I am eight years old. All around me, bookshelves stretch beyond sight in every direction: an infinite library. A librarian — my taskmaster — looms over me as I toil towards a fast-approaching deadline. I must catalog every book.
I remember this nightmare with startling immediacy. At eight years old, I feared infinity as tangibly as I feared wolves and crocodiles. I felt an intense emotional response to a mathematical abstraction, a response that hasn’t since dulled. In jest, I call myself an infinityophobe. But in all earnestness, I wonder — am I afraid of the concept itself, or of my inability to understand it?
Humans wield the unique ability to practice conscious science; we tame the world around us by explaining it. And yet, as mental gymnasts, we crumple in the face of the very large and the very small. The infinite and the infinitesimal remain obstinately alien.
As an eight-year-old, the enormity of infinity boggled my mind. As a college student, I struggled equally with the incomprehensibly miniscule — what my chemistry professor called the “sinister conspiracy theory of quantum mechanics.” Classical mechanics, with its tidy and manageable sense, works fine for the observable everyday world. But as you zoom in through matter and molecules, classic disintegrates into quantum. Shifty electrons change their fundamental nature as soon as we observe them. “Entangled” photons exchange telegrams that travel faster than the speed of light. In the probabilistic world of quantum mechanics, all bets are off.
Even without this quantum weirdness, we would struggle to grasp such small sizes. The largest object known to exhibit quantum behavior (at least according to my chemistry professor) — the famous “buckyball,” a roughly spherical lattice of sixty carbon atoms — has about one 70,000,000,000,000,000,000,000,000th the mass of an average grown man.
In fact, when it comes to size, our brains can manage a startlingly small range. We have trouble conceptualizing quantities we can’t see or experience. The earth, for instance, is about 4.5 billion years old. But as glibly as we may toss around that number, it is virtually meaningless to us. And that’s still a finite number.
To stretch ourselves even further beyond our mental limits, we can try to imagine the size of the universe. This is trickier, because scientists don’t know whether the universe is finite or infinite, so we must hold both notions simultaneously in our minds. Would it really matter, though, as far as the average person is concerned? Can we really comprehend the difference between tens of billions of light years and an infinite distance?
The infinite turns out to behave as weirdly as the infinitesimal — eerier even than my childhood infinity nightmares. Smaller infinities nest within larger infinities. Whole things are equal to their constituent parts. Poor Achilles must collapse from exhaustion before catching up to the plodding tortoise, because he can’t traverse the infinitely divisible space between them.
With increasingly sophisticated mathematics, we can manipulate multiple infinities and calculate quantum properties. Great minds have ventured to the frontiers of our comprehension, and returned more or less intact. But no amount of computational trickery will ever allow us to truly grasp either the size or the behavior of these extremes.
Bats map the world around them through sound, and it’s the same world that we map through sight. We use the tools that evolution provided us. Perhaps the weird extremes that bracket our understanding — the infinite and the infinitesimal — are not so fundamentally weird. Perhaps we’re just not neurologically equipped to feel our way through them.
For a lighter take on infinity, check out The Infinite Cat Project.
And, of course, Dunder Mifflin Infinity.