My eighteen-year-old son is a math whiz. He’s the kind of kid who learns Calculus 2 from some website, then lands a perfect score on the AP test without ever taking the class. Meanwhile, I’m still struggling with five times seven.
Was it really fifteen years ago when I bought a gallon bucket of plastic counting bears to teach him addition? Now he’s dragging me to the kitchen table at midnight and patiently drawing out diagrams on paper napkins, unpacking the glories of the numerical universe one step at a time.
He’s a born teacher, massaging higher math into the vernacular until my fog lifts, waiting for that moment when I gasp because I finally understand. All at once I see what he means and why it matters (Hallelujah!). I see why this concept is beautiful, why he wanted me to see it; then just as fast all the light passes away, and I’m back in the dark.
I don’t know math well enough to hold on to that kind of truth, but for a half second it was mine, and the flash of insight changes me, even as it fades. It wakes me up and makes me ache to get it back again. It lingers like a perfect melody that you heard in a dream but you can’t remember well enough to sing once you wake up.
Sehnsucht is a German word that comes close to what I mean, a yearning for something beyond our reach. C. S. Lewis referred to this desire as “the inconsolable longing in the heart for we know not what.” It’s more forward-looking than nostalgia, more solid and more hopeful than sentimentality. When Sehnsucht strikes me, it feels like I am rising up through the thick, choking waters of the earth into a cleaner air.
Philosophically, we can trace Sehnsucht back to George MacDonald, and back again to the German romantic philosopher Novalis. However, I want to save all that for another essay at another time — for now I’d like to just name the thing and move on to the heart of what I mean to say about it.
This week JD has been teaching me about the possibility of a fourth physical dimension. We have all been taught about the three obvious dimensions: length, width, and depth. You may have also heard that the fourth dimension is time. However, JD’s been digging around in theories about a fourth physical dimension — a realm like depth, but beyond depth.
I’ve sensed this possibility intuitively since I was a child, but I didn’t have good images for it until reading George MacDonald’s Lilith, and I didn’t think it was scientifically feasible until watching NOVA’s The Elegant Universe, a documentary about string theory.
I also felt shadows of a fourth dimension while studying the discipline of philosophy. The transition from old, historical rationalism (think logical proofs) through David Hume’s more modern empiricism (think scientific method) left me hungry. In fact, Hume knew his own gaps. He knew it was impossible to bridge the expanse between what is real and what we perceive to be real (AKA: the mind-body gap), which means that it’s very difficult to prove anything at all.
It’s not practical to linger in that futility very long. Those who take it too seriously seem to land in a sulking existentialism, but that’s not where it has to stop. Instead of pouting that we can’t nail everything down, it’s also possible to stay properly small and properly curious in a mysterious universe. As the old song goes, ‘tis a gift to be simple, ’tis a gift to be free, ’tis a gift to come down where we ought to be. Is it so terrible to recognize that our five little senses and our three pounds of brain might not have the ability to master all that is?
There’s something healthy about admitting the possibility that “Holy, Holy, Holy” isn’t redundant, but only the straight line of the edge of a cube? A multi-dimensional… trans-dimensional God … is likely to distort when we attempt to hammer him into a post-Enlightenment flatland.
Yesterday I listened to a mathematician describe the blasted, necessary intersection of a Klein bottle. A Klein bottle is an edgeless object that can exist in four-dimensional space, but when we try to form it in a three-dimensional world, we have to insert a seam to even hint at what the thing would be like in a larger reality. As I listened to this man describe how a three-dimensional model could never truly suffice, I heard in his frustration the frustration of a theologian trying to unpack God to a materialist.
Another four-dimensional object has been named the hypercube, a 4D version of a square. Generous researchers have tried to create computer simulations of the hyper cube, and there are several videos online that allow us to watch them rotate. When I see the rotation, I understand four-dimensionality for a moment, then lose it again. It’s so difficult to think beyond our own experience.
However, the stretch to comprehend this has still been good for me, and I’m going to share with you some of the shadows of possible truth that I’ve collected while stretching. Some of you are more mathy than I am, and you can write and correct me where I am in error, but we must start somewhere, so I’ll take a stab at it.
Actually, let’s make it a story. But instead of trying to tell this story in three and four dimensions, we’ll reel it back to two and three.
Once upon a time, there was a two-dimensional square who was given consciousness. Our poor square has only ever existed in a flat cognition. He has heard about three-dimensions, but to him this is the stuff of sci-fi conventions and religious cults. Most of his fellow squares have grown out of such nonsense entirely and have declared themselves atheists. They are perfectly content to spend their lives studying the edges and planes, and in fact, they feel rather bright to be doing it.
But our square is a bit of a romantic. That’s not his fault, he’s just survived a nasty breakup with a triangle, which has led him to wandering the flat streets, staying up too late, meeting with shady polygons in flat pubs — the sort of polygons who drink flat whiskey and talk about fairies, and fawns, and one particular mythical entity called a “cube of cheese.” You can judge him if you want, but the company does him good, even if he can’t take these dreamers too seriously.
As it so happens, you stumble into the pub on a night when this motley crew is discussing three-dimensionality and you decide to help. In fact, you have a couple of cubes of cheese in your pocket. How lucky.
You rotate one cube of cheese for the square to see, but as you do, you realize that he can only understand one side of the cheese at a time. You go over and over it, but he is clearly confused, and at one point you think he might start to cry. “This is why she left me,” he mumbles. She said I just didn’t “get” things.
“Women,” a rectangle groans.
A new idea comes to you. You take a knife and start slicing your 3-D cheese into little thin slices, like images from an MRI. You even slice the cheese diagonally, trying to help the square understand what depth means. The square says, “Oh! A cube is a million flat planes!’”
No, no. He’s closer but still missing it.
You begin to organize these slices of cheese like one of those flip books kids use to mimic motion. You show him the slices one at a time, more and more quickly, trying to help him see that they come together to make something whole. As these slices of cube pass over the square’s vision, he suddenly catches a glimmer of depth. All of these individual planes could merge into a single unity. A cube!He can almost see it. Almost. No… it’s gone.
Still, in that second, he felt what he had no real ability to perceive or validate. He caught the trajectory, an impression that woke up an ache in his wee little heart for a reality beyond him.
I am telling you this story because I think we are but squares in a universe of cheese cubes.
I am not a great scholar of science or philosophy, but I have studied enough to begin to feel the limits of the human mind. I can tell that there is a very real possibility that we we will not be able measure or reason our way through all that is. We can snicker at the idea of angels and gods, but the wisest among us are only nonagons.
Tim Keller, in his book The Reason for God, cites an old Indian story about five blind men and an elephant. The men are asked to describe an elephant by feel, so the man at the trunk says the elephant is only a sort of tube, while the man at the tail says it is a rope. The man at the ears says it is like a rug, and the man at the side believes it is a wall. On and on, and we finally understand that only someone with sight can perceive the whole of what an elephant is at once. It takes a consciousness that transcends our own limitations to put all the parts in true perspective.
It is not that we can know nothing, for God has given us senses, logic, and reason as gifts of imago Dei. Yet when an ache rises, when the universe hints to us that there is more still, and by a flicker, by a flutter, by a sudden flash of want and awe we are moved to Sehnsucht, perhaps these flat rocks are crying out to us. Perhaps this is the beginning of wisdom.