The Cosmological Argument (part 3)

August 26, 2007     Time: 00:18:24

We want to continue our discussion of the cosmological argument for the existence of God. We began to look at reasons for the second premise last time – that the universe began to exist. We examined the first philosophical argument for the beginning of the universe. This is the argument based upon the impossibility of the existence of an actually infinite number of things. I argued that given the impossibility of the existence of an actually infinite number of things and the fact that a beginningless series of past events would involve an actually infinite number of things (namely, past events), therefore the series of past events cannot be beginningless. Rather, it must have begun to exist. Since the universe is not distinct from the series of past events, it follows therefore that the universe began to exist.

This second philosophical argument is an independent argument for the beginning of the universe. Suppose my first philosophical argument doesn't convince you. Suppose that argument doesn't work. That's fine. This second argument is wholly independent of the first argument. It does not presuppose that an actually infinite number of things cannot exist. Rather, this second argument is based upon the impossibility of forming an actual infinite number of things by successive addition of one member after another. This argument is independent of the first and does not presuppose that an actually infinite number of things cannot exist. Rather, it is an argument that you cannot form an actually infinite number of things by adding one member after another.

Let's look at the premises for this argument.

1. An actually infinite collection of things cannot be formed by successive addition.

2. The series of past events is a collection of things formed by successive addition.

3. Therefore the series of past events cannot be actually infinite.

Let's look at these three premises in order.

First (1), “An actually infinite collection of things cannot be formed by successive addition.” By successive addition, I mean adding one member at a time after another. Adding one at a time, one after another. That is what I mean by successive addition. The idea is that you cannot form an actually infinite collection of things by adding one member after another one at a time.

Sometimes this is called the impossibility of counting to infinity. No matter how high you count, you can always count one more number before you will get to infinity. Therefore, you will never get to infinity. The series of natural numbers that you count will simply go on forever and ever with infinity as a limit but you never arrive at it. So the impossibility of counting to infinity is sometimes used to describe this first premise.

Other times this is described as the impossibility of traversing the infinite, or of crossing the infinite. It would be impossible to traverse an infinite distance because no matter how far you run, you can always take one more step before you arrive at infinity. If someone were running off into outer space he would never run an infinite distance because he would never arrive at infinity. He could always take one more step before he would get to infinity. Therefore, he could never reach infinity by taking one step at a time.

Indeed, when you think about trying to form an actually infinite collection of things, I think you can see that in principle this cannot be done by successive addition. This really is like trying to turn a potential infinite into an actual infinite by counting, and you cannot do that. No matter how far you count, you can always count higher. So you could never get to infinity by counting.

Some people have responded to this argument by saying that it is true that you cannot form an actually infinite collection of things by beginning at a point and trying to reach infinity, but you can form an actually infinite collection of things by never beginning and ending at a point.[1] You never begin – that is to say, you will have always been doing this – and then you will simply end at a point, and thus you will have formed an actually infinite collection of things in that way. But this second method of forming an actually infinite collection of things seems even more absurd than the first method. It would be like trying to count down all of the negative numbers from infinity. -3, -2, -1, 0. It would be like trying to arrive at zero by successively counting all of the negative numbers in order. It would sort of be forming an actual infinite by successive subtraction in a way, which seems just as absurd as trying to do it by successive addition. So this method seems even more implausible than the first. If you cannot traverse the infinite by running in one direction, how could you traverse it by simply turning around and running in the other direction? If a man comes running up to you and claims to have just finished crossing an infinite plane and he has just arrived here today, that seems just as absurd as someone trying to reach infinity by running there from this point. So it seems to me that this second method of never beginning but ending at a point is just as absurd as trying to begin at a point and get to infinity.

Indeed, there are actually deeper absurdities involved in this second method of never beginning but ending at a point that would not be involved in the other method. For example, suppose we meet someone who claims to have just finished counting down all of the negative numbers. He is just finished today. -3, -2, -1, 0! Phew! Well, now, immediately we might ask ourselves, why did he finish his countdown today? Why didn't he finish yesterday? Or the day before? If he has been counting from infinity then there has been just as many days prior to yesterday as there were prior to today. So if he is counting, say, one number a minute or one number per hour, given that there are an infinite number of minutes or hours prior to yesterday, why didn't he finish his countdown yesterday rather than today? Or why didn't he finish ten trillion years ago or any time in the finite past? At any point in the finite past, an infinity of days or hours will have already elapsed. So we could ask why doesn't the person finish his countdown then? It seems to me that there can be no reason at all given for this. At any point, an infinite amount of time would have already elapsed and therefore it seems that he would have already finished. If he could finish his infinite countdown by today then it seems that he would have finished at any point in the past which is absurd. But if he would have finished his countdown at any point in his past, what that means is that no matter how far you regress back in the past, you will never find the person finishing. Because at any point you reach, he will have already been done by then. But if at no point in the finite past could you find the man finishing his countdown then it is not true that he has been counting down from eternity. You could never find him finishing his countdown at any point in the past. Therefore, you reach the self-contradictory conclusion that he is not counting down from infinity after all.

