(1) The world necessarily had a beginning.
(2) The world necessarily lacked a beginning.
(3) The world is such that it could have a beginning or lack a beginning; but as a matter of fact it did have a beginning.
(4) The world is such that it could have a beginning or lack a beginning; but as a matter of fact it did not have a beginning.
(5) The world is not such that it could have a beginning, and it is not such that it could lack a beginning.
(1) is the sort of view one finds in Philoponus, the Kalam theologians, Saadia Gaon, al-Ghazali, and Bonaventure. The usual arguments for this are arguments for the impossibility of the relevant type of actual infinite, although there are a few others. (2) is usually associated with strong Aristotelians, e.g., Aristotle, Averroes, for reasons due to their analysis of motion; it is also Newton's view, for theological reasons (God exists necessarily; and constitutes space and time by necessarily existing always and everywhere). (3) is the view of Aquinas and probably Maimonides before him. I don't know of anyone who holds (4), although it's a possible opinion. (5) may seem self-contradictory, but it's actually a bit more clever than that. We find the position in Kant.
As I see it, this is Kant's argument. We can apparently refute both the conclusion that the world has a beginning and the conclusion that the world has no beginning. (Kant argues this at length.) Now, if this were a straightforward contradiction, reason would be in trouble; but in fact it is not. The reason the dilemma can be made to seem so serious, Kant argues, is that we tend to misread it. In particular, we tend to read
A) The world has a beginning.
as the contradictory of
B) The world does not have a beginning.
But we need not accept this assumption. Kant uses the analogy of good-smelling. We might tend to read
A') This body is good-smelling.
as the contradictory of
B') This body is not good-smelling.
But another way to read it is to read it as:
C') This body is such that it has a smell that is good.
and
D') This body is such that it has a smell that is not good.
These are not contradictories. They are contraries, which means that while they cannot both be true, they can both be false. In particular, they will both be false if this is true:
F') This body is not such that it has any smell at all.
So, Kant says, we need to distinguish the following three sets of claims:
A) The world is such that it has a beginning
B) The world is not such that it has a beginning.
C) The world is such that it has a beginning.
D) The world is such that it does not have a beginning.
E) The world is such that it either has a beginning or does not.
F) The word is not such that it either has a beginning or does not.
(A) and (B) are contradictories; but if we affirm (B), we aren't claiming that the world has no beginning (D), because (B) is also consistent with (F). Only if (E) is true will (B) imply (D); because only if (E) is true will (C) and (D) be contradictories. Kant famously claims that (E) is false, arguing instead for its contradictory (F): The appearances of the world are reason's synthesis in space and time; therefore, 'the world' as a total synthesis of appearances is not available to reason as an object but is presented to it as a task -- a task of continually regressing in time, as each prior moment requires another prior moment. Therefore, since 'the world' in this sense is not an actually existing object, but a task set for reason (i.e., is not a thing in itself but a representation), it is not the sort of thing that properly speaking has a beginning or lacks one. As Kant says, in this regress there can be no lack of conditions; but the synthesis never exists without regress (and further regress, ad infinitum). As he sums it up (A507/B535):
If the world is a whole existing in itself, it is either finite or infinite. But both alternatives are false (as shown in the proofs of the antithesis and thesis respectively). It is therefore also false that the world (the sum of all appearances) is a whole existing in itself. From this it then follows that appearances in general are nothing outside our representations--which is just what is meant by their transcendental ideality.
(Kemp Smith translation.) This is Kant's famous transcendental idealism.
There are whole worlds of problems with this approach, I think; but it must be admitted that it is quite clever. The ingenuity of it is typical of Kant, and enough to show why he's often considered a great philosophical mind.
[LATER NOTE: Rewrote a few sentences for clarity and to eliminate typos and revision-residue.]
