The Case for a Creator - Lee Strobel [53]
THE PATHWAY OF MATHEMATICS
The early Christian and Muslim scholars, Craig explained, used mathematical reasoning to demonstrate that it was impossible to have an infinite past. Their conclusion, therefore, was that the universe’s age must be finite—that is, it must have had a beginning.
“They pointed out that absurdities would result if you were to have an actually infinite number of things,” he said. “Since an infinite past would involve an actually infinite number of events, then the past simply can’t be infinite.”
It took a moment for that statement to sink in. I have always been a reluctant student of mathematics, especially such esoteric permutations as transfinite arithmetic. Before we could venture into any mathematical complexities, I reached over and pushed the “pause” button on my tape recorder.
“Hold on a minute, Bill,” I said. “If I’m going to track with you on this, you’re going to have to give me some illustrations to clarify things.”
Craig already had some in mind. “Okay, no problem,” he replied. When I turned the recorder back on, he continued.
“Let’s use an example involving marbles,” he said. “Imagine I had an infinite number of marbles in my possession, and that I wanted to give you some. In fact, suppose I wanted to give you an infinite number of marbles. One way I could do that would be to give you the entire pile of marbles. In that case I would have zero marbles left for myself.
“However, another way to do it would be to give you all of the odd numbered marbles. Then I would still have an infinity left over for myself, and you would have an infinity too. You’d have just as many as I would—and, in fact, each of us would have just as many as I originally had before we divided into odd and even! Or another approach would be for me to give you all of the marbles numbered four and higher. That way, you would have an infinity of marbles, but I would have only three marbles left.
“What these illustrations demonstrate is that the notion of an actual infinite number of things leads to contradictory results. In the first case in which I gave you all the marbles, infinity minus infinity is zero; in the second case in which I gave you all the odd-numbered marbles, infinity minus infinity is infinity; and in the third case in which I gave you all the marbles numbered four and greater, infinity minus infinity is three. In each case, we have subtracted the identical number from the identical number, but we have come up with nonidentical results.
“For that reason, mathematicians are forbidden from doing subtraction and division in transfinite arithmetic, because this would lead to contradictions. You see, the idea of an actual infinity is just conceptual; it exists only in our minds. Working within certain rules, mathematicians can deal with infinite quantities and infinite numbers in the conceptual realm. However—and here’s the point—it’s not descriptive of what can happen in the real world.”
I was following Craig so far. “You’re saying, then, that you couldn’t have an infinite number of events in the past.”
“Exactly, because you would run into similar paradoxes,” he said. “Substitute ‘past events’ for ‘marbles,’ and you can see the absurdities that would result. So the universe can’t have an infinite number of events in its past; it must have had a beginning.
“In fact, we can go further. Even if you could have an actual infinite number of things, you couldn’t form such a collection by adding one member after another. That’s because no matter how many you add, you can always add one more before you get to infinity. This is sometimes called the Impossibility of Traversing the Infinite.
“But if the past really were infinite, then that would mean we have managed to traverse an infinite past to arrive