Bonaventure argues that it is necessary for the world to have a beginning. He has a large number of arguments for this conclusion. I will just focus on one, since there it was recently defended by Benjamin Brown in American Catholic Philosophical Quarterly (Summer 2005). Bonaventure argues that an infinite series can never be ordered or traversed, and if the world had no beginning, an infinite series of days has been traversed. The classical response to this is that there can only be a traversal if there is a starting point and an ending point. So the traversal argument requires that there be some day that is infinitely distant in the past from the present day. If this were so, then traversing the past would face the same problem as traversing an infinitely divided line point by point: to reach any point from any other point, there would always be yet one more intermediate point, so it would not be traversable. In other words, in an infinitely divided line, there can be no such thing as a 'next point'. Such is the idea. However, the claim that the world had no beginning only requires us to say that for every day into the past we go, there is another day further in the past. Thus every day is only finitely distant from the present. (An objection that one sometimes meets with today, that an infinite is traversable in infinite time, is facile because it begs the question; I mention it merely because it seems to come up a lot.)
Brown argues that this objection to the traversal argument misses the point:
The nub of the question lies in how one should conceive of a beginningless past. Is it really analogous to the future, such taht it is infinite only potentially, not actually? If that were the case, then Bonaventure would think that an agreement was reached, for it would follow that no matter how old the world could be, it would still have a beginning, just as no matter how far into the future you go, there is still an ending. In other words, if there is no infinitely distant past point, which his objectors agree there cannot be, then every actual past point is finitely distant from the present one, so the past must be actually finite and only potentially infinite. In other words, if you deny that there is any point infinitely distant from the present, all you have is a finite past that could have been (but is not) further distant from the present, just like the future.
This argument is a bit obscure, so Brown helpfully breaks it down a bit. Let X be the infinite set of all past days, represented by {..., D-3, D-2, D-1, D-0}, where D-0 is today. Each element of this set must be touched on in succession, which requires: (1) that today be the last day; (2)that each element is distinct; and (3) that each element is touched on, but no two elements are touched on simultaneously. Now, this 'touching' can be done in any order one pleases, so long as these three conditions are met. However, even to get to D-0, an infinity of elements would have to be touched on first. The ellipse (...) represents an infinite series that has to have been completed step-by-step before we get to D-0. This is usually regarded as impossible.
Brown also argues that Bonaventure, contrary to the common view, is right to think that a beginningless world implies that there is a day infinitely distant from the present one. The actually infinite set of days, {..., -3, -2, -1, 0} can be re-arranged to {..., -6, -4, -2, ..., -5, -3, -1, 0} -- i.e., we can touch on all the even days first and then do all the odd days, finally ending with today. But on this rearrangement, -2 is infinitely distant from 0. Indeed, -2 is infinitely distant from any odd number. Brown thinks this shows that in the case of the days, as in the case of the divided line, there is no 'next'. It's very possible I'm missing some very subtle set theoretical point, but this strikes me as a very odd argument. The numbers of the set are not arbitrarily chosen; they are ordinals. Or, to be more exact, they are cardinals that index ordinals. We assign yesterday the number -1 because, and only because, it is the first day in the past; we assign the day before yesterday the number -2 because, and only because, it is the second day in the past. And so forth. Because of this, no matter how we rearrange the cardinal numbers in the set, they still index a rigid ordering. Place -2 anywhere you please, it still represents the second day in the past, and no other day. So it is simply false to say that if the past is beginningless the day indicated by -2 is infinitely distant from any odd day; by definition it is right next door to two odd days, those represented by -1 and -3, regardless of where we actually put the -2. I take it that something like this is Quentin Smith's point in his lovely paper Infinity and the Past that you can always re-arrange the set back to the standard {..., -3, -2, -1, 0}. Brown thinks this misses the point; but I don't see how.
In any case, it's an interesting issue. The most famous defense of the claim that a beginningless past is possible, the one I tend to agree with, is by Thomas Aquinas. Aquinas thinks that, as a matter of fact, the temporal world had a beginning; but he denies that there is any contradiction in the claim that it did not. His reasoning is, very roughly, as follows. For it to be possible for the temporal world not to have a beginning, two conditions have to hold:
(1) It must not contradict the nature of a temporal world for it always to have existed. (This is what Bonaventure denies.) Aquinas argues that the standard arguments for this fail, and he presents the best known of the classical responses to the traversal argument. He also argues that this is not surprising. Natures are universal; they do not contain the history of the thing that has the nature. But when a thing begins, if it does, is a matter of its history rather than its nature. Therefore there is no formal contradiction.
(2) There must be no contradiction on the part of the causes. That is, even if a thing is formally possible, if it is to be actually possible it must be either necessary or caused. If it is not necessary and there are no causes capable of causing it, it is not actually possible. Aquinas argues that it is not necessary (indeed, it is not necessary for the same reason it is not impossible). However, he insists that there is at least one cause capable of creating a temporal world without beginning, namely, the omnipotent, eternal God. Therefore there is no contradiction on the part of the causes.
