Infinity minus infinity
William Lane Craig on the infinite – Part 2 of ∞
This is the second of a series of posts on William Lane Craig’s treatment of the infinite. It’s a long one, but there are a lot of important topics to cover and points to make, some of which I have not seen anywhere else (especially the ideas in Section 4). So please get comfortable before reading on!
In the first post of the series, Infinite Dreams, I explored Craig’s motivation for rejecting the existence of actual infinite collections, and outlined the form of his argument against them. To briefly summarise:
- Craig mainly wants to argue that the past series of events could not be infinite in order to give philosophical support to the second premise of the Kalam Cosmological Argument: The universe began to exist.
- To do this, Craig attempts to argue that no collection could be infinite.
- And Craig attempts to show that no collection could be infinite by explaining why he thinks certain collections could not be infinite.
In the first post, I highlighted exactly why this kind of reasoning is unsuccessful: even if it was granted that the handful of examples Craig considered were indeed problematic, this doesn’t mean that all examples would be problematic.
However, there is an even bigger problem for Craig. In fact, none of the examples he considers lead to logical absurdities at all, and I intend to explain why this is the case in the next couple of posts in this series. Naturally, this doesn’t lead to the conclusion that there do exist infinite collections. But my purpose here is simply to show that Craig’s arguments against the existence of infinite collections are flawed.
Craig considers a fair few examples, so it is difficult to know where to start. However, in his article Reply to Smith: On the Finitude of the Past, Craig says:
[M]y strongest arguments in favour of the impossibility of the existence of an actual infinite [are] those based on inverse operations performed with transfinite numbers.
I don’t know if Craig still considers these to be his strongest arguments against infinite collections, but he has used these arguments in a lot of his recent debates, so I think it’s fair to suppose he still thinks they are strong arguments. In any case, I think it is probably fitting to deal with these arguments first.
1. Craig’s case against the infinite
Have you ever asked yourself where the universe came from? Why anything at all exists? Well typically, atheists have said that the universe is just eternal and uncaused. But there are good reasons, both philosophical and scientific, which call that assumption into question.
Philosophically, the idea of an infinite past is very problematic. If the universe never had a beginning, that means that the number of past events in the history of the universe is infinite. But the real existence of an actually infinite number of things leads to metaphysical absurdities.
For example, suppose you had an infinite number of coins, numbered 1, 2, 3, and so on to infinity, and I took away all the odd numbered coins. How many coins would you have left? Well, you’d still have all the even numbered coins, or an infinity of coins. So infinity minus infinity is infinity.
But now suppose instead that I took away all of the coins numbered greater than 3. Now how many coins would you have left? Three! So infinity minus infinity is three.
In each case, I took away an identical number of coins from an identical number of coins, and came up with contradictory results. In fact, you can subtract infinity from infinity and get any answer from zero to infinity.
For this reason, inverse operations like subtraction and division are simply prohibited in transfinite arithmetic. But in the real world, no such convention has any sway. Obviously you can give away whatever coins you wish. This, and many other examples, suggest that infinity is just an idea in your mind, not something that exists in reality.
But that entails that past events, since they’re not just ideas but are real, must be finite in number. Therefore, the series of past events cannot go back forever. Rather, the universe must have begun to exist.
Before critiquing Craig’s reasoning, it is worth pointing out that even if his concerns about infinite coin collections were ‘on the money’, this would say nothing about the possibility or impossibility of an infinite past. Unless Craig thinks it is possible to “take away” a (finite or infinite) number of moments from the past, his complaints about infinite coin collections (which are all based on supposed problems with taking away coins) do not translate directly to complaints about an infinite series of past events. If he tried to counter this by saying that we can imagine “taking away” past events, then there would be problems with an infinite future as well, since we could equally well imagine “taking away” future events.