So it seems to me that this idea of trying to form an infinite collection of things by successive addition is simply maladroit. It is impossible. It cannot be done.

Therefore, in modern set theory, any notions of successive addition has been done away with. When a mathematician says that there are, say, an infinite number of negative numbers, he doesn't imagine that these are posited successively. He just says that given the definition of membership, the membership of the set is given immediately, simultaneously all of the members are given. But it would be absurd to try to give them one at a time.[2] So if God were to create, say, an infinite number of marbles or an infinite number of books in a library, he would have to simply say, “Let there be!” BOOM! And all of the infinite marbles or books would come into existence simultaneously. But it would be impossible to try to form an actually infinite number of books or marbles by adding one member at a time, one after another.

So I think we have good grounds for affirming this first premise that an actually infinite number of things cannot be formed by successive addition.

Premise (2) says, “The series of past events is a collection of things formed by successive addition.” The past is not given whole and entire. Rather, events are formed sequentially one after another. Nor are events formed by subtracting events from the present. Sometimes we speak of an infinite regress of events into the past. But when you think about it, that is really not technically accurate. Our minds regress in the past as we think one year ago, two years ago, three years ago, and we mentally regress in the past. But the events themselves are not regressing in the past. They are progressing in the sense that one event occurs after another in the direction of the future. So the past is a collection of things formed by adding one event after another, one event occurring on the heels of another until we have arrived at today.

It is true, however, that some philosophers of space and time will deny this premise (2). We've talked about this at other times in this class as the difference between a dynamic and a static theory of time. On the static theory of time, all events in time past, present, and future are equally real. So if we let this represent the four-dimensional space-time continuum and this is the Big Bang at t=0 (the beginning of time), this is the end of the universe at some time in the finite future. On the static view of time all events in space and time are equally real. If this is 2004 (this cross-section of the spindle), well, the events existing, say, at ten trillion years ago are just as real as the events of 2004 and the events of, say, 30,000 AD are just as real as the events of 2004. Everything is equally real. Whereas on the dynamic view of time, the only thing that actually exists is the present slice of time. Our moment in time. Moments in the past and in the future are just unreal. They are potential. The future has not yet occurred. The past has already occurred and has lapsed away. Therefore they no longer or do not yet exist.

This second premise presupposes a dynamic view of time. Events in time are being formed one after another. They are being added layer by layer so to speak. You can avoid this argument by adopting this static theory of time in which you would say there just is no temporal becoming – everything (past, present, and future) is equally real. Therefore, the series of past events is not formed by successive addition. It just exists. I don't want to go into that at this point. Perhaps we could talk about that when I come back. In my book Time and Eternity I have laid out reasons why I think that the dynamic view is correct. Everybody admits that the dynamic view of time is the common sense view of time. The dynamic view of time, even static time theorists admit, is the view of the ordinary person. It is the common sense view of time. As I examined the arguments for and against the static and dynamic theory of time, I have to say that I don't find any compelling reasons to believe in the static view of time and to abandon the common sense view of time which is certainly real in our experiences. We experience temporal becoming. We experience the lapse of time, one event occurring after another.[3] I think we would have to have very, very powerful reasons for denying what we experience so vividly, so intimately, in the passage of time, in the lapse of time of one event after another. Indeed, the lapse of time, the reality of temporal becoming, is even more real to us than the existence of the external world. Because the external world could be an illusion – it could be a dream. But temporal becoming could not be an illusion because even the illusion of temporal becoming involves temporal becoming as the contents of your consciousness succeed one another in time. So there is in our inner life, in our inner mental conscious life, a kind of becoming of one thing after another that I think is virtually undeniable. Therefore, given that there are no compelling arguments for the static theory of time, I think we should stick with the dynamic theory of time which does say that the series of events in time are formed by adding one event after another and temporal becoming is real.

If temporal becoming is real – if events really do elapse in time one after another – and you cannot form an actually infinite collection of things by adding one member after another, then it follows (3) “The series of past events cannot be actually infinite.” The series of past events must be finite, and therefore the universe began to exist. If the series of past events were infinite, then we could never get to today. How could today ever arrive if, in order to arrive at today, you would have to traverse an actually infinite number of past events one event at a time. You would never get to today. Today would never arrive. And yet obviously today has arrived. That shows that time is finite in the past, it had a beginning point, and one event after another has been elapsing until we have arrived at the present moment which is today.

Therefore I am persuaded that this second philosophical argument, quite independently of the first one, gives us good grounds for believing that the past is finite and therefore the universe began to exist.

What we will do next time is look at scientific confirmation for these purely philosophical conclusions. Some people might look at these philosophical arguments as mere armchair cosmology, as it has been said. But in fact, I think we have quite good scientific confirmation of these purely philosophical arguments for a beginning of the universe. That is what we will examine together next time.[4]



[1] 18:20

[2] 10:00

[3] 15:03

[4] Total Running Time: 18:20 (Copyright © 2007 William Lane Craig)