Existence of God (part 7)

October 10, 2010     Time: 00:42:41


The Kalam Cosmological Argument continued.

Excursus: Natural Theology
§ II. Kalam Cosmological Argument
Lecture 2

We have been talking about the Kalam Cosmological Argument for God’s existence, and last week we looked at the first premise of that argument, that everything that begins to exist has a cause. And I gave three reasons as to why I thought this premise is more plausible than its opposite.


Question: Can you explain what the word “kalam” means?

Answer: It is an Arabic word, and it literally means “word” or “speech.” During the Middle Ages, during the high tide of Islamic theology, kalam came to have a more extended meaning to mean any statement or position of doctrine. The whole movement of medieval Islamic theology came to be called kalam. This is still used today. When I was in Turkey this spring speaking at the University of Ankara, I spoke at the Department of Kalam. These were practitioners of kalam, to whom I was speaking on the incarnation. It was just a wonderful experience. One of the gentlemen, after I finished, said, “You are a better practitioner of kalam than most of our philosophers!” So that was an encouragement to me. The reason why I called this the Kalam Cosmological Argument is because, although it has its roots in Christian theology, when Islam conquered North Africa it absorbed the Christian theological traditions that existed in places like Alexandria in Egypt. This argument was taken up into medieval Islam, and then it was highly developed, so some of the most sophisticated forms of this argument were on the part of these Muslim medieval theologians. So for that reason, the honorific title is well deserved.

The argument, however, also was defended by Jewish medieval theologians who rubbed shoulders with the Muslim theologians in Spain, while Spain was under Muslim control. And through the Jewish theologians, this in turn was mediated to the Latin-speaking, Christian, European West where people like St. Bonaventure adopted this argument and defended it. So the argument has a broad appeal having been propounded by Jews, Muslims, and Christians both Protestant and Catholic. It started with the Christians and their attempts to refute the Greek doctrine of the eternity of matter and the eternity of the universe. These early Christians wrote works against Aristotle. And when Islam took over North Africa, they absorbed this tradition. Since Muslims also believe in creation out of nothing, they found these arguments congenial. It is a nice bridge-building argument for sharing your faith with the Muslim, I must say, because you are on common ground here. Muslims just love it when I come and talk about this argument, but I usually then will try to transition to talking about Christ as well.

Question:Kalam” is also daily used to refer to the Scripture – to the Qur’an. So Muslims, whenever they refer to kalam, they all know it refers to the Qur’an.

Answer: Ah-ha! So in the same way that we call the Bible “The Word!” Very interesting! Thank you!

The Universe Began To Exist

Let’s go to that second premise, which is more controversial, that is, the universe began to exist. And what I want to do is present two philosophical arguments in support of this premise and then two scientific confirmations. So we have philosophical grounds or metaphysical grounds for believing this premise, and then we have scientific confirmation of those arguments from empirical evidence.

First Philosophical Argument

Let’s talk about the first philosophical argument, which is the argument based upon the impossibility of the existence of an actually infinite number of things. Remember we took as our point of departure the medieval Muslim theologian al-Ghazali’s statement of the argument.1 And he argues that if the universe never began to exist, then there must have been, prior to today, an infinite number of previous events. The series of events just goes back and back without beginning, and therefore the number of events that have transpired prior to today is infinite. But he argues an actually infinite number of things cannot exist because this will lead to various absurdities.

In order to understand al-Ghazali’s argument, it is very important that we grasp a distinction concerning the infinite, namely, the difference between an actual infinite and a potential infinite. Let me say something about the potential infinite first, as I think this is the more familiar usage. Al-Ghazali had no problem with the idea that there could be a potentially infinite number of things. What he wants to deny is that there can be an actually infinite number of things. When we say that something is potentially infinite, infinity serves merely as an ideal limit, which never actually exists but which one can endlessly approach. For example, any finite distance can be subdivided into one-half, then one-fourth, then one-eighth, then one-sixteenth, on to infinity. But you will never arrive at an “infinitieth” division. This infinity is merely potential – it serves as the ideal limit which you endlessly approach, but you never actually get there. The symbol for this sort of infinity is the so-called lazy-eight symbol, “∞.” That signifies of the idea of a potential infinite.