It should also be noted that Craig’s comments here seem to be at odds with his views on time. Craig is a presentist. He defends what is known as an A-Theory of time. Craig summarises his views on time, and contrasts them with B-Theory models, in an interesting (and very accessible) video, The Nature of Time. As stated in that video, his basic position is as follows:
The moments of time are ordered by past, present and future, and these are real and objective aspects of reality. The past is gone; it no longer exists. The present is real. The future has not yet come to be, and is not real. And so the future is not “out there”, ahead of us, down the line, waiting for us to arrive; the future is pure potentiality. Only the present is real.
The reader will of course have noticed that Craig says here that “[t]he past is gone; it no longer exists”, but says above that “past events … are real” when arguing against an infinite past. Since Craig’s problem is with infinite collections of real things, not with infinite collections of non-existent things, it seems that an infinite past should be no problem for him, even if it was granted that actual infinite collections were impossible. If he has a problem with an infinite past, then he may as well have a problem with an infinite future since, on his view, the number of future events is also infinite. (Craig believes in a never-ending afterlife and therefore, presumably, that the number of future events is greater than any finite number. Anyone who says that the number of future events is zero will be quickly proved wrong when the second hand creeps forward another tick.) Attempting to avoid this problem by claiming that future events are not real will not help, because neither are past events real on Craig’s view, so it seems that there could only be a problem with an infinite past if there was also a problem with an infinite future. (Christian philosopher Wes Morriston makes a similar point in his article Beginningless Past, Endless Future, and the Actual Infinite. Craig’s response to this article, as well as Morriston’s response to Craig’s response, can be downloaded from Morriston’s webpage.) Craig does actually address the point I am making here in an online article, Is a Beginningless Past Actually Infinite?, but it is my view that Craig’s response is unsatisfactory. Although it would take us too far afield to give a detailed critique of Craig’s article at the moment, I do intend to come back to it in a future post in this series.
But these preliminary considerations are not central to my argument here, so let’s put them aside for now and move on to the topic of subtraction.
2. Subtraction as the inverse of addition
When you learnt arithmetic back in primary/elementary school, you probably did many worksheets with exercises like these:
- 2 + 3 = __
- 5 + 7 = __
- 3 + 1 = __
As you got older, you would have seen more advanced exercises such as:
- 1 + __ = 5
- __ + 6 = 9
- 3 + __ = 8
Each of these exercises has a unique solution. For example, the answer to the problem 1 + __ = 5 is uniquely determined by the numbers 1 and 5. We generally denote the answer by 5 – 1 and, in fact, we define 5 – 1 to be “the solution to the equation 1 + __ = 5″. If two different people solved the equation 1 + __ = 5 in two different (but valid) ways, they would always get the same answer, so there is no ambiguity in defining 5 – 1 in this way. So the second set of exercises could be rewritten as:
- 5 – 1 = __
- 9 – 6 = __
- 8 – 3 = __
This is one way to define the subtraction operation: as the inverse operation of addition.
Now, if you have some intuition with infinity, you would probably agree with the following sums:
- ∞ + 1 = ∞
- ∞ + 2 = ∞
- ∞ + ∞ = ∞
All of these (and many more) mean that the equation
- ∞ + __ = ∞
does not have a unique solution. There are several values that could correctly fill the blank. In fact, there are infinitely many values that could correctly fill the blank: 0, 1, 2, 3, 4, … , ∞ all work fine. Since there is no unique solution, we cannot unambiguously define ∞ – ∞ to be “the solution to the equation ∞ + __ = ∞”.
But is this a problem?
It actually isn’t, although Craig thinks it is, as we’ll see later. As Millican pointed out in his debate with Craig, there are other operations that run into similar “problems”. For example, when you learned multiplication, you would have worked on exercises like these:
- 2 × 5 = __
- 3 × 9 = __
- 4 × 6 = __
And then you would have moved on to more advanced ones like:
- 3 × __ = 21
- __ × 8 = 32
- 2 × __ = 18
Again, there are unique answers to each of these problems. And this allows us to define 21 ÷ 3 to be “the solution to the equation 3 × __ = 21″. And these three exercises could be rewritten as:
- 21 ÷ 3 = __
- 32 ÷ 8 = __
- 18 ÷ 2 = __
But what happens when we do multiplication with zero? Well, zero multiplied by any number is still zero:
- 0 × 0 = 0
- 0 × 1 = 0
- 0 × 6 = 0
And these serve to show that the equation
- 0 × __ = 0
does not have a unique solution. Because of this, it is impossible to define 0 ÷ 0. But does this mean that we should be dubious about the number 0? Should we suppose that it is impossible for someone to have no coins? I certainly don’t think so! Amazingly, we’ll see that Craig does have some kind of problem with the number 0, though the reasons he gives are baffling to say the least.
