Kant distinguishes reason (Vernunft) from understanding (Verstand). Understanding applies categories to experience and yields genuine knowledge. Reason, however, seeks to go further: it wants to complete every series, reach every ultimate ground, explain every conditioned thing by an unconditioned one. This drive is natural and unavoidable — but when reason applies itself to objects beyond all possible experience (the world as a whole, the ultimate nature of the soul, the existence of God), it generates illusions it cannot escape.
The antinomies expose this predicament in its starkest form. Kant identifies four pairs of contradictory theses about the universe: whether it has a beginning in time or is infinite, whether matter is ultimately divisible or composed of indivisible atoms, whether freedom exists or all events are determined by natural law, and whether there exists a necessary being or all existence is contingent. For each pair, he produces a rigorous proof of both the thesis and the antithesis — demonstrating that pure reason, operating without the check of experience, can reach contradictory conclusions with equal validity.
This is not a defect in Kant's arguments — it is the point. The antinomies show that the questions themselves are malformed. They treat the world as a completed whole available for theoretical inspection, when in fact the world as a whole is never given in experience. Reason has over-reached: it has applied concepts that are only valid for phenomena to supposed noumena.
Kant's resolution differs for the two pairs. The first two antinomies — about the extent and divisibility of the world — are resolved by showing that both thesis and antithesis are false: the question of the world's extent in time or space does not arise, because the world is not a thing-in-itself but only a series of appearances that can always be extended further. The third and fourth antinomies — about freedom and a necessary being — are resolved differently: both thesis and antithesis may be true, because they concern different domains. Causality governs the phenomenal world; freedom may be a feature of the noumenal world of things-in-themselves. This resolution opens space for Kant's practical philosophy: the freedom required for moral responsibility is preserved even within a deterministic natural order.
The antinomies are set out in the Transcendental Dialectic in Chapter II of the section on Transcendental Illusion. Kant regarded the antinomies as the clearest demonstration of the limits of theoretical reason, and the resolution of the third antinomy as crucial for his entire practical philosophy.