Spending several days reading about the development of Algebra was one of the most monotonously fascinating things I did in that lab. It was boring work to say the least, but there was still something rather enticing about it as I turned through the pages. I saw a connection between those age old philosophers and Evenstar, between their assistants and Amethyst. It brought a sense of wonder as I let myself become engrossed in the fanciful images that it conjured in my mind. They weren’t unlike that of the lab I was a part of aside from the more recent advances such as glass. It was interesting watching them work through the day, trying to figure out what I knew so intuitively now.
It was prophetic, in a way, of how I would one day look back on this situation and laugh at my own stupidity. The analog equations, those infernal methods that would tease our minds with solutions to unimaginable problems were so elusive to us. They laid right before our muzzles, and yet we were too blind to see them. The relationship of a line and its slope would unlock worlds beyond our wildest dreams, but we were unable to bring them together.
As I read further, I noticed the deception and lies that had been twisted into the lives of the philosophers. Many had once believed that they were the harbingers of the truth, but they were not unlike the ponies who held them in the highest regard. It was the lies which fascinated me most. In one rather striking instance, one had claimed to venture thousands of miles to an ocean to meditate, but there was something strange about his account. The time he was gone and the distance he could travel would never make sense. Even though ponies had said they saw him in cities walking toward the ocean along his journey, he could not have made the entire distance even at top speed. There always seemed to be something off about their work, and that forced me to raise a few questions of my own about the research we were performing.
The analog equations were the topics of discussion that constantly shot through the room. Despite having many dead ends, there was an underlying fervor to their work. Both Amethyst and Evenstar felt as though they had one last chance to prove his grandiose ideas. They worked tirelessly for nearly two weeks as I was occupied with my reading, sometimes leaving before them and arriving several hours after they did the following day. It was incredible to see them work with such intensity. Evenstar seemed to have cracked, in a way, as his focus was solely on those equations. He barely gave himself time to eat and forced Amethyst to work round the clock to keep up with him.
Every now and again I would sit back in my chair and let the two of them work as I just watched. It was so fascinating to see them play off of one another as they scrambled around the room. Evenstar would fire off a list of equations, which Amethyst would scribble on the chalkboard, erasing old ones as she went. They worked with an ever increasing intensity that enraptured me.
Over the first half of the week, the deliberated on how to approach the problem. Initially, Evenstar remarked that he and Amethyst would work from the back of the problem, attempting to solve the problems of the world with the abstract concept of the analog equations. This, however, proved to be even more difficult than developing the equations. I honestly think he just let his excitement get the better of him, but either way, the duo soon chose to focus on developing the equations again.
It wasn’t nearly as exciting, as prior they had struggled for months with seemingly no progress, but somehow the looming sense of dread was just a bit less oppressive. Earlier they had worked in a state of near depression about their work, but now… now they chose to embrace it. Every error they took only fueled their drive to work at the problem once more.
Every time I would sit back and watch, I noticed the two of them glancing in my direction on occasion. Evenstar’s consternation wasn’t easily masked, and I saw him sigh a time or twenty as he returned to whichever task he was working on. However, Amethyst was different altogether. While Evenstar turned toward me whenever he wasn’t working, she looked at me when she needed help. I didn’t pick up on it at first, but one instance tipped me off to the pattern.
Amethyst and Evenstar were, as always, going on about something that I was only halfheartedly paying attention to. Between their accelerated speech and my fleeting interest with my reading material, there wasn’t much I could focus on. However, the room grew silent and that’s what pulled my full attention to the two.
“That’s ridiculous,” Evenstar said as he cast a disapproving glare at the freshly drawn equation. His eyes scanned over it, each time slower than the last.
Amethyst only stared in response. The equation that drew our attention was one involving infinity. Only in desperation had Evenstar ever tried to employ the abstraction in his mathematics.
“There’s no way this is valid. Just look at it!” The more Evenstar ranted about its impossibility, the more troubled he grew.
“Why can’t it be valid?” Amethyst asked as she set the eraser aside. “It seems just as valid as one plus one equals two…”
“It can’t be,” Evenstar muttered as he furrowed his brow and drew close to the chalkboard. “But I can’t figure out why.”
