(Book: On Guard, by William Lane Craig. Chapter 4: “Why did the universe begin?”)
We’re on a trip back to 12th-century Persia. Dr. Craig needs someone to rescue the Christian God from the perils of scientific advancement, and he thinks he has found a champion in a Muslim philosopher named Ghazali. (It’s amazing that Dr. Craig is able to successfully sell Muslim philosophy to conservative post-9/11 American Christians, don’t you think?) According to Ghazali, whatever begins to exist has a cause (premise 1), the universe began to exist (premise 2) and therefore the universe has a cause (conclusion).
Being a medieval Muslim philosopher, Ghazali did not have access to discoveries about particle physics and so on, so to prove premise number 2, he (and Dr. Craig) rely on philosophy. The argument is a bit flawed, however, in that it relies on a fundamental misunderstanding of what an infinity is. Curiously enough, it also contradicts the Christian doctrine of eternal life.
Ghazali argued that if the universe never began to exist, then there have been an infinite number of past events prior to today. But, he argued, an infinite number of things cannot exist… Ghazali recognized that a potentially infinite number of things could exist, but he denied that an actually infinite number of things could exist….When we say that something is potentially infinite, infinity serves merely as an ideal limit that is never reached. For example, you could divide distance in half, then into fourths, then into eights… The number of divisions is potentially infinite, in the sense that you could go on dividing endlessly. But you’d never arrive at an “infinitieth” division. You’d never have an actually infinite number of parts or divisions.
If there cannot be an actually infinite number of events in the past, then there also cannot be an infinite number of events in the future. Believers can get around this by saying that most of those events are only potential events because we haven’t experienced them yet. According to Christian cosmology, however, God exists outside of space and time, and is therefore not subject to the “haven’t experienced it yet.” All times are supposed to be immediately real to God, and thus if there cannot be an infinite number of them, then sooner or later there must come a day that will be the last day in the life of “immortal” believers and even God Himself. This means that, according to Dr. Craig’s arguments in Chapter 2, Christian life is devoid of all true meaning, value and purpose.
The fundamental mistake that Dr. Craig makes is a failure to really grasp a point that he alludes to above: there is no “infinitieth” division or anything else. Infinity means “not finite,” i.e. not constrained by limits, not a fixed quantity. There’s an old joke that says two plus two equals five, for sufficiently large values of two. (Sounds like something an engineer would say, eh?) The number two is a fixed quantity, a finite amount. It has limits. If it’s going to be two, then it cannot be more than two nor less than two. It’s a fixed quantity.
Infinity, by contrast, is not a fixed quantity, and that’s where Ghazali and Dr. Craig get themselves confused. I’m sure they know that, in dictionary terms, infinity is infinite, i.e. not finite, i.e. not a fixed quantity. Yet when it comes to practical applications, they expect it to have the same behaviors and characteristics of a finite quantity. They treat it like it was just some very large—but still fixed—number.
The way in which Ghazali brings out the real impossibility of an actually infinite number of things is by imagining what it would be like if such a collection could exist and then drawing out the absurd consequences.
Dr. Craig’s favorite example is the “Hilbert Hotel” imagined by mathematician David Hilbert. See if you can spot all the ways Dr. Craig assumes that infinity is a very large finite number.
[L]et’s imagine a hotel with an infinite number of rooms, and let’s suppose… that all of the rooms are full… There isn’t a single vacancy throughout the entire infinite hotel; every room already has somebody in it. Now suppose a new guest shows up at the front desk, asking for a room. “No problem,” says the manager. He moves the person who was staying in room #1 into room #2, the person who was staying in room # 2 into room #3…and so on to infinity. As a result of those room changes, room #1 now becomes vacant, and the new guest gratefully checks in, but before he arrived, all the rooms were already full!
Then an infinite number of guests show up, and the manager solves the problem by moving 1 to 2, 2 to 4, 3 to 6, 4 to 8, and so on, leaving all the odd-numbered rooms vacant. Since there are an infinite number of odd numbers, everybody gets a room even though the hotel was already full. Absurd, right?
One of the qualities of finite numbers is that you can compare two finite numbers to see if they are equal. You can say, for example, that the number of rooms in a finite hotel is equal to a finite number of guests (assuming one guest per room, for simplicity). Another quality of finite numbers is that you can subtract one from another. You can say, for example, that the number of guests is equal to the number of rooms, and therefore there are zero empty rooms left over.
Infinity, however, is not a finite number. It does not obey the same set of rules as a finite number, because it is not a fixed quantity. Thus, you can have an infinite number of guests in a hotel with an infinite number of rooms, and still have an infinite number of empty rooms available, allowing the manager to do his “magic.” You might think that infinity minus infinity must be equal to zero, since they’re both the same quantity, but that’s a misperception—you’re thinking of both infinities as being fixed quantities that you can compare and subtract.
I skipped over a section where Craig tries to dispense with Cantor and modern mathematics in order to make his point, but I think now would be a good time to go back and look at that.