Now by contrast with this, an actual infinite is not growing toward the infinite as a limit – rather it is infinite. It is a collection having a number of members which is in excess of any natural number 1, 2, 3, 4 . . . . Any finite number you can think of, the number of members in this collection is greater than that. It is complete; there are an actually infinite number of things existing in the collection. The symbol that is used for this type of infinity in modern mathematics is the Hebrew letter aleph “א.”

Al-Ghazali has no problem with the idea of a potentially infinite number of things because this just means an ideal limit – you just go on and on and on, and you never actually get there. But he wants to argue that an actually infinite number of things cannot exist because various absurdities would result. So there could not be an actually infinite number of coins, or an actually infinite number of chairs, or an actually infinite number of particles, or something of that sort. That would mean that if you cannot have an actually infinite number of things, you couldn’t have an actually infinite number of events. The number of past events would have to be merely finite. If the number of past events is finite, then the universe cannot be beginningless. Rather the universe must have begun to exist, which is the second premise of the argument.


Question: Would you call a numbering system an actual infinite because you could theoretically keep on counting forever?

Answer: This is disputed by philosophers of mathematics. Some mathematicians think that the natural numbers 1, 2, 3, 4 . . . are merely potentially infinite in the sense that you construct the numbers by adding 1 and that could go on potentially infinitely. But other mathematicians say, no, the set of all natural numbers just exists – it is just complete, and they are all there. Therefore, there are an actually infinite number of natural numbers. That is a disputed question. The vast majority of mathematicians would say that there are an actually infinite number of natural numbers, but there is a minority called “intuitionists” or “constructivists” who would say, no, it is merely potentially infinite.2 But the majority view would be that the number of numbers is actually infinite.

Question: What would be another example of an actual infinite?

Answer: Right! This is a very good point – it is hard to think of any examples in physical reality and nature of anything that is actually infinite. But I think we can imagine, say, that the universe goes on forever spatially and that per unit-area there is one star. If there is an infinite number of areas, there would be an actually infinite number of stars. It is not as if you are adding new stars all the time, but you can imagine, if the universe is spatially infinite, that there would be an actually infinite number of stars. Now there is no evidence that that is the case. In fact, I think the contrary is true. But that would give you at least an idea of what we are talking about when we say an actual infinite. But I certainly agree with you that it will be futile to try to refute this argument by saying to al-Ghazali, “Look, here is an actual infinite!” I think in every case he will be able to re-interpret that and say, no, that is merely potentially infinite or finite. I think it would be futile to try to refute him by finding a counter-example and saying here is an example of something that is actually infinite.

Question: Since this seems to be concerning an infinite number of things, would this apply to God, then? Could we say that God is infinite?

Answer: OK, this is a good question that is often asked. This argument is worded in such a way as to stave off this misunderstanding: “An actually infinite number of things cannot exist.” If we think of God as infinite, he isn’t a collection that is composed of an actually infinite number of things. So the infinity of God, as that phrase is used by theologians, isn’t a mathematical concept. It is not a quantitative concept, if you will; it is a qualitative concept. The infinity of God means that God is, for example, eternal, necessary, morally perfect, omnipotent, omniscient, omnipresent; he has all of these superlative attributes. But the infinity of God is not a quantitative, mathematical concept. So it simply doesn’t fall under the case of an actually infinite number of things’ existing.

Question: Would time be a potential infinite?

Answer: This is a really good question! Al-Ghazali is going to argue that time is finite in the past because otherwise we are going to have an actually infinite number of past hours or days or whatever. But what about the future? Is the future actually infinite? This depends upon your theory of time. In this class, we have differentiated two different theories of time which we called the A-Theory and the B-Theory. You may remember that the A-Theory is the view that the future does not exist in any sense. The future is pure potentiality. It is not as though your supper tonight is sort of existing up ahead out there, and you are waiting to get to it somehow. Rather on the A-theory things come into being, and they go out of being. On the A-Theory, if the future should go on forever, it would be an example of a purely potential infinite, so time would be potentially infinite. For any point in time you pick toward the future, there will only be a finite number of events that have occurred, but you keep adding new events every day or every second or whatever, so that the future is potentially infinite. That is not problematic. On the B-Theory, this is the view that everything in time exists. Whether past, present, or future, everything is equally real.3 Now if time goes on forever in the B-Theory, then you would have an actually infinite future; you would have an actually infinite number of future events and an actually infinite number of past events. So what this argument would require on the B-Theory is that time would come to an end. Time would have to have a beginning, and time would have to have an end. It would have to be like a yardstick that has the first inch and a thirty-sixth, or final, inch, and that is the end of the yardstick. On the A-Theory, time can go on forever because it is merely potentially infinite. On the B-Theory, time could not go on forever, if this argument is correct. It would have to be finite in the future.

