The Doctrine of Creation (part 8)

[Opening prayer]

Last time we began to look at arguments for creation out of nothing. You will recall that I said this tradition originated within Christian thought in the efforts of early Christian philosophers to refute Aristotle’s doctrine of the eternity of the universe. When Islam swept over North Africa, it absorbed this intellectual tradition and developed it in a highly sophisticated way. This tradition was then fed back into the Latin speaking medieval West through the Jewish population living in Muslim Spain at the time.

We want to look at some of the arguments for creation out of nothing beginning with the philosophical arguments. The first philosophical argument that I would like to look at with you is going to be the argument based on the impossibility of the existence of an actually infinite number of things. In order to understand this argument, we first need to understand the concept of an actual infinite or an actually infinite number of things. To say that a collection has an actually infinite number of things means that the number of items in the collection is greater than any natural number: 0, 1, 2, 3, out to infinity. Those are the natural numbers – the integers that we all are familiar with that go out to infinity. To say that something has an actually infinite number of members is to say that it has a number of members which is greater than any natural number. The symbol that mathematicians use today to represent the idea of an actual infinity is the Hebrew letter aleph (א) which represents the first transfinite number. It is the number of numbers in the set of natural numbers (0, 1, 2, 3, and so on out to infinity). The number of numbers in that set is aleph – the first of the infinite or transfinite numbers. To say something is actually infinite is to say that it is a number which is greater than any natural number. The symbol for that is א.

To be distinguished from an actual infinite is the idea of a potential infinite. A potential infinite is a collection that is always finite at any time but it is growing toward infinity as a limit. It is always finite at any time you pick, but it is always endlessly increasing toward infinity as a limit. The symbol for this type of infinity is the lazy-eight or the sideways-eight. It is called a lemniscate (∞). For example, I could divide any finite distance like the distance of the points on this podium in half, and then I could divide it in half again, and then I could divide it in half again. This would be a potentially infinite series of division. No matter where you stop me I will have only made a finite number of divisions but I can keep on going toward infinity as a limit. I never arrive at the limit. I never reach this sort of infinity. It is merely a limit which you endlessly approach. The argument here is not against the existence of a potential infinite. Rather, the argument is that an actually infinite number of things cannot exist.[1]

START DISCUSSION

Student: [inaudible]

Dr. Craig: Physically that is true. You are quite right. Physically you will get down to these sort of minimum distances. But we are talking mathematically here. To say that between any two points there is always another point is to say that this series is what mathematicians call dense. Between any two points you pick there will always be another point. You will never come to the place where you can’t make another division. It would be like the series of fractions converging toward zero as a limit. 1/2, 1/4, 1/8, 1/16. You will never get down to a smallest fraction. There is no such thing as a smallest fraction. You are right, physically you would encounter quarks or something at subatomic scales that couldn’t be divided. But mathematically we are talking here about just an endless progression that goes to infinity as a limit.

Student: [inaudible]

Dr. Craig: It is used in set theory and transfinite arithmetic.

Student: [inaudible]

Dr. Craig: I didn’t say that, although that is true. If you summed all the natural numbers they would add up to infinity. What I said is it is the number of members in that set. Take a set like {1, 2, 3, 4}. How many members are there in that set? There are four. So four is the number of the members in that set. If you take all of the natural numbers – all of the counting numbers – how many numbers are in that series? The answer is aleph-null. It is the first letter of the Hebrew alphabet. It is just an arbitrary symbol, they could have used anything. It is the concept I want you to grasp, not the symbol. The idea is that infinity in this case is complete. It is entire. It is not merely a limit that you are striving toward endlessly but you never reach. Rather the idea of an actual infinite is that it is complete. There really are an actually infinite number of things in the collection, as opposed to a potential infinite which is something that is always finite but just always endlessly increasing on and on and on forever toward infinity, but it never gets there. The actual infinite is there, so to speak.

Student: [inaudible]

Dr. Craig: I don’t think that. It is not contradictory. If that were true you would have to throw out set theory. I was taught my arithmetic by set theory. When we had the new math back when I was in grade school, that is what they taught us – sets and things. That is not inconsistent. It is not logically contradictory.

Student: [inaudible]

Dr. Craig: Ah, well, that is a different question. We are just talking about the concept, not whether or not it exists.