Before we come to examine Craig’s response to these ideas, it is worth noting that anyone who has studied mathematics at university level will have seen a few more examples of this phenomenon: the squaring operation, multiplication of matrices or in certain rings. These are all operations that are not invertible in general. But it doesn’t mean that there are irreparable problems with arithmetic in these settings. It simply means that these operations are not invertible in general. No more, and no less. This would only be a problem if you had a predetermined view that every operation must be invertible.
As it happens, my area of mathematical expertise is Semigroup Theory, which is the study of algebraic systems with an operation that is generally not invertible (this can be compared to Group Theory, the study of algebraic systems with an invertible operation). A few of my most recent papers concern infinite semigroups – I have a feeling Craig would be appalled if he read them!
3. Craig’s comeback
Here is what Craig has to say in response to all of this:
Dr Millican says that infinity minus infinity is not defined in transfinite arithmetic – there’s more than one solution to the equation [ie, the equation ∞ + __ = ∞]. And that is precisely the problem when you try to translate this into the real world. You can slap the hand of the mathematician who tries to subtract infinity from infinity, but you can’t stop someone from taking away a certain number of coins. And the contradiction is that you have identical quantities, you subtract identical quantities, and you come up with non-identical results. And it needs to be understood that infinity in this case is not that sideways lazy 8, it’s the number aleph null, which is a number. It is the cardinal number of infinity. And it would be identical minus identicals yields non-identicals, and I submit in reality that’s absurd.
As for zero, I think zero is very problematic, frankly. I mean, suppose someone said “There’s an elephant in the quad” and I said “Well, I don’t see any elephant in the quad”, and he says “Well, there is an elephant in the quad and its number is zero”. Well, I think that’s very problematic. I don’t think there is such a thing as zero. It means just the absence of something.
Now there’s a lot to say about this speech from Craig. I’ll deal with the obvious things first, and then get to the heart of Craig’s misunderstanding.
Lazy 8. First, Craig’s comment about the symbol ∞, or the “sideways lazy 8″, is neither here nor there. Mathematicians use the symbol ∞ for a variety of different purposes, and one of them is to denote the cardinal number “aleph null”, or , which is also sometimes denoted ω (the lower case Greek letter omega), although the ω symbol is usually only used when referring to as an ordinal number rather than a cardinal number. Aleph null is not “the cardinal number of infinity” – in fact, this expression doesn’t make any sense at all. Rather, aleph null is the cardinal number of the set of natural numbers. Interestingly, there are other infinite cardinals: , , and so on. In fact, is a bigger infinity than , and is bigger still than . For any infinity, there is an even bigger infinity. Yes, this means there are infinitely many different kinds of infinities! I’m surprised that Craig never complains about this!
So, even though Craig might be upset about it, I’m just going to use the symbol ∞ to mean the cardinality of the natural numbers. Mathematicians typically do this when they are not talking about several kinds of infinity at once.
