the actual infinite postulated by the
antithesis position (and rejected by Kant), he consistently distinguishes
between an actually infinite series (set) and the merely
potentially infinite regress in it.
This unconditioned may be conceived in either of two ways. It may be
viewed as consisting of the entire series in which all the members without
exception are conditioned and only the totality of them is absolutely unconditioned.
This regress is to be entitled infinite ... the series a parte priori is
without limits or beginning, i.e. is infinite, and at the same time is given in
its entirety. But the regress in it is never completed, and can only be called
potentially infinite. (B445)
After rejecting the finite series of the thesis position, Kant
introduces a third possibility: a progress or regress is possible not
only in infinitum but also in indefinitum. An advance or retreat can
be carried on not only to infinity but also "to an indefinite extent"
(B*540-1). This distinction is at first introduced as a difference in the
terminology used by philosophers and mathematicians. "Mathematicians
speak solely of a progressus in infinitum. Philosophers,
whose task is to examine concepts, refuse to accept this expression
as legitimate, substituting for it the phrase progressus in indefinitum."
(B538-9)
With regard to a progress this difference in the language of
philosophers and mathematicians is unimportant. For instance in
the continuation of a straight line, "the distinction between an infinite
and an indeterminately great advance (progressus in indefinitum)
would be a mere subtlety" (B539). This applies to every unlimited
progress, that is, for every such advance from a condition to
what it conditions "this possible advance proceeds to infinity in the
series of appearances," for "we are referring only to what we can
do" (B*539). Here we can imagine that the series really goes on forever
since it is not presupposed as "given (datum)" but only added on
as "giveable (dabile)." In this case reason does not demand a given
totality in the series, and, accordingly, no conflict of reason with
itself can arise.
39 "The true (transcendental) concept of infinitude is this, that the successive
synthesis of units required for enumeration [Durchmessung] of a quantum can
never be completed." (B460) An "actual infinity" is rejected (B549). An actual
infinity would be "a given quantity that can never be wholly given, which is selfcontradictory"
("Progress of Metaphysics," Ak 20, 289; W 3,625).