END DISCUSSION

Here is the argument:

1. An actually infinite number of things cannot exist.

2. A beginningless series of events in time involves an actually infinite number of things.

3. Therefore a beginningless series of events in time cannot exist.

Let’s say a word about each of these premises.[2]

An actually infinite number of things cannot exist. When it is argued by these philosophers that an actually infinite number of things cannot exist, what they mean is it cannot exist in reality. It doesn’t mean that the concept is inconsistent; that it involves a contradiction. They are not saying this is logically impossible, but what they are saying is that it is really impossible. You can’t really have an infinite number of things. In order to show this they devised all sorts of paradoxes and absurdities that would result if you could have an actually infinite number of things.

Let me just use one of my favorites – the so-called Hibert’s Hotel, which comes from the great German mathematician David Hilbert (perhaps the greatest mathematician of the 20^th century). To warm up, Hilbert invites us first to imagine a finite hotel; that is to say a hotel that has a finite number of rooms in it. Further he says let’s suppose that all of the rooms are occupied. There is no vacancy in the hotel. Now suppose a new guest shows up at the front desk asking for a room. “Sorry,” the proprietor says. “All the rooms are full.” And the guest has to be turned away.

But now Hilbert says let’s suppose what would happen if there were an infinite hotel. That is to say a hotel which has an infinite number of rooms. And let’s suppose once again that all the rooms are full. This is absolutely crucial to understand. There is not a single vacancy throughout the entire infinite hotel. Every room in the hotel is occupied. Now suppose that a new guest shows up at the desk asking for a room. “Of course! Of course!” says the proprietor. He immediately shifts the guest who was in room #1 into room #2. He shifts the guest who was in room #2 into room #3. He takes the guest who was in room #3 and puts him in room #4. And so on out to infinity. As a result, room #1 now becomes vacant, and the new guest gratefully checks in. But remember, before he arrived all the rooms were full.

Moreover, according to the mathematicians, there are no more people in the hotel after the new guest checked in than there was before. The number is just infinite. But how could there not be any more people in the hotel? The guy just took his keys and walked down the hall. How could there not be one more person in the hotel? But according to the mathematicians it is just the same number – it is infinite.

In fact, Hilbert says let’s suppose that an infinite number of new guests show up asking for a room, and all of the rooms are already occupied. All the rooms are full. “No problem! No problem!” says the proprietor. He shifts the guest who was in room #1 into room #2. He takes the guest who was in room #2 and puts him in room #4. He takes the guest in room #3 and shifts him into room #6. He puts every former guest into the room number double his own out to infinity. As a result, all of the odd numbered rooms become vacant, because any number multiplied by two is always an even number. So you have an infinity of empty odd numbered rooms, and the infinity of new guests gratefully check in. And yet, again, before they arrived all the rooms were full.

In fact the proprietor could repeat this operation an infinite number of times, over and over again, and always be able to accommodate infinities of infinities of new people in the hotel. And yet, according to the mathematicians, there would never be any more people in the hotel, even though there is just as many new guests that checked in as there were old guests who were already in the hotel. It is just the same number – infinity.

But Hilbert’s Hotel is even weirder than the great German mathematician made it out to be. Because it occurred to me as I thought about Hilbert’s Hotel when I was studying this in my doctoral work in Birmingham, “What would happen if some of these guests started to check out?” Suppose the guest in room #1 checks out?[3] Isn’t there one less person in the hotel? Not according to the mathematicians. [laughter] In reality you can’t prevent a guest from checking out. He can climb out the window if you tried to bar him at the door. You can’t keep real people from checking out of a real hotel. And yet if he did check out there would still be the same number of people in the hotel – no fewer people. But don’t ask housekeeping who has to go make the bed. In fact, suppose you had all of the odd numbered rooms check out? In this case an infinity of people will have left the hotel, but according to the mathematicians there will be no fewer people in the hotel than there were before. But don’t talk to the people in housekeeping who have to go clean all those rooms.

You might think that by means of this sort of operation the proprietor could always keep his hotel full of guests because what he could do is – say he doesn’t like having a half-empty hotels (all the odd numbered rooms vacant). By simply dividing every number he can shift the people as before so that the hotel will be full again. Every room will be occupied. You might think in this way the proprietor could always keep the hotel full. But in fact that would be mistaken. Ask yourself, what would happen if the guests in room numbers, say, 4, 5, 6, out to infinity checked out? At a single stroke the hotel would be reduced from infinity to finitude. There would be only three people left in the hotel. The infinite would be transformed into the finite. Yet, exactly the same number of people checked out this time as when all of the odd numbered guests checked out. In the one case, infinity minus infinity is infinity. In the other case infinity minus infinity is three! In fact, you do come up with a logical contradiction here; namely, you subtract identical quantities from identical quantities and you get non-identical results. That is why inverse operations (that is to say, operations of subtraction and division) are prohibited using these transfinite quantities. You cannot say, “What is aleph-zero minus aleph-zero?” There is no such quantity. That operation is simply forbidden. But as someone earlier rightly pointed out in his question, while you can prohibit the mathematician from doing certain operations with numbers, you can’t stop people from checking out of the hotel. You can’t keep people in reality from leaving the hotel if they want to.