With Evenstar now obstructing the equation, I let my eyes wander around the room. The board around the duo was filled with an oddly familiar set of graphs, equations, and diagrams. A few moments later, and I realized they were talking about the proof we had worked on only a few weeks prior. They had found themselves back at the analysis of the infinitely small distances the challenge had suggested.
“This doesn’t work,” Evenstar growled as he turned from the board. “If we solve it, we can prove that one can be equal to two, which can’t happen.”
“But look at the graph!” Amethyst protested as she pointed with the chalk. “The function arrives at 0 when x equals infinity!”
“Amethyst,” Evenstar started with a sigh, “I know it looks like it does, but we can’t be sure that the graph itself is accurate at that point.”
Her eyes squinted ever so slightly as she started looking around the room. She seemed to be searching for something to use as a demonstration, but nothing lent itself to her aid. Her intensity began to dwindle and, in what I can only assume as an act of desperation, she turned to me. It took her a half second to realize what her mind told her when our eyes locked, but when it did her mouth hung agape.
“Wait…” she whispered as she spun toward the board. “This equation doesn’t make sense because it includes infinity, right?”
“In essence, yes…” Evenstar said as one of his eyebrows began to rise. “What are you getting at.”
Amethyst tried to regain her composure, aside from a wry smile that spread across her face. She glanced to me and then to Evenstar. “We’re trying to use a bucket to carry air.”
“What?” Evenstar sounded as incredulous as I felt. “We’re doing math.”
She shook her head and started again. “Sorry, bad analogy,” she dismissed. “But I think I see what Starswirl meant!”
As you can imagine, the mention of my name snapped my mind to attention. I was not expecting to be referenced, and the sudden realization that I had been immediately put me on the defensive. I stared, wide eyed, at the two. Evenstar was as confused as I was, but Amethyst had a sparkle in her eye. She had figured something out.
“What are you talking about?” Evenstar asked as he shot a glance at Amethyst. “He hasn’t spoken all day.”
“When you were gone, Starswirl and I were talking about the shell spell I’ve been working on. When we started talking about the infinitesimals, he said something that I didn’t understand then. Now, however, it makes perfect sense!”
“Oh?” Evenstar asked as he took a step back. “And what might that be”
“We’re using old math.”
That’s when it clicked. My mind immediately recalled the time when we were talking about her magic. I had mentioned something about not being able to calculate using infinity because the math we were using was wasn’t built for such concepts. However, my uneducated mind at the time still demanded that something be done instead of giving up. I figured, if something wasn’t compatible with what we were observing, we should be able to make something that was. To develop a new branch of mathematics.
Amethyst’s gaze was locked with mine andm, had I been given a mirror, I would have seen that my face and hers were both stunned. Such monumental revelations weren’t meant for us, Evenstar was supposed to have them, but we had it instead.
“Then what do you propose we do?” Evenstar asked as he surrendered from the board. “What ideas do you have?”
Amethyst motioned for me to join her, which resulted in me bumping several tables as I galloped to her side. As I arrived before the board, she spoke. “I don’t know, but I think we do.”
I was too caught up in the moment to realize the implications to what she said. Instead, my mind was already swimming with equations, variables, methods and the philosophers who developed algebra, curiously enough.
“Alright,” I said as I took a seat and faced the board. “Why does this break math, Evenstar?”
“One of the quick tests I use to prove that something is invalid is the zero equals one test.”
“Alright, can you talk it out?”
“Certainly,” Evenstar said as he pointed toward the equation. “Right now, our function is Y equals one divided by X, which is simple enough. However, if we set X to infinity, we can’t divide X by infinity.”
My ears twitched at the mention of ‘can’t.’ “Why can’t we?”
“As I’ve said before: Infinity is a concept, not a number.”
“Alright, then we’ll fight fire with fire,” I muttered as I continued to stare down the board. “What happens when X equals ten?”
“Y equals one over ten,” Amethyst said.
“What about one hundred”
“One over one hundred.”
“And what about ten thousand?”
“One over ten thousand…”
Evenstar interrupted. “What are you getting at, Starswirl?”
