For many of us uncomfortable with infinity, the word number can be defined as "that which makes numb," as Rudy Rucker wryly notes in his book Infinity and the Mind (Birkhäuser, 1982). This is especially true when a number is so outlandishly enormous that it smacks, however remotely, of the infinite. Galileo himself felt this way. "Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness," he wrote in his Dialogues of Two New Sciences of 1638. "Imagine what they are when combined." Rather not, thanks—makes me numb.

This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set—anything with an infinite number of things in it—is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is—you got it—infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2infinity is.

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.