I think in the end these illustrations go to show that the idea of an actually infinite number of things is inherently absurd, and therefore an actually infinite number of things cannot exist. Since nothing hangs on the infinite being a hotel (you could have done this illustration with books in a library and checking the books out of the library rather than people checking out of a hotel – nothing hangs on the hotel illustration), I think these kinds of absurdities show in general that the real existence of an actually infinite number of things is really impossible. It is absurd.

That would be, I think, the best way to defend that first premise.

START DISCUSSION

Student: [inaudible]

Dr. Craig: He asks, “What about the infinity of God? Wouldn’t this prove that God is finite? That God cannot be infinite.” That is why I emphasized so strongly that you understand the concept of the actual infinite. When theologians or philosophers talk about God’s infinity, they do not mean a mathematical quantity. They are not talking about a mathematical quantitative concept of being composed of an actually infinite number of pieces or parts or having an actually infinite number of members. If you will, the infinity of God is a qualitative notion, not a quantitative notion. The infinity of God means he is eternal, omniscient, omnipotent, perfectly holy, necessary, self-existent, and all of the other superlative attributes of God. So I don’t think there is any contradiction at all with affirming the infinity of God and saying that an actually infinite number of things cannot exist because God’s infinity is not a quantitative concept involving an actually infinite number of finite pieces or members.[4]

Student: [inaudible]

Dr. Craig: That is a good question you are raising. I think that these problems are not just practical problems. The problem with Hilbert’s Hotel is not that there wouldn’t be enough wood to build such a big building or something of that sort. These are conceptual problems, I think, that are inherent in the very idea. Hilbert’s own view, if you are asking about that, was that the realm of mathematics and of set theory is a creation of the mind of the mathematician. He thought it was the most fertile and creative invention of the human mind, but that it has no existence in the real world. It is purely conceptual.

Student: [inaudible]

Dr. Craig: You raised a couple of things. He points out that there is a whole series of these infinite numbers, and that is quite right. There is aleph-zero, aleph-one, aleph-two, and so on. There is a whole series of these transfinite numbers. That is true. But if aleph-zero can’t exist neither can the higher infinite numbers. That would again not say anything to undercut this field of mathematics but simply to say that this is a purely conceptual realm, not something that exists in reality. You also said what about what we talked about last week when we talked about the existence of angels? Remember Thomas Aquinas, I said last week, thought that angels existed in a kind of special time called the avum which is a sort of time that is like ours but it is different from ours in that it is not measured by any physical clocks. What I want to argue here is that what you cannot have would be a beginningless series of events. Whether those would be events in the mind of an angel or they would be physical events would be immaterial; so long as you have distinct countable events you are going to run into the problem that is raised in that you would have an actually infinite number. Let me just hold on that until we get to point 2.

Student: [inaudible]

Dr. Craig: Yeah, she asked about Hawking. He would be another example. Stephen Hawking, the great Cambridge astrophysicist who has Lou Gehrig’s Disease and is wheelchair bound, also uses mathematical concepts like imaginary time to try to avoid the beginning of the universe. Imaginary time would be using imaginary numbers for the time variable in his equations. Imaginary numbers are numbers like the square root of -1. There isn’t any real number that is the square root of -1 because any real number times itself is always a positive number. These imaginary numbers are used in electrical engineering and quantum physics as mathematical devices or tricks to make the equations work – to make the equations tractable. But when you get to the final result, you always convert back into real numbers in order to have a physically significant result. I think the mistake that Hawking makes is that he invests these mathematical ideas with physical reality when in fact they are unintelligible. What sense would it make to talk about the imaginary number of people in my Sunday School class today? What was the imaginary attendance? What would it mean to talk about the imaginary passage of two minutes – the passage of two imaginary minutes of time? These are physically unintelligible concepts. These would all be examples of how things can exist as mathematical ideas or concepts, but that doesn’t automatically mean they can translate into reality.

