Drunkard's Walk - Leonard Mlodinow [48]
The set of concepts central to both calculus and Bernoulli’s work is that of sequence, series, and limit. The term sequence means much the same thing to a mathematician as it does to anybody else: an ordered succession of elements, such as points or numbers. A series is simply the sum of a sequence of numbers. And loosely speaking, if the elements of a sequence seem to be heading somewhere—toward a particular endpoint or a particular number—then that is called the limit of the sequence.
Though calculus represents a new sophistication in the understanding of sequences, that idea, like so many others, had already been familiar to the Greeks. In the fifth century B.C., in fact, the Greek philosopher Zeno employed a curious sequence to formulate a paradox that is still debated among college philosophy students today, especially after a few beers. Zeno’s paradox goes like this: Suppose a student wishes to step to the door, which is 1 meter away. (We choose a meter here for convenience, but the same argument holds for a mile or any other measure.) Before she arrives there, she first must arrive at the halfway point. But in order to reach the halfway point, she must first arrive halfway to the halfway point—that is, at the one-quarter-way point. And so on, ad infinitum. In other words, in order to reach her destination, she must travel this sequence of distances: 1/2 meter, 1/4 meter, 1/8 meter, 1/16 meter, and so on. Zeno argued that because the sequence goes on forever, she has to traverse an infinite number of finite distances. That, Zeno said, must take an infinite amount of time. Zeno’s conclusion: you can never get anywhere.
Over the centuries, philosophers from Aristotle to Kant have debated this quandary. Diogenes the Cynic took the empirical approach: he simply walked a few steps, then pointed out that things in fact do move. To those of us who aren’t students of philosophy, that probably sounds like a pretty good answer. But it wouldn’t have impressed Zeno. Zeno was aware of the clash between his logical proof and the evidence of his senses; it’s just that, unlike Diogenes, what Zeno trusted was logic. And Zeno wasn’t just spinning his wheels. Even Diogenes would have had to admit that his response leaves us facing a puzzling (and, it turns out, deep) question: if our sensory evidence is correct, then what is wrong with Zeno’s logic?
Consider the sequence of distances in Zeno’s paradox: 1/2 meter, 1/4 meter, 1/8 meter, 1/16 meter, and so on (the increments growing ever smaller). This sequence has an infinite number of terms, so we cannot compute its sum by simply adding them all up. But we can notice that although the number of terms is infinite, those terms get successively smaller. Might there be a finite balance between the endless stream of terms and their endlessly diminishing size? That is precisely the kind of question we can address by employing the concepts of sequence, series, and limit. To see how it works, instead of trying to calculate how far the student went after the entire infinity of Zeno’s intervals, let’s take one interval at a time. Here are the student’s distances after the first few intervals:
After the first interval: 1/2 meter
After the second interval: 1/2 meter + 1/4 meter = 3/4 meter
After the third interval: 1/2 meter + 1/4 meter + 1/8 meter = 7/8 meter
After the fourth interval: 1/2 meter + 1/4 meter + 1/8 meter + 1/16 meter = 15/16 meter
There is a pattern in these numbers: 1/2 meter, 3/4 meter, 7/8 meter, 15/16 meter…The denominator is a power of two, and the numerator is one less than the denominator. We might guess from this pattern that after 10 intervals the student would have traveled 1,023/1,024 meter; after 20 intervals, 1,048,575/1,048,576 meter; and so on. The pattern makes it clear that Zeno is correct that