Immanuel Kant makes a distinction between two categories: phenomena and noumena. The phenomenal realm is what we perceive and, according to Kant's interpretation, is limited to the mind only. By contrast, the noumenal realm is what a thing is in and of itself. Noumena cannot be known by the mind, or the perceiver. For Aristotle and Thomas Aquinas, what can be known through phenomena corresponds to the noumena. Kant denies this possibility. (It should be noted that Aristotle and Thomas did not use these terms, but their acceptance of alethic, or truth-bearing, realism entails correspondence.)
As a result, Kant rejects the traditional arguments for God's existence, at least insofar as they have any bearing on the noumenal realm. Nevertheless, traditional theistic proofs can be limited to the scope of the phenomenal realm, which is of significance. Of course, one need only make such limitations if Kant's distinction is actual, and not in the mind only. However, for the purposes of this post, Kantianism will be accepted for the sake of argument. While he already accepts a moral argument for God's existence for pragmatic reasons in his Critique of Practical Reason, I want to argue that the historical proofs of theism are still salvageable on Kant's worldview.
Let's start from the basic premise that we perceive (phenomena) that dependent things exist, even if this cannot be established to hold true in the noumenal realm:
1. Dependent things exist. (Premise)
2. Every dependent thing has a sustaining cause. (Premise)
3. Either the regress of dependent and sustaining causes is infinite, or else there exists an independently existing first cause in the order of sustaining causes. (Implied by 1 and 2)
4. There cannot be an infinite regress of dependent and sustaining causes. (Premise)
5. Therefore, there exists an independent first cause in the order of sustaining causes. (From 3 and 4)
I take (2) to be true by definition insofar as X is understood to be causally sustained by Y if and only if X cannot exist apart from Y. That's only a rough definition, but it will do for now.
The key premise, as with so many cosmological arguments, is (4). I've already argued at length that such an infinite regress cannot exist, even if the universe's past is conceivably eternal, which would imply an infinite regress of originating causes. One of the reasons I affirm (4) is because it would take infinite time for an infinite regress of causes to be instantiated. Given that there are finite periods of time even on the assumption that time is infinite, it follows that during any finite interval the regress of sustaining causes must also be finite.
Another way of thinking about it is that it is impossible to form an actual infinite by successive addition whenever one begins counting. Since at each finite interval the regress of sustaining causes begins anew, we can deduce that during this interval there can only exist finite causal regresses. At t1, the regress begins: 1, 2, 3, . . . n. At t2, the regress begins again: 1, 2, 3, . . . n. And so on. Before t2 can arrive, t1 must reach infinity, which is impossible because there will always and indefinitely be another number to be traversed before arriving at infinity.
One objection is that between any two natural numbers - say, 1 and 2 - there are infinitely-many numbers (1.5, 1.25, . . . n). The problem with using such examples is that the sum of these numbers is itself a finite number. Moreover, the interval between 1 and 2 has a definite beginning (1) and end (2). These two considerations show that the objection actually presupposes what it sets out to disprove, making the objection self-defeating.
The truth of (1) through (4), together with the argument's logical validity, necessitates the truth of (5): there exists an independent first cause in the order of sustaining causes. It doesn't make any sense to ask what causes the first cause, since the first cause exists independently, and the causal premise of (2) is limited to what exists dependently.
There are a few more points to make. First, one might be satisfied with this conclusion and seek no further implications. The atheist is right to point out that the existence of a first cause does not automatically lead to theism. If the theist wishes to make any connection between the first cause and God, additional metaphysical considerations will need to be taken into account. However, that is beyond the purview of what I want to look at in this post.
As I said, one could go no further than concluding that a first cause exists, and one wouldn't even need to go beyond the phenomenal realm. Nevertheless, I'm interested in ways to argue that the proof extends to the noumenal realm as well. This will likely be the area of philosophy I most explore next. For now, let's argue transcendentally, or from the impossibility of the contrary:
Prove A: Something independent exists in the noumenal realm.
Assume ~A: Nothing independent exists in the noumenal realm.
~A --> B: If nothing independent exists in the noumenal realm, then only dependent things exist in the noumenal realm.
~B: B is inscrutable.
Hence, ~~A: by modus tollens.
Therefore, A: by negation.
The noumenal realm does not escape the law of excluded middle. Either X is dependent or independent, and there is no third alternative. If A is denied on the grounds of inscrutability, then so is ~A. For Kant, this demonstrates the collapse of human reason, but such a conclusion cannot be affirmed without making a claim about the noumena. This also means that agnosticism about A and ~A is self-defeating, and so the only rational conclusion is that the phenomena maps onto the noumena.
If this argument is correct, and I'm only tentatively affirming it, one is compelled to reject the strong Kantian doctrine in favor of some form of metaphysical realism.