Just another very simple example of this would be: suppose I have three apples on my desk. I take away five apples. How many are left?[5] The mathematician has no trouble. 3 minus 5 is -2. But in reality that is absurd because there is no such thing as negative apples. You can’t take away 5 apples from 3 apples and have -2 apples left over. This is just another illustration that the ability to do things with mathematical numbers doesn’t automatically mean that this is possible in reality.

That leads me to say that many, many people will respond to this first premise by saying this has been invalidated by modern set theory that springs from the German mathematician Georg Cantor who showed how we can talk about infinite sets. He developed all of this about aleph and the series of alephs and so forth. But you see none of that says anything against this first premise because we are not denying that the actual infinite is a fruitful and consistent mathematical idea. Not at all! What one is denying is that it can be translated into the real world of people, and pennies, and chairs, and hotel rooms. Then it results in absurdities.

Student: [inaudible]

Dr. Craig: Well, 2 isn’t infinity, but in geometry, if you think of a line as composed of points, then between any two points you pick there is an infinite number of points in between. But again geometry is mathematics and, as I said, physically there aren’t an infinite number of points between these two edges of the podium, but mathematically you can talk about a line being composed of an infinite number of points.

Student: [inaudible]

Dr. Craig: No, that would be a contradiction to say 2 is infinity. What you would say is that there is an infinite number of points between 1 and 2 and that 1 and 2 are the endpoints of that series. Maybe what you are having difficulty grasping is that in this case the infinite has an end. That is not problematic in mathematics that the infinite should have an end. Here is an infinite series of points. It has one end in 1 and has another end in 2. So it has endpoints. But there is an infinite number of points in between. That sounds odd but mathematically that is not a problem.

Student: [inaudible]

Dr. Craig: I think once you grasp the illustrations, this can actually be a lot of fun sort of thinking up these kind of thought experiments to show the odd things that would result if you could have, say, an infinite number of baseball cards or an infinite number of pennies or something of that sort.

END DISCUSSION

In the interest of time, let me just go on to the second premise. The second premise is that a beginningless series of events in time involves an actually infinite number of things. I think this is obvious. If prior to today there has been an infinite series of events going into the past and the universe never had a beginning, then how many events have their been prior to today? Well, an actually infinite number. You can’t say that it was potentially infinite because for the number of events in the past prior to today to be potentially infinite you would have to say that it is finite but growing toward infinity in a backwards direction, which is impossible. The past isn’t growing backwards. The series of events is going forward as each event happens one after another. The arrow of time is from past to future not from present to past. If the series of past events is beginningless then the number of events in the past would be infinite, and not merely potentially infinite; it will be actually infinite. So a beginningless series of events in time would involve an actually infinite number of things, namely an infinite number of past events. Again, you can give illustrations for this. Imagine, for example, that each event is a domino falling. You have a series of dominoes that fall, say, at one per second or something like that, and it goes into the past. How many dominoes have fallen prior to the present domino falling? It would be an actually infinite number of dominoes if the series of dominoes never had a beginning.[6] So you can make the events be the falling of the dominoes if you will.

START DISCUSSION

Student: [inaudible]

Dr. Craig: That is what the argument is. The argument is “therefore a beginningless series of events cannot exist.” In a sense, you are right. We are trying to say, “If the universe never began to exist – if the series of events in the past never began to exist – then there will have been an actually infinite number of past events.” But an actually infinite number of things cannot exist, so it follows that a beginningless series of events in time cannot exist. That is to say, the series of events in time must have had a beginning. Since the universe is not distinct from the series of events in time then that means the universe began to exist, which is an argument for creation out of nothing. There had to be a first event, and therefore a first uncaused cause. That is the idea. You can’t have an infinite regress. You have to have a finite number of events that goes back to a first uncaused cause.

Student: [inaudible]

Dr. Craig: This is a very good question. What would this imply about God and his relation to the world? If this argument works, it means that God cannot have endured through an infinite number of prior moments before creation. You shouldn’t think of God as sitting around idly twiddling his thumbs from creation until finally a finite time ago he created the world. Rather, St. Augustine in dealing with this question said, What was God doing prior to creating the world? He said the answer should not be that he was preparing hell for those who pry into mysteries, which is what some people said. He said, no, what this means is that God brought time into existence at the moment of creation. So the creation of the universe is also the creation of time itself. So there is no moment before creation. To say “What was God doing one hour before creation?” is to use words without meaning. There is no such moment as one hour before creation because time was brought into being at the moment of creation. On this view God would exist timelessly without the universe, and then creation will be the coming into existence of space and time itself.