It is frequently alleged that this sort of argument has been invalidated by developments in modern mathematics: it was fine for al-Ghazali’s day in the 11th century, but today it has been overtaken by today’s modern math. For example, the use of infinite sets in modern mathematics is commonplace. Things like the set of natural numbers {0, 1, 2, 3 . . .} is said to be an infinite set. It has an infinite number of numbers in it. The number of members in the set of natural numbers is not merely potentially infinite; rather in modern set theory, the number of members in the set of natural numbers is actually infinite. There is an infinite number of numbers in the set. And many people will say that these developments have undermined al-Ghazali’s argument because now we see that it is perfectly reasonable to talk about an actually infinite number of things.

However, I think this objection is misconceived. What these developments merely show is that if you adopt certain axioms and certain rules, then you can talk about actually infinite quantities in a perfectly consistent way without contradicting yourself. And all this accomplishes is showing that you can set up a certain universe of discourse about infinite quantities. By a universe of discourse, I simply mean you can talk about these things in a self-consistent framework. But this does absolutely nothing to show that mathematical entities really exist or that an actually infinite number of things can really exist. If al-Ghazali is right, this universe of discourse that is employed in set theory and modern mathematics is just a sort of a fictional realm. It is like the world of Sherlock Holmes. In that fictional world, Holmes lives on Baker Street, has a companion Dr. Watson, does all sorts of exploits, has a housekeeper named Mrs. Hudson, and so forth. Those are all true in that fictional world created by Sir Arthur Conan Doyle. But none of those things actually exists. And similarly, all that modern mathematics shows is that if you set up certain axioms and rules, you can talk about this universe of discourse in a consistent way without contradicting yourself. But it doesn’t show that it is something that exists in reality rather than just something that exists in your mind.

Moreover, it is worth pointing out that al-Ghazali’s argument is not that the existence of an actually infinite number of things is a logical self-contradiction. He is not claiming that to talk about an actually infinite number of things is somehow self-contradictory. Rather his argument is that the existence of an actually infinite number of things is really impossible. To give you an analogy, take the statement, “Something comes into being uncaused out of nothing.” There is no logical contraction in that statement. There is no self-contradiction in saying, “Something came into being uncaused out of nothing.” Nevertheless, I think it is plausible to say that that statement is really impossible. It is really impossible for something to come into being uncaused out of nothing, even though there is no strict logical contradiction in saying so. Al-Ghazali is not claiming that the concept of an actually infinite number of things is a self-contradictory concept; he is simply arguing that it is really impossible for an actually infinite number of things to exist.4 We will see why in a moment.

These modern mathematical developments, far from undermining al-Ghazali’s argument, actually can support it. It can help it by giving us insight into the nature of the actual infinite and helping us to see what it would be like if an actually infinite number of things really could exist. Far from undermining his argument, these modern mathematical developments actually are quite helpful and give insight into his argument.


Question: About the idea of an actually infinite number of things’ existing, you gave the example of the stars. For it to be actually infinite, would that mean that all the space in the universe would have to be taken up by the stars?

Answer: No, it wouldn’t mean that all the space had to be taken up because you would just need to have one star for every, say, cubic million miles or something like that. For every cubic million miles, there is one star in it. Well, if you have an infinite number of those cubic volumes, you have got an infinite number of stars. But obviously, there is a lot of empty space, too, because you could put more things in them. So don’t think it would be just jam-packed. That is one of the paradoxes of infinity, in fact. It doesn’t really fill up everything, even though you might have an infinite number of things. Here is another example. Think of a wall that has an infinite number of bricks in it. The wall just goes off to infinity, and it has an infinite number of bricks in the wall. Does that mean it would fill all of reality? Well, no – you can have another wall going in the other direction that would also have an infinite number of bricks in it, too. The two walls could come to a kind of doorway, and on the left is one infinite wall and on the right is another one. You would have two infinities there. It would not fill up everything, even though an infinite volume would be occupied by each wall.