It’s very frequently alleged that this kind of argument has been invalidated by developments in modern mathematics…
These developments in modern mathematics merely show that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way, without contradicting yourself. All this accomplishes is showing how to set up a certain universe of discourse for talking consistently about actual infinities. But it does absolutely nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist. If Ghazali is right, then this universe of discourse may be regarded as just a fictional realm…
This from a guy who disparages materialists for not believing that souls and spirits are real! Discoveries that you can establish by rigorous application of mathematics are just “fiction,” but angels and demons and entire realms suggested by superstition and wishful thinking—they’re all real. And if you don’t believe it, then you must be a materialist.
Ah well. One of many points that Dr. Craig is overlooking here is that there are two alternatives open to us. One, we can embrace a mathematical understanding that lets us discuss infinity in a meaningful and consistent way. Or two, we can embrace a mathematical understanding that FAILS to allow us to discuss infinity in a meaningful and consistent way. By rejecting modern mathematics (option 1) as “fiction,” Dr. Craig is explicitly selecting the second alternative, and the “absurdities” he finds are merely a reflection of his poor choice of mathematical frameworks.
Think about it. What Dr. Craig is saying is that there cannot be an actually infinite number of integers greater than zero. In the real world, you can’t actually have numbers progressing onwards towards infinity. Somewhere out there is some finite positive integer that is The Biggest Number (because if there weren’t, that would be an actual infinity). But can such a number really exist? If you could add one to it (as finite numbers allow you to do), then it wouldn’t be the biggest number. The plus-one version would be the biggest number. But if you add one to that number…?
This kind of difficulty is what is properly called absurd, because the contradiction arises even though you’re correctly treating finite numbers like finite numbers. Dr. Craig’s “absurdity,” by contrast, is merely a reflection of the fundamental error that arises when you try to treat infinity like a finite number. It’s not that there’s anything deficient about infinity, it’s just that Dr. Craig is deficient in his understanding and application of mathematical principles.
Not all of his arguments are deficient, however. His second argument is that you cannot have an infinite number of events in the past, because if you did, you could not arrive at “now,” since you have an infinite distance to traverse before you get here. That’s not actually a bad argument, though it’s moot, given the origin of time at the Big Bang. He does a pretty good job of defending this against the “every point in the past is only a finite distance from the present” objection (fallacy of composition), though he doesn’t seem to appreciate the fact that a finite past leaves God without any time in which He could create time itself.
He got the main point right, though. Chalk one up for Dr. Craig. His next example fails again, however: suppose you have two planets that have been orbiting the same star for all eternity, and one takes twice as long as the other to complete one orbit. If they’ve been orbiting forever, which one has completed more orbits?
The answer is that the number of their orbits is exactly the same: infinity! (Don’t let someone try to slip out of this argument by saying infinity is not a number. In modern mathematics it is a number, the number of elements in the set [0, 1, 2, 3, ...].)
Comparing two infinities as though they were fixed quantities. You can see it in his sly aside: infinity is a number, and not just any number. It is a specific number, the number of elements in the set of all natural numbers. By rejecting modern mathematics as a “fiction” that doesn’t actually exist in the real world, we are left with the kinds of numbers that are fixed quantities that can be compared to each other, leading to the absurdity of saying “2x = x” for x > 0. Yet the absurdity arises, not from any problem with infinity, but from the error of dismissing modern mathematics as some kind of fiction. Infinity is fine, only Dr. Craig’s math is deficient.
His final philosophical argument is to suppose you’ve met a man who is just counting down from infinity to zero. 3, 2, 1, 0.
Whew! Why, we may ask, is he just finishing his countdown today? Why didn’t he finish yesterday, or the day before? After all, by then an infinite amount of time had already elapsed. So if the man were counting at the rate of one number per second, he’s already had an infinite number of seconds to finish his countdown. He should already be done! In fact, at any point in the past, he’s already had infinite time and so should already have finished. But then at no point in the past can we find the man finishing his countdown, which contradicts the hypothesis that he has been counting from eternity.
Comparing two infinities again, as though they were finite quantities. Well, at least he’s consistent. This line of reasoning does seem to have some merit in the sense of not being able to reach the present from an infinite past. Dr. Craig is just doing the math wrong.
The bottom line is that there are good reasons to suppose that time does not extend infinitely far back into the past. Unfortunately Dr. Craig’s arguments, and specifically his idea that there cannot be any real infinity, are not among them. Nor does Dr. Craig seem to appreciate the implications of having a finite past, because if time itself is finite, that makes it less likely the universe will require a supernatural creator. There’s less territory that a natural explanation needs to cover, and less reason to suppose that there was ever a time when the universe did not exist in some form.
Ghazali’s arguments try to show that the universe has a beginning, but by showing that the past is finite, all he achieves is to demonstrate that the natural explanation does not need to explain an infinite number of past events. There may be an origin (i.e. a minimum value for time), but there is no beginning, i.e. no transition from a point in time where the universe did not exist to a point in time where it did. For that, you need a time when the universe did not already exist. Even with medieval Muslim help, Dr. Craig is no closer to providing that evidence than he was when he started.