“This is rather uncalled for,” Evenstar muttered as he strained against the lengths of rope around his forelegs. When Amethyst initially suggested we physically restrain Evenstar, I didn’t believe her to be serious.
Once he coupled this newfound energy with his small dose of physical clumsiness, we restrained him for the benefit of us and the lab.
“I think it was quite called for,” Amethyst muttered as she carefully lifted the several items that had been tossed from the desk and returned them to their place. “I’ve never seen you like that.”
“To be honest,” Evenstar started, “Neither have I.”
I let out a small snicker. The whole chaotic event had happened in only a minute’s time, but it felt as though it had lasted an hour. It seemed that everything suddenly slowed down, time itself bending to the chaotic stumbling of Evenstar. He had bumped a few tables, and the one that we sacrificed to bring him to a halt had been all but destroyed from my well placed sidestep. I don’t think I hurt him, but even if I had I would have snickered all the same.
“Is something funny?” Evenstar asked as a wry smile spread across his lips.
“Oh many things are. Just look at us. We had to tackle a fully grown stallion because he was giddlily prancing through the tables because of some numbers.”
“Galloping,” Evenstar sternly corrected. “Prancing is for fillies.”
“So is knocking over tables,” Amethyst chuckled.
“I was pushed!” He howled, causing us to erupt in laughter. It wasn’t a scene that we saw frequently in the laboratory. When we did it was nothing short of magical though. These moments had led to the three of us to become good friends. Evenstar’s initial cold and calculated exterior had given way to a warmer natured old stallion. He was precise, but he was a lot more easygoing than I initially imagined.
It took several minutes for us to regain our composure, but as the laughter settled, Evenstar’s confidence returned. It was evident that we were on the precipice of something great, that much I could determine from his eyes.
“Well,” I started. “What did you discover?”
“Hmm?” he hummed, “Ah yes, the proof. You see, Starswirl, I think I’ve finally convinced myself that you are indeed correct about your assumption. These limits seem to be the key to unlocking the world of the infinite. Math seems to no longer stand in its way.”
“How so?” Amethyst asked as she stepped to the chalkboard.
“Well, I haven’t discovered anything. It was Starswirl that made the discovery. I have just simply proved it, or at least, I think I have.”
Amethyst readied her chalk.
“Alright, here’s the idea. What is the limit of Y equals X as X approaches two?”
“Well it’s two… Obviously.” I said, a bit confused by his simple question.
“Exactly. So what about X equals one or three?”
“That’s one or three…” I still had no idea what he was getting at.
“So, what distance is one away from two?”
“It’s… one unit away? How is this important?” Amethyst asked as she let the chalk fall to its tray.
“Just watch,” Evenstar said. “So when X equals Y, if a point is within one unit of 2 on the X axis, it is, by definition, one unit away from 2 on the Y axis.”
As Amethyst drew that graph on the board, it slowly started to dawn on me what Evenstar had pointed out. There was a measurable, consistent distance between X and Y that could be calculated. “So then all we have to do is make that distance arbitrary, and then we have our proof!”
“I…” Evenstar sputtered as his head snapped toward me. “Well yes, of course! And then, once we find our solution to that function, we can then, retroactively, prove all other functions! By making that arbitrary distance small enough, we can get as close as we want to our answer!”
With those two statements, we had fundamentally solved the problem of infinity. We had used such a strange concept against itself in order to further the world’s understanding of itself. Looking back on that exchange, the time when we first cracked infinity, I never truly understood the impact that we had just made. Not only did it open doors for a new way of evaluating functions, we opened doors for an entirely new branch of mathematics. I was still an infant in my understanding of these concepts, but as time went on, we realized the implications of our discoveries.
Amethyst hurriedly wrote the proof on the board, outlining the indescribably huge amount of time spent on its formulation in one small equation. We all stood in silence as she let the small piece of chalk drop to the tray. It was a reverent silence, one that urged our core to remain nearly motionless in the presence of such a monumental yet tiny concept in mathematics.
“Well…” Evenstar said several minutes later, “That’s step one.”
“Step one?”
“Yes. We can’t tell a soul about this proof until we’ve ensured that it is sound. Then, once that has been completed we must begin drafting a proof to the Academy for a formal review. Once it is approved, then we may request publishing. There’s a long road ahead, but luckily, the hardest part is behind us.”
Amethyst spoke up. “Also, we have a long list of other things to accomplish ahead of us. Evenstar has done this process before, and since I can use magi-”
The motion of the conversation ground to a halt. I hadn’t realized it, but Amethyst had struck a sort of nerve in my heart. My ears began to ring as I mulled over the implications of her sentence. They didn’t need my help. I was just a pawn. Every fear and self-destructive thought that I had toiled over only a few months prior suddenly came racing back to my mind.
“Yeah… no problem.” I said as I choked back my tears. “I’ll keep everything else… clean and organized.”
“Starswirl, I didn’t mean it that way…” Amethyst pleaded.
I wanted to believe her, and I did in my mind, but my heart fought against my reason. It was the first time since I had started that I was reminded that I was somehow lesser than they. Their understanding and control of magic far surpassed my own, and despite the intentions of her words, they stung deep.
“I know, it’s fine.” My lie wasn’t well crafted, but it kept her from prodding. The last thing I wanted was to let her know of all the chaos that began swirling in my soul.
The two set off on their work, hurriedly scavenging for paper and books, but I could only sit and watch as they began the next step. Their focus was now directed on formalizing the proof, something that Amethyst had studied and Evenstar had practices. They had experience here, just like their experience in magic, and I didn’t want to disturb them.
So, to keep myself busy as they busied themselves, I turned my attention back to the book I had been assigned to. I had just reached the chapter where the philosophers were addressing the public about their discovery of algebra. To be honest, I had expected the discovery of their studied system to mark the end of the story, but rather it came near the beginning.
At first I had no idea why, and as the philosophers began their lectures, I finally understood why. Algebra wasn’t just a new set of equations or some other generic solution to a problem. No, instead it was just the beginning for the ponies who took the system of algebra and used it to its potential. The creation of algebra, that fundamental ruleset for discovering variables in equations, wasn’t blazing a new trail in science, it was like finding a completely uncharted continent. The more they professed and lectured about their discoveries, the more they realized what potential it had. Months went by with the two of them becoming extraordinarily famous for their work, and yet, they had no idea of the vast number concepts that would emerge from their discovery. Algebra had changed the face of the planet.
I tore myself from the book as Evenstar galloped across the room to retrieve something. “You don’t even know what you’ll do to this world… do you?” It was an interesting thought, one that I hadn’t considered until now. The research into the Analog Equations wasn’t really research into equations at all. It had the potential to be so much more than a simple equation, and the more I thought about it, the more I wondered why he had called them equations in the first place.
We were looking for a system, a way of relating two seemingly independent components. There wasn’t any equation involved, rather, it felt as though looking at the entire system as an equation was something that would limit our appreciation of its power. “You don’t know what you’ve stumbled across… I can only hope that you get to see the fruits of your labor, Evenstar.”