Question: (inaudible)

Answer: That is in dispute. In a sense, this is an argument to try to show that the past is not actually infinite – that the past had a beginning, is finite, and therefore there must be something that brought the universe into being. Now in terms of future time, think again what I said about the difference between the A-Theory and B-Theory. Someone like al-Ghazali, I think, would be an A-Theorist, and so he would think that time is infinite in the future, but only in the potentially infinite sense, which is unobjectionable.

How would al-Ghazali show the real impossibility of the existence of an actually infinite number of things? It will be by imagining situations in which an actually infinite number of things do exist and then drawing out the absurd consequences from such a thing. Let me share with you one of my favorite illustrations, which comes from the great German mathematician, David Hilbert, who was perhaps the greatest mathematician of the 20th century. This concerns a thought experiment that Hilbert developed that has been called Hilbert’s Hotel.

Hilbert first invites us to imagine an ordinary hotel with a finite number of rooms. Let’s suppose that all of the rooms are full. There is a guest occupying every room in the hotel. If a new guest shows up at the desk asking for a room, the hotel manager says, “Sorry, all the rooms are full,” and the new guest is turned away. But now, Hilbert says, let’s imagine instead a hotel with an actually infinite number of rooms. Let’s suppose once again that all the rooms are full. This needs to be clearly understood. There is not a single vacant room throughout the entire, infinite hotel. Every room in the hotel has a flesh and blood person living in it. There is no vacancy; every room is already occupied. Now suppose a new guest shows up at the hotel asking for a room.5 “No problem!” says the manager, and he moves the person who was in room 1 into room 2. He moves the person who was in room 2 into room 3. He moves the person who was in room 3 into room 4, and so on out to infinity. As a result of these transpositions, room 1 now becomes vacant, and the new guest is easily accommodated. And yet, before he arrived, all the rooms were full!

Now, Hilbert says, let’s press this a little further. Suppose an infinite number of new guests show up at the desk asking for a room. “No problem!” says the proprietor. And he moves the guest who was in room 1 into room 2. He takes the guest who was in room 2 and moves him into room 4. He takes the guest in room 3 and puts him in room 6. He takes the guest from room 4 and puts him in room 8. He moves every person into the room number that is double his own – 1 into 2, 2 into 4, 3 into 6, 4 into 8, and so on, out into infinity. Since any natural number multiplied by 2 (or doubled) is always an even number, all of the guests wind up in even-numbered rooms. As a result all of the odd-numbered rooms now become vacant, and the infinity of new guests is easily accommodated! And yet before they arrived, all of the rooms were already occupied!

In fact, the proprietor could do this an infinite number of times and always be able to accommodate more guests by these sort of transpositions. And yet, each time, the hotel is already full. As one student remarked to me upon hearing this in class, “If Hilbert’s Hotel could exist, it would have to have a sign posted outside saying, ‘No Vacancy. Guests Welcome’.”

But Hilbert’s Hotel is even stranger than the great German mathematician made it out to be. Ask yourself this question, which occurred to me as I thought about Hilbert’s Hotel: what would happen if some of the guests started to check out? What would happen then? Suppose all of the guests in all of the odd-numbered rooms checked out – 1, 3, 5, and so on, out to infinity. In this case, an infinite number of people have left the hotel. Indeed as many people have left the hotel as still remain in the hotel in the even-numbered rooms. And yet, according to the mathematicians, there are no fewer people in the hotel. The same number of people is still in the hotel, namely, just an infinite number.

Now suppose the manager doesn’t like having a half-empty hotel – it looks bad for business, having half the rooms vacant. No problem! By moving the guests in the reverse order than he did before – 2 into 1, 4 into 2, 6 into 3, etc. – he turns his half-empty hotel into a hotel that is completely booked and every room is full, bursting at the seams.

