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?
Depending on your field of study, you’ll view the question through different lenses. A philosopher might quote one of Zeno’s paradoxes. “Suppose this,” he begins; a rabbit needs to run a straight course that’s 100 meters long. Before it can finish this course, it needs to run ½ that distance. However, before completing 50 meters, the rabbit needs to first run ½ of the 50 meters. If we repeat this pattern ad infinitum, the poor rabbit can never cross the finish line, since it must first travel half the distance. In fact, since any finite distance can be divided in half, the rabbit could never start at all since there is no “first step” and infinity has properly trapped the rabbit in a state of non-movement. We’ve implied that all motion is an illusion, since there is no possible “first step” for us to take to initiate our journey to anywhere. To the haters who claim that his proposition is rubbish, the philosopher simply asks “how?” As the disbelievers struggle to disprove the obviously fallacious claim that motion is an illusion, the philosopher pats himself on the back. He has channeled the energy of Zeno and Parmenides, both of whom were essentially arguing that contrary to the information from our senses, change is mistaken.
If a logician were to come across the philosopher, he might protest at this point, “that’s impossible!” He stands and simply takes a couple steps to prove that motion couldn’t be an illusion and looks at the philosopher daringly. Of course, in order to explain properly why he thinks this is rubbish, he might point out that as the distance the rabbit needs to complete becomes increasingly smaller, the time it takes for the rabbit to complete that distance becomes smaller as well. As we make our way towards a distance that’s number infinitely small, the time needed to complete that distance becomes infinitely small as well. The philosopher has ignored the fact that he cannot simply choose to divide space into infinitely small parts but ignore the fact that time is there as well.
“But!” The philosopher jumps up, “Suppose we want to carefully watch the rabbit race.” We take infinity many photos of this rabbit as it races from 0 to 100 meters at duration-less moments in time so that we know the precise location of the rabbit in each of these photos. In each photo the rabbit is still, but always in a slightly different position. The rabbit can’t be move to where it is not, because these photos are duration-less and no time elapses for the rabbit to move there. Similarly, the rabbit can’t be where it was before, because we know that the rabbit was running the race. Therefore, in every instant in time, there was no motion. If time is composed of these instants and every instant is motionless, then motion is impossible. At this point, we are no longer talking about dividing space, we’re dividing time.
“But we’re not just dividing time!” The logician protests, “We’re dividing it into points!” With a twist, the philosopher has turned the rabbit race into the arrow paradox. We’re arguing about the possibility of constructing a “whole” based on infinitely small parts. To rephrase—is a line composed of points? Points themselves are dimensionless, yet we can say that a bunch of these dimensionless points can create lines which can create dimensions. However, if we channel Betrand Russel at this point, we can uncover a logical flaw in the philosopher’s paradox. Russel says that “the only point where Zeno probably erred was in inferring (if he did infer) that, because there is no change, therefore the world must be in the same state at one time as at another. This consequence by no means follows. . . .” In a single photo, the rabbit is still, thus there are no distinctions between “at rest” and “in motion”. The moment where the distinction occurs is when we consider the position of the rabbit in multiple photos. Or: Position is a function of time. There is no additional process of ‘moving’ between these different photos, so it’s not a paradox at all that time is composed of immobilities.
A mathematician would scoff at the silly fight happening between the logician and the philosopher. Both he and the physicist know that infinity is not some matter of debate, but simply another tool to be used in calculating equations. Even before the concept of “infinity” was established, people could still describe the idea of a large magnitude. The mathematician quotes Archimedes and draws a square on the blackboard. “Suppose there’s a square with a side length 1,” he begins, “And we divide this square into 4 equal segments.” We choose one of these corners and divide it into 4 segments again. Then we choose one segment of the 4 to divide into 4 segments again. We repeat this process ad infinitum. Even though infinity may seem like a huge concept, as long as it is bounded, it can be shown to converge to a single value. In the case of the square, we can only divide a square inside the larger square. No matter how many times we continue to divide, the overall area of the square still remains 1 and each division can be reduced to 1/4 + 1/16 + 1/64 + … = 1/3. And in the case of the rabbit, we know that no matter how many times we divide the track, the total distance will still be 100 meters. Time will always be passing relative to the time of the world, so the rabbit will definitely be able to cross the line.
In all cases of the rabbit though, we are taking a physical interpretation into a mathematical one. However, the physical world cannot be described using infinity, as our world itself is finite. Whenever we invoke the power of infinity, we’re probably using it as a placeholder for an idea of “really, really big”. However, the mathematical and theoretical world can freely use it. We can easily divide infinity into types. We have the countable, the uncountable, the convergent infinity, the alephs—all of which allow us to do really cool things in our calculations. However, when using a tool as powerful as infinity to describe the real world, we’ll have to be careful that we are invoking it responsibly.