Infinity is a well explored concept across many disciplines. As of yet, I’ve managed to only interact with infinity behind the safety net of definite integrals and unbounded limits. So I thought I’d take a deeper look at infinity though the lens of Zeno’s paradoxes. But first, a short discussion about English.
When I was 6, “red” was the color of my favorite rain jacket. That was it.
When I was 14, “red” was no longer simply a color, it was an emotion, an object, a physical force. I could smell the redness of a cherry orchard and hear the redness of a laugh.
In fact, if I were imaginative enough, I could make the word “red” mean anything I wanted. “You are a poet!” my teacher cried, “Bend your words, twist them, pepper them with your thoughts; present to the world a feast of poetry.” English was quite a willing and malleable medium, which was great, but this foray into non-literal completely defeated the sure-ness of words that I had believed in as a kid. With poetry, even the most innocuous sentence like “today was sunny” could have ten different meanings.
Now, I’ve held on to the belief in the certainty of mathematics since my days of Algebra I. Unlike poetry, the result of a definite integral doesn’t change with the whims of the one grading the problem. Math was unmoved by emotion, and (to borrow the words of Bertrand Russell) I felt that it was the turtle that could hold all other turtles.
However, take a hard look at this concept called “infinity”. Infinity is to mathematics what poetry is to English—it takes the previously solid math of algebra into the fuzziness of philosophy. Though I knew nothing of its intricacies, I often invoked the threat of infinity upon my childhood friends when I “triple times infinity dared” them to do something dumb. Infinity was a big deal, since we didn’t know anything of what it was other than the fact that it was really, really big. But what is the actual value of infinity? How can a mere concept of “really, really, big” become something we can realistically use?