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.

Excerpted from ‘The Drunkard’s Walk – How Randomness Rules our Lives’ by Leonard Mlodinow