The Potential Well

I fell down a well and started hallucinating.

I don't know whether it was the nature of the well, my impact, or perhaps the fact that for the first time in my life I was completely isolated, unable even to reach for my phone to assuage my boredom and panic. Whatever the cause, I realized that I understood, for once, the limitations of our mathematics.

My hallucinations began as line after line of equations danced before my eyes. They were simple and innocuous at first, single-variate algebra, then multivariate, with the functions becoming more advanced. Then calculus was introduced, starting with the notion of limits, which now seemed perfectly sensible and not at all frightening to me. After the limits came the single-variate derivatives and integrals, and very soon I was watching partial differential equations flash by at a dizzying rate. I solved every physical problem that could be solved, from the simple and elegant Maxwell equations to the Bessel function that governs the undulating ripples of a circular drum.

Then it all changed. I departed from the simple physical approximations; the drum of water morphed into a oddly-shaped pool, and the single equation with a simple solution became a superposition of many functions, all together approximating—no, I was beyond the realm of approximation—representing wholly and perfectly—the confines of this new vessel of water. It rippled in a complex pattern that would be solvable only in approximation to even our best supercomputers, but to me the solution was perfectly clear; the motion of the fluid was pulled by the strings of an equation that would have taken up the pages of every single physics textbook combined before it was exhausted.

I suddenly found myself in a quandary. I could use my newfound abilities to think up a way out of this well, or I could continue this exploration to the very end of all mathematics and physics, finally understanding the true nature of the universe. Perhaps this was little different from the quandary faced by prospective students who looked between engineering and physics for their future careers.

Then I had it—I would do both! I would put my newfound knowledge to the test by determining the exact frequency I would need to sing at to induce the pressure differentials of the waveforms to lift me out of the well!

I set about the task. Scanning the walls of the well, I accounted for each and every brick and crack and nonuniformity both above and below the water. Every brick and every nook and cranny and protrusion brought its own set of expressions to the symphony I was weaving. The thought brought a smile to my face: unlike our greatest physicists and mathematicians, my modelling was as real as the world I lived in. It accounted for everything. Whereas we have historically only known the mathematical expressions for simple objects, modelling of the resulting waveforms assuming ideal shapes like the infamous spherical cow, I could finally create the world entirely anew within the realm of symbols.

In the pedestrian ways of the rest of humanity, in order to increase the accuracy of model outputs, the results from several simple shapes would have to be combined—thus the results could only ever approximate reality. Infinite series encroached on the elegance of a finite solution. It was like computing the digits of Pi by increasing the number of sides of a polygon; the true value of Pi is approached, but by necessity never reached. It cannot be, unless the number of sides is infinite—but one would continue forever in increasing the number of sides, exhausting all the paper and ink in the universe, before reaching the last digit—because it does not exist.

Fortunately I didn't have to approximate. In my present state I finally glimpsed the solution to our problem of irrational numbers, seeing that it was all so perfectly simple; a mere trick of our means of representation. We were bound by number lines when the universe operated on no such thing. Circle as a concept descended into crude matter and morphed into the relationship between points and center, the whole becoming scattered in a numerical universe. I took circle and put it into my equation. I similarly found the gravitational constant to simplify by redefining our units, as is done when using Planck units, whilst also collapsing our remaining constants into this tidy new system.

The well water sloshed at my waist as I rotated, surveying my prison. The damp stones glistened. Moss quivered in a gentle downdraft, foretelling the subtle chill that whispered past my ears just seconds later. The coolness, mixed with the sound of the wind rustling through the leaves far above, briefly scattered the mathematics that whirled like dervishes about my head.

I took a breath of the freshened air, enjoying the feeling of experiencing without knowing, but it only lasted a second. In a flash, the dervishes danced ever faster, faster, spinning a million tales of all the possible states of the universe that would have led to that exact breeze causing that exact sound. A dizzying number of white symbols, chalked onto the blackboard of the dark well, spoke of every leaf on every tree, picking apart the countless waveforms that together gave rise to the *whoosh-ssshhh* that I heard in that moment.

I had to get out before I traced the path of every particle in the universe! I had to sing, produce the note that would lift me out of the well, before it was too late!

I fell to my knees, the water climbing my chest, settling on my neck, cool, but swiftly warming. I opened my mouth, took a deep breath, and sang.

The water shook, trembled, then sloshed. Waves built up around me. I modulated my tone and the well amplified it until it was nearly deafening. I added undertones and overtones and the water started bubbling—no, boiling! The walls of the well undulated. With every change I altered my call subtly, so subtly. The mathematical hallucinations blurred, images turning to simple notions—intuitions—in my mind. They were now a part of me.

I paused to take a breath, slipping it in before the water could settle. Yet in that momentary silence I recognized something in my tune. It was a sound I had heard many years ago on top of a mountain. But enough of that—I had a task to do. I sang and sang, and the water danced in a mesmerizing pattern, and the air and the water become so compressed in some places and so rarified in others that they produced sparks of light, little balls of plasma winking into and out of existence.

I started to rise, though my legs did no work. I was so close, so very close! Just a few more alterations to the waveform and I would have my freedom! And yet I couldn’t do it. Though I knew in my mind how to make my escape, I was limited by my vocal cords. Even with the addition of certain Tuvan throat singing techniques my efforts fell short. I was slowly sinking back into the water despite working harder, singing louder, than ever before. Lower, lower, into the shadows, until—

The outline of a villager eclipsed the sky. He reached for the bucket and was about to lower it when he froze. What he beheld was my levitating form, surrounded by pulsating waves that danced to my ululations, lit by the fires of my own standing waves. I did not have to reconstruct the firing of his neurons to understand that this was more than he had bargained for when setting out to fetch a pail of water. He vanished the next moment, screaming and shouting about a madman in a well.

Then I was alone again in the dark and the damp. I shivered as the temperature plummeted, pushing away visions of thermodynamics as I leaned against the wall. My equations had failed me. I slapped the water in anger. The ripples radiated away from my hand, and reflecting against the wall, they returned to me, tugging on my shirt like a group of small children. No, I told myself. I had failed my equations.

I looked up. How would I get out now? The well was so very deep, and I was a simple, weak human, more aware than ever of the limitations of my body.

Yes, I was just one human. What could I do? The answer to that question came in a flash, bringing with it my best idea yet. I was still making a few half-hearted vocalizations, but I stopped them instantly and began the call that would finally spell my salvation.

“Help!” I screamed. “Help me, I’ve fallen down a well and I can’t get out!”