Kant

Immanuel Kant (1726–1804) gave extensive consideration to infinite regresses in the deliberations of his Critique of Pure Reason (1781). He was primarily interested in the concept of a regress of conditions required for making a claim appropriately. And here his influential discussion deserves to be quoted in full:

[Consider] the problem: how far the regress extends, when it moves in a series from something given as conditioned to its conditions. Can we say that the regress is in infinitum, or only that it is indeterminately far extended (in indefinitum)? Can we, for instance, move from the men now living, through the series of their ancestors, in infinitum; or can we only say that, so far as we have gone back, we have never met with an empirical ground for regarding the series as limited at any point, and that we are therefore justified and even obliged, in the case of every ancestor, to search further for progenitors, though not indeed to presuppose them?

We answer: when a series is something that is given as a whole in empirical intuition, then the regress in the series of its inner conditions proceeds in infinitum; but when only a member of the series is given, starting from which the regress has to proceed altogether, then the regress is merely of indeterminate character (in indefinitum). Accordingly, the division of a body, that is, of a limited portion of matter, must be said to proceed in infinitum. For this matter is here given in empirical intuition as a whole, with all its possible parts included. Since a part is a condition of this whole is its part, and the condition of this part is the part of the part, and so on, and since in this regress of decomposition an unconditioned (indivisible) member of this series of conditions is never met with, there is never any empirical ground for stopping in the division. For here the further members of any continued division are themselves empirically given prior to the continuation of the division. The division, that is to say, goes on in infinitum. On the other hand, since the series of ancestors of any given man is not given in its absolute totality in any possible experience, the regress proceeds from every member in the series of generations to a high member, and no empirical limit is encountered which exhibits a member as absolutely unconditioned. And since the members, which might supply the condition, are not contained in an empirical intuition 140of the whole, prior to the regress, this regress does not proceed in infinitum. Rather it proceeds only indefinitely far, searching for further members additional to those that are given, and which are themselves again always given as conditioned.

In neither case, whether the regress be in infinitum or in indefinitum, may the series of conditions be regarded as being given as infinite in the object. The series involves not things in themselves, but only appearances, which, as conditions of one another, are given only in the regress itself. The question, therefore, is no longer how great this series of conditions may be in itself, whether it be finite or infinite, for it is nothing in itself; but how we are to carry out the empirical regress, and how far we should continue it.

Here we encounter an important distinction in regard to the rule governing such procedure. When the whole is empirically given, it is possible to proceed back in the series of its inner conditions in infinitum. When the whole is not given, but has first to be given through empirical regress, we can only say that the search for still higher conditions of the series is possible in infinitum. In the former case we could say: there are always more members, empirically given, than I can reach through the regress of decomposition; in the latter case, however, the position is this: we can always proceed still further in the regress, because no member is empirically given as absolutely unconditioned; and since a higher member is therefore always possible, the enquiry regarding it is necessary. In the one case we necessarily find further members of the series; in the other case, since no experience is absolutely limited, the necessity is that we search for them. For either we have no perception which sets an absolute limit to the empirical regress, in which case we must not regard the regress as completed, or we have a perception limiting our series, in which case the perception cannot be part of the series traversed (for that which limits must be distinct from that which is the limit). ¹