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Infinity—Nothing to Trifle With

I’m preparing a presentation for the Atheist Alliance of America conference in Seattle, August 7–10. I’ll be posting less often for a week. Thanks for your patience.

The topic of infinity comes up occasionally in apologetics arguments, but this is a lot more involved than most people think. After exploring the subject, apologists may want to be more cautious.

William Lane Craig scatters tacks on the road

Philosopher and apologist William Lane Craig walks where most laymen fear to tread. Like an experienced actor, he has no difficulty imagining himself in all sorts of stretch roles—as a physicist, as a biologist, or as a mathematician.

Since God couldn’t have created the universe if it has been here forever, Craig argues that an infinitely old universe is impossible. He imagines such a universe and argues that it would take an infinite amount of time to get to now. This gulf of infinitely many moments of time would be impossible to cross, so the idea must be impossible.

But why not arrive at time t = now? We must be somewhere on the timeline, and now is as good a place as any. The imaginary infinite timeline isn’t divided into “Points in time we can get to” and “Points we can’t.” And if going from a beginning in time infinitely far in the past and arriving at now is a problem, then imagine a beginningless timeline. Physicist Vic Stenger, for one, makes the distinction between a universe that began infinitely far in the past and a universe without a beginning

Hoare’s Dictum is relevant here. Infinity-based arguments are successful because they’re complicated and confusing, not because they’re accurate.

One of Craig’s conundrums is this:

Suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., –3, –2, –1, 0. We could ask, why did he not finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished by then.… In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will have already finished.

More on infinity

Before we study this ill-advised descent into mathematics, let’s first explore the concept of infinity.

Everyone knows that the number of integers {1, 2, 3, …} is infinite. It’s easy to see that if one proposed that the set of integers was finite, with a largest integer n, the number n + 1 would be even larger. This understanding of infinity is an old observation, and Aristotle and other ancients noted it.

But there’s more to the topic than that. I remember being startled in an introductory calculus class at a shape sometimes called Gabriel’s Horn (take the two-dimensional curve 1/x from 1 to ∞ and rotate it around the x-axis to make an infinitely long wine glass). This shape has finite volume but infinite surface area. In other words, you could fill such a container with paint, but you could never paint it.

A two-dimensional equivalent is the familiar Koch snowflake. (Start with an equilateral triangle. For every side, erase the middle third and replace it with an outward-facing V with sides the same length as the erased segment. Repeat forever.) At every iteration (see the first few in the drawing above), each line segment becomes 1/3 bigger. Repeat forever, and the perimeter becomes infinitely long. Surprisingly, the area doesn’t become infinite because the entire growing shape could be bounded by a fixed circle. In the 2D equivalent of the Gabriel’s Horn paradox, you could fill in a Koch snowflake with a pencil, but all the pencils in the world couldn’t trace its outline.

Far older than these are any of Zeno’s paradoxes. In one of these, fleet-footed Achilles gives a tortoise a 100-meter head start in a foot race. Achilles is ten times faster, but by the time he reaches the 100-meter mark, the tortoise has gone 10 meters. This isn’t a problem, and he crosses that next 10 meters. But wait a minute—the tortoise has moved again. Every time Achilles crosses the next distance segment, the tortoise has moved ahead. He must cross an infinite series of distances. Will he ever pass the tortoise?

The distance is the infinite sum 100 + 10 + 1 + 1/10 + …. This sum is a little more than 111 meters, which means that Achilles will pass the tortoise and win the race.

Some infinite sums are finite (1 + 1/2 + 1/4 + 1/8 + … = 2).

And some are infinite (1 + 1/2 + 1/3 + 1/4 + … = ∞).

(And this post is getting a bit long. It will be concluded in part 2.)

To say that that having a degree in theology
makes you qualified to say that God is real,
is like claiming that having memorized all the Harry Potter books
makes you qualified to say that unicorns exist.
– Unknown

Photo credit: Wikipedia

(This is an update of a post that originally appeared 7/12/12.)

About Bob Seidensticker

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