I held up my hoof as my mind tried to bring the words to my mouth. I could see it ever so dimly, but I saw a solution. “It’s… going there.” Was all I could muster before silence took over once more. I shook my head and focused. I knew this was important, but the words refused to come.
“Where is it going?”
“It’s going to Zero, but… it never gets there.”
“What?” The two of them asked in unison.
“One tenth is small, but one hundredth is smaller, right?” I didn’t wait for them to respond. “Well, if we continue down the function, we see that it’s getting closer to the axis, but it never quite makes it. It keeps on taking steps, each one smaller than the last, for an infinite amount of time.”
“But we can’t rely on infinity,” Evenstar said. “So now we’re dealing with two infinite sums.”
“Yeah… we are, aren’t we…” I muttered as my confidence began to dwindle. I was so close, something was there, but it started to flicker. A silence filled the room as the idea began to falter. It was decaying right before my eyes, and I had nothing that I could do to stop it.
“You know, it’s funny…” Amethyst started, trying to break the silence. “We always talk about infinity like we know what it is, but whenever it shows up in equations, we don’t know what to do with it. I feel like we’re right back where we started.”
Come on… Think! I tried to drown her out, to keep my ideas in line, but her voice forced me to consider her words. So what if you find it funny, that’s not the point! The harder I thought, the less prevalent the idea became, it’s ethereal form vanishing in the background noise of my own mind. Sweat began to bead on my brow as I tried to force the final conclusion to come through, but it was to no avail. The concept that had tantalized me so powerfully had all but slipped from my mind, and the only thing I could do was watch it slowly fade away.
Then, when I released the notion from my mind, a new one appeared. We always talk about infinity like we know what it is… Her words rang through my mind. There was something there, something important. “Do we really know what it is?”
“Not exactly,” Evenstar sighed. “We just tell ourselves it’s something important and then dismiss its implications.”
“How?”
“Hmm?” Evenstar hummed at my retort.
“How do we dismiss it?”
“I’m not really sure…” he started, “But I think we just don’t even try considering something like it.”
“If we don’t try, why does math have to?”
“Because math’s math,” Evenstar said with a frown. “It has to deal with calculable values.”
“But why?” I’m not even sure I knew where I was going with this train of thought. I was trying to grasp the nature of the problem, but I was asking a looping question that seemed to get me nowhere in a hurry.
“That’s just the way math works, Starswirl. You should know that by now.”
I did know that, but I refused to take it at face value. I wanted a way to put my ideas into numbers. I saw that line, I saw it go to zero at its infinite end, but for some reason, I didn’t know why I couldn’t explain it. I needed a method, and that was where I was stuck. You couldn’t methodically count to infinity; it just didn’t happen. Infinity can’t be used it’s just a concept. “Can math even work with concepts?”
“Well, all of math is concepts,” Evenstar said with a chuckle. “I think you’re onto something, but I think you’ve lost yourself somewhere.”
I did not appreciate his observation. However, I kept my mind focused. “So we can conceptualize infinity, but math can’t?”
“Unfortunately you’re right. Math has no idea what to do with infinity.”
“But we can…” I whispered. “Wait a minute… Does math actually conceptualize anything?”
“No, that’s up to us to do. Math is just a notation.”
Amethyst chucked. “I believe some philosophers would like to argue the opposite. There are many ponies that suggest otherwise.”
“Alright point taken. But as for mathematics itself, we’re the ones who do the work, not the chalk,” Evenstar said. “Either way, I think we need to take a step back from this and approach it from another angle.”
As they began to dismiss themselves, I latched onto those last few words. We’re the ones who do the work, not the chalk. We do the work… so math doesn’t have to… That’s it.
“Are you going to resume your reading, Starswirl?” Evenstar asked.
I turned to the others and stared at them wide eyed in a state of wonder. “I… I think I’ve got it. We don’t have to make math do the work, we can do it instead.”
“Starswirl?”
“Amethyst…” I started as the world around me seemed to fade. “Can you write something on the board?”
“Uhhh… sure…” she said as she lifted a piece of chalk and readied herself.
“As X approaches infinity, Y goes to 0… because we can imagine it.”