Kant thought that certain of his antinomies (God and Freedom) could be resolved as "Postulates of Practical Reason". He used them to describe the equally rational-but-contradictory results of applying the universe of pure thought to the categories or criteria, i.e. applying reason proper to the universe of sensible perception or experience (phenomena). Empirical reason cannot here play the role of establishing rational truths because it goes beyond possible experience and is applied to the sphere of that which transcends it.
Kant's antinomies are four: two "mathematical" and two "dynamical". They are connected with (1) the limitation of the universe in respect of space and time, (2) the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist), (3) the problem of free will in relation to universal causality, and (4) the existence of a necessary being.
The first two antinomies are dubbed "mathematical" antinomies, presumably because in each case we are concerned with the relation between what are alleged to be sensible objects (either the world itself, or objects in it) and space and time. The second two are dubbed "dynamical" antinomies, presumably because the proponents of the thesis are not committing themselves solely to claims about spatio-temporal objects.
The mathematical antinomies
The first antinomy (of space and time)
- The world has a beginning in time, and is also limited as regards space.
- The world has no beginning, and no limits in space; it is infinite as regards both time and space.
The second antinomy (of atomism)
- Every composite substance in the world is made up of simple parts, and nothing anywhere exists save the simple or what is composed of the simple.
- No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple.
The dynamical antinomies
The third antinomy (of spontaneity and causal determinism)
- Causality in accordance with laws of nature is not the only causality from which the appearances of the world can one and all be derived. To explain these appearances it is necessary to assume that there is also another causality, that of Spontaneity.
- There is no Spontaneity; everything in the world takes place solely in accordance with laws of nature.
The fourth antinomy (of necessary being or not)
- There belongs to the world, either as its part or as its cause, a being that is absolutely necessary.
- An absolutely necessary being nowhere exists in the world, nor does it exist outside the world as its cause.
Geoff Goddu (https://facultystaff.richmond.edu/~ggoddu/Modern/272h-k1.html) offers this version:
4th antinomy: Thesis: A necessary being is either part of or cause of the world.
Proof: 1. The sensible world contains a series of alterations. 2. Every alteration requires a condition without which that alteration would not be possible. 3. Every condition presupposes a complete series of conditions up to the unconditioned which is itself, absolutely necessary. 4. A necessary being is part or cause of the world.
4th antinomy: Antithesis: A necessary being is not (a) part of the world or (b) cause of the world.
Proof (a): 1. If a necessary being is part of the world, then either the beginning of the series of alterations is absolutely necessary or the series has no beginning and the whole series is absolutely necessary. 2. The series of alterations can have no beginning. 3. The whole series of alterations cannot be absolutely necessary. 4. A necessary being is not part of the world.
Proof (b): 1. If a necessary being is cause of the world then it exists outside the world. 2. If a necessary being is a cause of the world, then it is in time and so not outside the world. 3. A necessary being is not cause of the world.
- Encyclopædia Britannica, 11th ed. (1911), Vol. 2.
- S. Al-Azm, The Origins of Kant's Argument in the Antinomies, Oxford University Press 1972.
- M. Grier, Kant's Doctrine of Transcendental Illusion, Cambridge University Press 2001.
- M. Grier, "The Logic of Illusion and the Antinomies," in Bird (ed.), Blackwell, Oxford 2006, pp. 192-207.