Student: [inaudible]

Dr. Craig: The question was: can we have relations “prior” to the beginning? What about relations in the Trinity? We talked about this (for those of you who were with us when we did the attributes of God); what this would imply is that the relations among the persons of the Trinity must be timeless. That is to say, they do not require time in order to elapse. There is a wonderful doctrine among the early church fathers called perichoresis in Greek. The doctrine of perichoresis in the church fathers is that the three persons of the Trinity are wholly transparent to one another in the sense that all that the Father loves, the Son and the Spirit love. All that the Father wills, the Son and the Spirit will. All that the Father knows the Son and the Spirit know. Similarly with the other persons of the Trinity. There is nothing new that could arrive within the Godhead that the Son would say, “Oh, I didn’t know that before” or the Spirit would say, “I didn’t love that before and now I do.” Rather, there is this complete inter-penetration of the persons of the Trinity of one another so that I think of these relations as timeless changeless relations among the persons of the Trinity. The relations of knowledge, love, and will that are perfect and changeless and timeless. I think we can get a human analogy for this if we think of two lovers who are, say, just sitting across the table from each other at the restaurant looking into each others’ eyes, not saying a word, and will sometimes say that they were lost in that timeless moment.[7] I think that is kind of a human analogy of the love relationship between the persons of the Trinity. It is a timeless changeless inter-pentetration of love, knowledge, and will among the persons of the Trinity. The wonder of creation and of redemption is that this perfect being complete in himself would create us finite persons and invite us into the fellowship of this inter-trinitarian love relationship as adopted sons and daughters of God. It really inspires worship, I think, when you think of all that that entails – the condescension on God’s part.

Student: [inaudible]

Dr. Craig: Yes. He asked could you use a potentially infinite past to calculate some quantity in physics? Yes, absolutely. Very often I will read in physics papers about the boundary conditions at infinity in the past. What they have in mind here is that you project back to this limit and say, “What would obtain at that limit?” You can talk about boundary conditions or even that the universe was created at this limit of infinity. But, again, that is simply a mathematical way of beginning in the present and then thinking back to infinity as a limit. But clearly, in reality, that is not the way the events happen. The events happen one after another forward in time, not going back. But, again, this is just another illustration how you can do things with mathematics that may not correlate with reality.

Student: [inaudible]

Dr. Craig: Look at your outline, point (b): scientific confirmation, Big Bang cosmology. We will hit that later. These are purely philosophical arguments that were already known in the Middle Ages, though I have given you an updated version using modern set theory and Cantorian concepts of transfinite arithmetic to show you that even though these ancient arguments are very old, nevertheless they are fully defensible today in light of modern mathematics and set theory. But, as you rightly note, modern cosmology has arrived at very similar conclusions. When you trace the expansion of the universe back in time, you come back to a point in the finite past (about 13.7 billion years ago) before which the universe literally did not exist. It would be meaningless to say, “What was there one hour before the Big Bang?” That is as meaningless as saying, “What was God doing one hour before creation?” There is no such moment because on modern cosmology space and time come into being at the moment of the Big Bang and therefore there is no prior moment just as philosophically or theologically we shouldn’t think of God sitting around idly prior to creation waiting to create the world. Rather, space and time come into being at the moment of creation.

Student: [inaudible]

Dr. Craig: She asks, “Is it meaningful to ask what is beyond the universe?” Notice how subtle that question is here. She is a very subtle thinker. She doesn’t say “What was before the universe?” That would be meaningless, right? But what is beyond the universe? I think we can give sense to this. One way would be to talk about what is causally prior to the universe. I would say the answer to that is God. God is not temporally or chronologically prior to the universe, but he is causally prior to the universe. In that sense you would say God is beyond the Big Bang. Not before the Big Bang but beyond it in a causal sense. He is the cause; the Big Bang is the effect. I think you can make sense of it in that way. You could also say that there is a beyond the universe if the universe were embedded in some higher dimension. This is what Hugh Ross has suggested. Our 4-dimensional universe is embedded in some higher dimension. I think that is wrong. I think that is just a misconception. But nevertheless it is a conceivable way of construing beyond – as a higher dimensional reality. I would want to say that God is not before creation, but he is beyond creation in the sense that God is the cause and the creation is the effect.

END DISCUSSION

With that we will close today. Next time we will come back and look at the second argument for creation which is the argument based on the impossibility of traversing an actual infinite, or crossing an actual infinite.[8]

[1] 5:05

[2] 10:14

[3] 15:01

[4] 20:14

[5] 25:00

[6] 30:00

[7] 35:09

[8] Total Running Time: 40:42 (Copyright © 2008 William Lane Craig)