Kant's treatment of rational cosmology makes reference to these various doctrines (and others opposing them), but it does so on Kant's own terms. Reason, the basis of this kind of cosmology, is said to be the faculty which brings to completion the activity of the understanding. The activity of understanding is synthetic, and synthesis is completed through the categories. Thus Kant tried to organize rational cosmology around the division of the categories.
Further, the world of which rational cosmology treats is made up of synthesized appearances. In this way, the object of cosmology differs from that of psychology and theology. In the case of psychology, the object is the substratum of inner intuition, and thus not an appearance at all. In the case of theology, the object is the ground of all possibile reality, an object much too great for us to encounter in experience. Only cosmology attempts to build on principles of the understanding by attempting to bring the synthesis of appearances to completion.
Before we can examine how the synthesis is supposed to be completed, we must first examine the activities of synthesis themselves. These are grouped according to the four types of category.
Quantity: Tracing back what happens at a time to previous times which serve as its condition, and tracing things in space to contiguous spaces which serve as their conditions.
Quality: Tracing the division of objects to the smaller objects of which they are composed.
Relation: Tracing the chain of causes from a present effect to a previous cause.
Modality: Tracing the chain of dependence from something contingent to that upon which it depends.
Kant states that the completion of these activities can be understood in two distinct, and opposed, ways. One way would be to find a first member of the series, such that it is a condition for the rest, while it requires no condition. Another way would be to show that the series is infinite, in which case the series is completed by showing an infinite totality of conditioned and conditions. The first way is that of dogmatism, which completes the series by appeal to an object which is distinct from all objects of experience. The second is that of empiricism, which insists that the series be kept strictly within the bounds of experience. The fundamental problem with dogmatism is that it satisfies reason with ideas beyond the scope of the understanding. Conversely, empiricism accommodates the understanding with ideas too small for reason.
The arguments for the theses advanced by dogmatism and the antitheses advanced by empiricism are of the same type -- indirect. Each side assumes the position of the other and attempts to show that it is inconsistent with a high-level principle it takes to be true. Since each side can reduce the other to conflict, either the last argument given is the most persuasive, or one walks away in disgust, converted to skepticism.
Now let us turn to the first Antinomy, specifically the part concerning a beginning of the world in time. The dogmatic Thesis is that the world has a beginning in time, and the empiricistic Antithesis denies that the world has a beginning.
The argument for the thesis begins with the assumption that the world has no beginning in time. This means (says Kant) that the task of tracing backward through time is a process that cannot be completed: it is an infinite task. But this conflicts with a principle which states that when a conditioned (the state of things at the present time) is given, the conditions are given with it. The present is what it is now by virtue of all the past states which have led up to it. Since all the past states are given, they must be reachable by a synthesis. But the assumption implies that such a synthesis cannot reach all the conditions, so the assumption is false. The world has a beginning in time.
The argument for the antithesis begins with the assumption that the world has a beginning in time. If so, then there was a time at which it did not exist (an empty time). In an empty time there is no sufficient reason for something to come to be, since there is nothing to distinguish one empty time from another. But this conflicts with a principle which states that whatever comes to be has a sufficient reason for its existence rather than non-existence. The assumption implies that there is none, so it is false. The world has no beginning in time, but has existed from infinity.
The argument for the antithesis is adapted from Leibniz, who however drew quite a different conclusion from it. He claimed that time is meaningless apart from a world of existing things. Thus he concluded not that the world has existed from infinity, but the weaker thesis that the world has no beginning in time. For Kant, however, an empty time is possible, as announced in the Aesthetic. So if there is none, infinite time is filled up.
Kant held that there are advantages and disadvantages to both sides. Dogmatism has the advantage in that it supports the practical (action-connected) interests of reason. A first beginning, simple components of the world, uncaused causes, and a necessary being are all requisite for the interests of morality and religion. Empiricism offers nothing in their support. It also has the advantage of poplularity. People think they understand a first member of a series better than they understand an infinite series. And philosophers can pronounce on matters that are unfettered by the restrictions of experience. Empiricism does not appeal to the imagination in going about its way. Finally, first things serve the "architectonic" interests of philosophical system-building. They provide single starting points from which all else flows. The only place where empiricism has anything to offer is in what we might call epistemic integrity. Since it countenences no deviation from experience, its judgments are not subject to the questioning inevitably called forth by the bold assertions of dogmatism. On the other hand, dogmatism offers the security of first principles, the "foundation" that epistemology has always sought. So there is a bit of a standoff here.