Zero. Next, I hope it is abundantly clear to everyone reading that Craig’s comments about zero are hopelessly incoherent. I happen to agree that the conversation Craig presents is problematic. But this only illustrates a problem with the kind of person that might say such things, not a problem with the number 0. I don’t really know what Craig was thinking, here. Perhaps he meant his hypothetical person to say that “the number of elephants in the quad is zero”, rather than “there is an elephant in the quad and its number is zero”. But this wouldn’t be a problem at all. I’m quite happy for someone to tell me the number of elephants in the quad of the University of Birmingham is zero. Unless there was an actual elephant there, this would be a true statement. As it happens, the number of elephants sitting in my lap as I type this is also zero. Thank goodness for that! But if Craig really has a problem with the number zero, then subtraction becomes problematic even when restricted to whole numbers. If Craig gave all his coins away, how many would he have left? Zero. We know that
- 5 – 5 = 0
- 7 – 7 = 0
- 9 – 9 = 0.
And all of this is perfectly consistent. There is no problem whatsoever in me saying that I have 0 daughters, and 0 degrees in theology.
In the end, Craig’s comments about zero are completely nonsensical, but they’re not central to his argument, so I’ll let them go. I’m prepared to accept that this was an off-the-cuff statement from Craig, and that he might have said something more coherent if he had a bit more time to think about his words. So let’s move on to the more important problem.
4. Subtraction as “taking away”
Another way to define subtraction is via the operation of “taking away”. Sometimes “5 – 3″ is read as “5 take away 3″. And, to help you calculate 5 – 3 when you were learning, your teacher might have said something like:
- If you had 5 apples, and I took away 3 of them, how many would you have left?
You might have actually done an experiment like this with apples several times, and seen that the answer is always 2. In fact, it doesn’t matter which 3 apples you take away: you will always have 2 left. But there’s more. If you started with 5 bananas and took any 3 of them away, you’d have 2 left. And this illustrates a general point. If you start with a collection of 5 objects of any kind, and take any three of them away, then you will always have 2 left.
All of this seems completely unremarkable. But it illustrates a couple of basic theorems in set theory. To state them, I’ll first need to introduce a tiny bit of notation.
Let A be a set (ie, a collection of objects). The objects contained in A are called the “elements” of A. We write |A| for the number of elements in A. For example, if A is the set consisting of all your fingers, then |A| = 10 (probably).
A set B is called a “subset” of A if each element of B is an element of A. We write B ⊆ A to indicate that B is a subset of A. For example, if A is the set consisting of all your fingers, and if B is the set consisting of all the fingers on your right hand, then B ⊆ A. Essentially, B ⊆ A means that A is a set that extends B, in the sense that it contains everything that B contains, and possibly more.
If B ⊆ A, then the “complement of B in A” is defined to be the set of all elements from A that are not in B. We write A \ B for the complement of B in A. Continuing with the above example, if A is the set consisting of all your fingers, and if B is the set consisting of all the fingers on your right hand, then A \ B is the set of all the fingers on your left hand.
And now we can state a couple of theorems of set theory:
Theorem 1. Suppose A is a finite set. Suppose B ⊆ A and C ⊆ A, and that |B| = |C|. Then |A \ B| = |A \ C|.
Theorem 2. Suppose A and B are finite sets and that |A| = |B|. Suppose C ⊆ A and D ⊆ B, and that |C| = |D|. Then |A \ C| = |B \ D|.
These might look complicated (mathematical notation can sometimes have that effect). But all the theorems really say is that if you have two collections, each with the same finite number of objects, and you then remove the same number of objects from each collection, then you will end up with the same number of objects in each case. Theorem 1 is the reason why you get the same number of apples when you remove 3 apples from 5, no matter which 3 you remove. And Theorem 2 is the reason why you get the same answer if you used bananas instead of apples.
Because of these theorems, we are able to define subtraction in a more direct way, without having to refer to addition at all. For example, we can define 9 – 5 to be “the number of objects left when you take 5 objects away from any collection of 9 objects”. Because of the theorems, we know we are going to get the same answer no matter which collection of 9 objects we start with, and no matter which 5 objects we take away.
But here is where it gets interesting. You’ll notice that Theorems 1 and 2 specifically require the sets A and B to be finite. The theorems cannot be proved for infinite sets. In fact, the theorems are false in the context of infinite sets. Instead, we have interesting situations like the one Craig referred to with the infinite coin collection. Here is another theorem from set theory:
Theorem 3. Suppose A is an infinite set. Then there exist subsets B ⊆ A and C ⊆ A such that |B| = |C| but |A \ B| ≠ |A \ C|.