You might think that by these maneuvers the manager could keep this strange hotel always fully occupied, so that he would never have empty rooms. But in fact you would be wrong! Suppose, instead, this is what happens. Suppose the guests in rooms 4, 5, 6, 7, 8, and so on, out into infinity, check out. In that case, the hotel will be virtually empty. There will be only three guests left in the hotel – in rooms 1, 2, and 3. The guest register will be reduced to but three names, the infinite will have been converted to finitude. And yet, in this case, exactly the same number of people have left the hotel as when all of the odd-numbered guests checked out!

Could such a hotel really exist? I don’t think so. It seems to me that Hilbert’s Hotel is absurd. Since nothing hangs on the illustration’s involving a hotel, you could substitute any sort of physical reality for it. I think the argument could be generalized to show that the existence of an actually infinite number of things is absurd.6


Question: It seems like you have to keep adding more people at the end when you do this. I don’t quite get it.

Answer: I know that is what it seems. We think of the hotel as being a potential infinite, don’t we? But what happens when he moves these people from 1 into 2, 2 into 3, 4 into 6? It seems as though somewhere out there, at infinity, somebody is falling off the edge, or there is a new room being created. But that is not it! That would be like a potential infinite. But in an actual infinite – and this is unimaginable – those rooms just go on forever, there is no end room, they are all occupied, every room in the hotel is full, and yet, just by moving people around, you can magically create more space or fill up space just by moving people from room to room.

Followup: I must be missing something. If you are saying that in the beginning the thing is totally full out to infinity, then that means there is nothing left that is open.

Answer: Yes, that is right! You are not missing something; you are just seeing the very, very paradoxical nature of the actual infinite that Hilbert meant you to see by developing this illustration. He wants you to feel uncomfortable with this illustration. The person who just sort of blows it off, I think, hasn’t really thought about it in the way you have. There isn’t any more room for people, where did these people go? Well, they just go into the rooms that we’ve said and just by moving them around, somehow it works out. The question is – could such a thing really exist?

Followup: I think this argument is kind of a fallacy. It doesn’t make too much sense. It seems like it contradicts itself.

Answer: No, it is not contradictory. That is what I want to emphasize. There is no logical contradiction here, except with regard to the subtraction business, where I said you subtract the people when they check out. As I will say in a minute, there you really do get a live contradiction. But at least on the other ones, I think what it suggests is that it is just metaphysically, or really, impossible for this to happen. But there isn’t any sort of self-contradiction, at least in the first part of the story, as Hilbert tells is.

Question: I have to say I am confused in general with the examples you are using. You used an example with space, and you refer to time. Space has volume, but time has nothing to do with volume. What is the relationship there? I do not understand.

Answer: I guess it would just be that things in time or in space can be counted. Therefore, we can talk reasonably about, for example, how many U.S. Presidents have held office. We can talk about what day of the month it is. Or how old you are. All of those involve counting things, which are things that have existed in time. Or we can count how many chairs there are in this room, or how many particles there are in the universe. Things of that sort. It seems to me that whether things exist in time or in space, the key here is that these things can be enumerated and that, therefore, we can talk about there being an actually infinite number of them. Have there been, for example, an actually infinite number of past U.S. Presidents? Well, no, obviously not! It is merely a finite number.

Question: Are we saying that space is an actual infinite? Or is there a limit to the outer boundaries of space?

Answer: If al-Ghazali is right, he would say space is finite. There is a limit; but this is a delicate question. Either there would be a limit or, what modern cosmologists would say, the shape of space is such that it is finite but doesn’t have a boundary. Think of the surface of the Earth – it is finite, but there is no boundary. If you start at the North Pole and go in one direction, you come back to your starting point again. Physicists say that it is perfectly plausible that our three-dimensional space is finite but unbounded in the same way. There is not a limit, but it is not infinite; it is finite.

Question: Mathematically, to be able to take an infinite set of anything, cut it in half or multiply it by two and get the same number, seems to imply something to do with creating and destroying matter. Does that make sense or am I in left field?7

Answer: In the example that we gave from Hilbert, there is no creating of new matter because we just imagine that there is a hotel that already has an infinite number of people in it, and there is a crowd of people outside the door waiting to get in, and there is an infinite number of people in the crowd. What Hilbert shows is how the manager can accommodate that infinite crowd into his hotel, even though the hotel is already fully occupied. So it is not a matter of creating new matter in this story at least.

