(quote from
A Defense of the Cosmological Argument for the Existence of God)
The more controversial premise is the second one: that the universe began to exist. And in support of this premise I present two philosophical arguments, and then two scientific confirmations of those arguments. The first philosophical argument, (2.1), is the argument based on the impossibility of an actual infinite, and it runs like this: (2.11) An actual infinite cannot exist. (2.12) An infinite temporal regress of the events is an actual infinite. (2.13) Therefore, an infinite temporal regress of events cannot exist.
Now in order to grasp this argument it's important to distinguish an actual infinite from a potential infinite. An actual infinite is a collection of things having a proper subset which has the same number of members as the original collection itself. An actual infinite is not like a potential infinite, which is a collection which is at every point in time finite but is growing toward infinity as a limit. My argument is simply that an actual infinite cannot exist. I do not deny the existence of a potential infinite.
Why do I hold to (2.11)? Well, very simply this: if you try to translate the idea of an actually infinite number of things into reality, you wind up with all sorts of absurdities and, in the end, logical contradictions. For example, what is infinity minus infinity? Well, mathematically you get self–contradictory answers, unless you impose some wholly arbitrary rules to prevent this. This shows that infinity is just an idea in your mind, not something that exists in reality. David Hilbert, who is perhaps the greatest mathematician of this century, states, "The infinite is nowhere to be found in reality. It neither exists in nature, nor provides a legitimate basis for rational thought…. The role that remains for the infinite to play is solely that of an idea."{5} So as I understand the actual infinite, it is simply a conceptual idea; it is not something that exists in reality. (2.12) says, an infinite temporal regress of events is an actual infinite. I think this is fairly obvious. If the universe never began to exist, then the number of past events is actually infinite. And therefore it follows: (2.13) that an infinite temporal regress of events cannot exist. Therefore, the temporal regress of events is finite and must have a beginning. Since the universe is not distinct from the temporal series of past events, it therefore follows that the universe began to exist.
The second philosophical argument is the argument based on the impossibility of the formation of an actual infinite by successive addition. This argument is independent of the first. It's claiming that even if an actual infinite can exist, it cannot be formed by successive addition. And this argument goes this way: (2.21) A collection formed by successive addition cannot be actually infinite. (2.22) The temporal series of past events is a collection formed by successive addition. (2.23) Therefore, the temporal series of past events cannot be actually infinite. The first step in the argument, a collection formed by successive addition cannot be actually infinite, is true by the very nature of infinity. You can never get to infinity by addition because you can always add one more. Sometimes this is called the impossibility of counting to infinity, or another way it's referred to is the impossibility of traversing the infinite. Now if the past were infinite, it would be as though someone had claimed to have just finished counting down all the negative numbers ending in "0," and surely this is absurd. If you can't count to infinity, how can you count down from infinity? If you can't traverse an infinite distance by running in one direction, how can you traverse it by simply turning around and running in the opposite direction?
Indeed, the idea that the past could be actually infinite is absurd. I think this is well illustrated by the Tristram Shandy paradox of Bertrand Russell. Russell imagines Tristram Shandy, a character in a novel by Sterne, who writes his autobiography so slowly that it takes him a year to write down the events of a single day. Russell says that if Tristram Shandy were to live forever, then his autobiography would be completed because there would be an infinite number of years and an infinite number of days, so that every day would be written about. It seems to me that this conclusion is incorrect because the future is a potential infinite only. Tristram Shandy would never arrive at actual infinity. The number of days and hence the number of years of his life would always be finite but potentially increasing toward infinity as a limit. But suppose we turn this story around and imagine that Tristram Shandy has been writing from eternity past. Then the number of years and the number of days would in fact be actually infinite, and you could say that Tristram Shandy would have completed his autobiography. But if you say that Tristram Shandy would have completed his autobiography, then the question arises: Why did he finish it today rather than yesterday, or the day before, or the day before that? By any time in the past, an infinite amount of time had already elapsed, so that if Tristram Shandy would finish his autobiography given infinite time, he should have already finished at any point in the finite past. But that means that no matter how far back in the past you regress, you will never find Tristram Shandy writing, which contradicts the hypothesis that he has been writing his autobiography from eternity. And thus the notion of an infinite past, it seems to me, is absurd.
That leads to the second premise, that a temporal series of past events is a collection formed by successive addition. Again, I think this point is obvious. The series of past events is a collection which has been formed by one event occurring after another, by successive addition. But that leads to the conclusion: therefore, the series of past events cannot be actually infinite. It must be finite, and the universe must have begun to exist.
Again, this argument was agreed to by both Hume and Kant. In his Enquiry, Chapter 12, Section II, paragraph 125, Hume writes, "An infinite number of real parts of time, passing in succession, and exhausted one after another, appears so evident a contradiction, that no man whose judgment is not corrupted, instead of being improved, by the sciences, would ever be able to admit of it."{6} Hume tries to elude what he calls the absurdities and contradictions of this by embracing a nominalist view of numbers and abstract objects. Now I’m very much in sympathy with that view, but clearly it does nothing to solve the problem of how a temporal series of real past events could have been formed by successive addition and yet be infinite. And, as is well known, Kant as well, in the thesis of his first antinomy concerning time, also endorses this argument. Kant writes,
If we assume that the world has no beginning in time, then up to every given moment an eternity has elapsed and there has passed away in the world an infinite series of successive states of things. Now the infinity of a series consists in the fact it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world–series to have passed away, and that a beginning of the world is therefore a necessary condition of the world's existence.{7}
And it can't be emphasized enough that, according to Kant, this is an undeniable requirement of reason. Reason forces you to that conclusion, on Kant's view. Of course, he also believed that reason forced you to adopt the antithesis as well, but I think that the argument for the antithesis is simply a faulty argument. It erroneously assumes that time necessarily precedes the beginning of the universe; but on a non–Newtonian relational view of time, time begins simultaneously with the first event. So there's simply no problem about when the universe would have begun to exist in the empty time prior to the beginning of the universe. So again, it seems to me that this argument is a forceful and persuasive argument, which both Hume and Kant, in effect, concede.
