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We now proceed to determine clearly our notion of a synthesis
which can never be complete。 There are two terms commonly employed for
this purpose。 These terms are regarded as expressions of different and
distinguishable notions； although the ground of the distinction has
never been clearly exposed。 The term employed by the mathematicians is
progressus in infinitum。 The philosophers prefer the expression
progressus in indefinitum。 Without detaining the reader with an
examination of the reasons for such a distinction； or with remarks
on the right or wrong use of the terms； I shall endeavour clearly to
determine these conceptions； so far as is necessary for the purpose in
this Critique。
We may； with propriety； say of a straight line； that it may be
produced to infinity。 In this case the distinction between a
progressus in infinitum and a progressus in indefinitum is a mere
piece of subtlety。 For； although when we say； 〃Produce a straight
line；〃 it is more correct to say in indefinitum than in infinitum；
because the former means； 〃Produce it as far as you please；〃 the
second； 〃You must not cease to produce it〃； the expression in
infinitum is； when we are speaking of the power to do it； perfectly
correct； for we can always make it longer if we please… on to
infinity。 And this remark holds good in all cases； when we speak of
a progressus； that is； an advancement from the condition to the
conditioned； this possible advancement always proceeds to infinity。 We
may proceed from a given pair in the descending line of generation
from father to son； and cogitate a never…ending line of descendants
from it。 For in such a case reason does not demand absolute totality
in the series； because it does not presuppose it as a condition and as
given （datum）； but merely as conditioned； and as capable of being
given （dabile）。
Very different is the case with the problem： 〃How far the regress；
which ascends from the given conditioned to the conditions； must
extend〃； whether I can say： 〃It is a regress in infinitum；〃 or only
〃in indefinitum〃； and whether； for example； setting out from the human
beings at present alive in the world； I may ascend in the series of
their ancestors； in infinitum… mr whether all that can be said is；
that so far as I have proceeded； I have discovered no empirical ground
for considering the series limited； so that I am justified； and
indeed； compelled to search for ancestors still further back； although
I am not obliged by the idea of reason to presuppose them。
My answer to this question is： 〃If the series is given in
empirical intuition as a whole； the regress in the series of its
internal conditions proceeds in infinitum； but； if only one member
of the series is given； from which the regress is to proceed to
absolute totality； the regress is possible only in indefinitum。〃 For
example； the division of a portion of matter given within certain
limits… of a body； that is… proceeds in infinitum。 For； as the
condition of this whole is its part； and the condition of the part a
part of the part； and so on； and as in this regress of decomposition
an unconditioned indivisible member of the series of conditions is not
to be found； there are no reasons or grounds in experience for
stopping in the division； but； on the contrary； the more remote
members of the division are actually and empirically given prior to
this division。 That is to say； the division proceeds to infinity。 On
the other hand； the series of ancestors of any given human being is
not given； in its absolute totality； in any experience； and yet the
regress proceeds from every genealogical member of this series to
one still higher； and does not meet with any empirical limit
presenting an absolutely unconditioned member of the series。 But as
the members of such a series are not contained in the empirical
intuition of the whole； prior to the regress； this regress does not
proceed to infinity； but only in indefinitum； that is； we are called
upon to discover other and higher members； which are themselves always
conditioned。
In neither case… the regressus in infinitum； nor the regressus in
indefinitum； is the series of conditions to be considered as
actually infinite in the object itself。 This might be true of things
in themselves； but it cannot be asserted of phenomena； which； as
conditions of each other； are only given in the empirical regress
itself。 Hence； the question no longer is； 〃What is the quantity of
this series of conditions in itself… is it finite or infinite？〃 for it
is nothing in itself； but； 〃How is the empirical regress to be
commenced； and how far ought we to proceed with it？〃 And here a signal
distinction in the application of this rule becomes apparent。 If the
whole is given empirically； it is possible to recede in the series
of its internal conditions to infinity。 But if the whole is not given；
and can only be given by and through the empirical regress； I can only
say： 〃It is possible to infinity； to proceed to still higher
conditions in the series。〃 In the first case； I am justified in
asserting that more members are empirically given in the object than I
attain to in the regress （of decomposition）。 In the second case； I
am justified only in saying； that I can always proceed further in
the regress； because no member of the series。 is given as absolutely
conditioned； and thus a higher member is possible； and an inquiry with
regard to it is necessary。 In the one case it is necessary to find
other members of the series； in the other it is necessary to inquire
for others； inasmuch as experience presents no absolute limitation
of the regress。 For； either you do not possess a perception which
absolutely limits your empirical regress； and in this case the regress
cannot be regarded as complete； or； you do possess such a limitative
perception； in which case it is not a part of your series （for that
which limits must be distinct from that which is limited by it）； and
it is incumbent you to continue your regress up to this condition； and
so on。
These remarks will be placed in their proper light by their
application in the following section。
SECTION IX。 Of the Empirical Use of the Regulative Principle
of Reason with regard to the Cosmological Ideas。
We have shown that no transcendental use can be made either of the
conceptions of reason or of understanding。 We have shown； likewise；
that the demand of absolute totality in the series of conditions in
the world of sense arises from a transcendental employment of
reason； resting on the opinion that phenomena are to be regarded as
things in themselves。 It follows that we are not required to answer
the question respecting the absolute quantity of a series… whether
it is in itself limited or unlimited。 We are only called upon to
determine how far we must proceed in the empirical regress from
condition to condition； in order to discover； in conformity with the
rule of reason； a ful