Sometimes students or laymen will react to Hilbert’s Hotel by saying these absurdities result because the concept of infinity is beyond us and we don’t understand the nature of infinity. We can’t grasp it, and that is why we get these absurdities. But that reaction is just quite frankly naive and mistaken. As I said a moment ago, infinite set theory is a highly developed and well-understood branch of modern mathematics. These paradoxes result, not because we don’t understand the infinite; they result because we do understand the nature of the actual infinite. Hilbert was a very smart guy, and he knew how to illustrate very well the bizarre nature of an actually infinite number of things. Do not think that these absurdities result from a lack of understanding. Quite the contrary, as I say, modern mathematics can give us insight into the nature of the actual infinite, so that we understand how these sorts of things would result.

Really, the only thing that the critic of the argument can say at this point is to just bite the bullet and just say, “Well, Hilbert’s Hotel is not absurd after all. I guess you can have a hotel that is fully occupied with an actually infinite number of rooms that can always be occupied by more people!” Sometimes critics will try to justify this move by saying, if an actual infinite could exist, then these sorts of situations are exactly what you ought to expect. If an actual infinite number of things could exist, then you should expect that it will be like this. But I think that justification is inadequate. Of course, Hilbert would agree that if an actual infinite number of things could exist, then you could have a hotel such as he described. The hotel wouldn’t be a good illustration if that weren’t the case! In order to be a good illustration, it has to be the case that if an actual infinite number of things can exist, then this is what would happen and what it would look like. That is not the question. The question is, is such a hotel really possible?

Moreover, the critics can’t simply bite the bullet when it comes to the guests’ checking out of the hotel because here, you really do have a logical contradiction. Namely, you subtract identical quantities from identical quantities and you get self-contradictory answers. In the one case, infinity minus infinity is infinity, and in the other case, infinity minus infinity is three. The former case is when you subtract all the odd numbers from all the natural numbers, and you get the even numbers left. The latter case is when you subtract all of the numbers 4 and greater from the natural numbers, then you get three left over. In fact, you can get any answer from 0 to infinity for infinite minus infinity. That is why these operations are prohibited in mathematics: because they lead to self-contradictions. You subtract identical quantities from identical quantifies and you get non-identical answers. Therefore you do have a bona fide contradiction here.

In mathematics, you can slap the hand of the mathematician who tries to perform subtraction with infinite quantities. But you can’t stop people from checking out of a real hotel if they want to. If you bar the door, they will jump out the windows! We are talking about real people and real hotels. And there you cannot simply say, “That is against the rules! You can’t do that or a contradiction will result!” It seems to me, in that case, you do have a genuine contradiction as a result. So it is not enough simply to say that this hotel is not impossible because if an actual infinite could exist then this is what we ought to expect.8

That really is the only response that a critic can give, and yet I think it falls short.


Question: Isn’t it a fact that infinity is not a member of the set of all natural numbers?

Answer: That is correct. That is a very good point. Nor is it a successor to that, the series of natural numbers 0, 1, 2, 3 . . . . Don’t think of infinity as being the last number of that series. Don’t think that infinity is in some way connected to it. If you will, infinity stands outside the series, over and above it and is the number of members in the series. Aleph (“א”) is a number. It is an infinite number just as natural numbers are finite numbers.

Followup: I would say that it is a symbol for the number of members. It is not a number itself. That is the important thing, isn’t it?

Answer: You are making a correct distinction, but it is the distinction between a number and a numeral. What I have written here on the board, “א” is a numeral, that is, a symbol. And you can have different numerals; for example, Roman numerals I, II, III, IV. Those are different numerals, but those numerals are symbols for the same numbers as 1, 2, 3, 4. So when I say infinity is a number, I am talking about the quantity that is the number of members in the series. But the use of “א” is just a symbol. It is a numeral. But, as I say, the question is, do numbers and things like that really exist? Well, I don’t think so; it seems they are just fictions or concepts, if you will.

Therefore, in conclusion, al-Ghazali’s argument is a good one. There are no actually infinite quantities of things existing in reality.9




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9 Total Running Time: 42:41