This theorem says that given any infinite collection, it is possible to remove an equal number of objects from the collection in two different ways, resulting in a different number of objects left over. Craig has kindly supplied us with an example of this theorem in action. His analogy with coins can be simplified with mathematical notation. So, let
- A be the set of all positive integers: 1, 2, 3, 4, 5, 6, …
- B be the set of all odd positive integers: 1, 3, 5, 7, 9, 11, …
- C be the set of all integers greater than three: 4, 5, 6, 7, 8, 9, …
Then we see that B ⊆ A and C ⊆ A. Also, we clearly have |B| = |C|, since both B and C are infinite. However:
- A \ B is the set of all even positive integers: 2, 4, 6, 8, 10, …
- A \ C is the set of the first three positive integers: 1, 2, 3.
So here we have |A \ B| = ∞, while |A \ C| = 3. In other words, |A \ B| ≠ |A \ C| even though |B| = |C|.
Examples like these show that we would not be able to define ∞ – ∞ as “the number of objects left when you take infinitely many objects away from any infinite collection of objects”, since you could get different answers depending on which objects you chose to take away. But this is the only problem, if it is indeed deemed a problem (remember, we can’t define 0 ÷ 0 either, and this is not regarded as problematic by any mathematician in the world, as far as I know). It is worth quoting Craig again on this point, to remember exactly what he said about this:
You can slap the hand of the mathematician who tries to subtract infinity from infinity, but you can’t stop someone from taking away a certain number of coins. And the contradiction is that you have identical quantities, you subtract identical quantities, and you come up with non-identical results… and I submit in reality that’s absurd.
I agree that a mathematician who thought it was possible to define ∞ – ∞ should have his or her hand slapped (metaphorically, of course). And I agree that you can’t stop someone from taking away a certain number of coins, no matter how big their coin collection is. But there is no contradiction here at all. To think otherwise is to grossly misunderstand what is going on. The fact that ∞ – ∞ has no unambiguous meaning does not prohibit someone with an infinite coin collection from giving away infinitely many of their coins. All it means is that the number of coins they have left after doing so will depend on which coins they gave away. If Bob has an infinite collection of coins, then he may give infinitely many of them away if he likes. However, if he tells Wendy that he has done so, she will not be able to determine how many coins are left – at least not without knowing which coins were given away. But there is nothing to prohibit Bob from giving away as many coins as he wants, even though Craig seems to be saying there is.
The above discussion makes it clear that Craig is (deliberately or accidentally) equivocating on the phrase “take away”. For example, the sentence “you can’t take away infinity from infinity” could be interpreted in (at least) two different ways:
- you can’t define ∞ – ∞, or
- you can’t take infinitely many objects away from an infinite collection.
As we have seen, Statement 1 is true, but Statement 2 is false. More importantly though, Statement 1 has no bearing on Statement 2, even though Craig claims that it does.
Craig’s problems seem to arise from the view that Theorems 1 and 2 should apply to infinite collections (if they exist), and not just finite collections. But they don’t, as the example of the infinite coin collection clearly illustrates. But far from demonstrating that it is therefore impossible for a person with an infinite coin collection to give away infinitely many coins, all it shows is that it would not be possible to determine how many coins were left after such a generous act, unless we knew precisely which coins had been given away. In fact, all these considerations show that life would be very interesting for someone with an infinite coin collection, even if most economic models couldn’t cope with an infinitely wealthy tycoon.
But, more importantly, the impossibility of unambiguously defining ∞ – ∞ has no bearing whatsoever on the possibility or impossibility of the “real existence of an actually infinite number of things”. Craig’s argument against the existence of actual infinite collections therefore fails. Maybe such collections are impossible. Maybe time couldn’t be infinite in the past. But one thing is abundantly clear: Craig’s arguments go nowhere towards demonstrating it.