Memoirs of a Magic Earth Pony

by The Lunar Samurai

XXXVII: Epsilon

“You have my attention,” Evenstar said breaking the silence that had descended upon the three of us. “This is an interesting way to approach the problem, and it does seem to give a clear answer, but there’s nothing here that we can do beyond conjecture.”

Amethyst spoke up. “If it’s just conjecture, what’s the point?”

“Because…” Evenstar whispered as he brought his hoof up to the board. “I’ve never thought about using infinity like this before. Starswirl, I think you’re onto something.”

That sentence sent a chill down my spine. No longer was I the second rate earth pony with little to no magical understanding. I was finally doing what I had wanted to all along. The concept was my own, one that I had found drifting through my mind, not cherry picked from somepony else.

“However,” he continued, rupturing my train of thought, “This is just a theorem. We can’t use it until we know it’s true.”

“But isn’t it obvious?” I asked in half astonishment. I had yet to realize that the two of them did not see so clearly what I had conceived, and that my incredulous responses would not assist in their understanding.

“It seems to be true for this case, but what about others? There may be substance here, but there is a chance that this method only works with one over X, or a similar problem. It’s not mathematically sound as of yet.”

“What do you mean, yet?” Amethyst asked as she cocked an eyebrow. “Is somepony else working on it?”

“I can’t say for sure, but it’s promising enough for us to take a look. Who knows, maybe this will help us with the analog equations.”

His words were so nonchalant that they hung in the air before they made sense to my mind. Evenstar was offering to sacrifice time on his life’s work to pursue a fleeting idea I just had the chance to solidify in my mind. There was a sense of curiosity about him, something that I rarely saw in his tired eyes. The conjecture was beginning to enrapture him, just like it had with me. With every minute that we considered starting to work on them, the more alert the both of us became.

“You sure you want to do this?” Amethyst asked as she glanced at the board. “The Analog equations—“

“I know,” Evenstar interrupted. “They need to be our main focus, but I can’t seem to shake the feeling that this will somehow be useful to us. It’s describing infinity, but that’s just an infinitesimal by any other name. The notion that Starswirl may have come across a method for calculating with infinity is not to be taken lightly.”

Then, without warning, he turned to me. “So, Starswirl? What do you think?”

“What?” I asked as I retreated with a few steps. “I mean, I don’t know. It’s not like I have any real say in the matter.”

“Why not?”

“You’re the researcher, not me. This is your lab; you do what you feel is most important.”

“Starswirl,” Evenstar said as his lips stretched into a subtle smile, “I may be the researcher here, but this is your idea. I can’t see what you saw, I can’t know what you know, so I can’t, in good faith, choose either way without getting your input.”

I looked to Evenstar and then to Amethyst. Their eyes were full of questions, as I am sure mine were, and I was the only one who could answer. Evenstar needed what I saw to bring it to reason, but he needed to know what I thought about it. I had the fate of the analog equations in my hooves.

“Well?” Amethyst prodded, forcing me to lock eyes with her. I could see she was almost as apprehensive as I, her eyes whispered of worry and excitement wrapped into a tumultuous chaos in her mind. I’m not sure how I knew, but I felt it as she stared into my own eyes.

“I guess…” I started as I tried to collect my thoughts. “If it won’t take too much time, I think we should at least try.”

“Excellent,” Evenstar said with a nod. “Remember, Starswirl, inspiration will strike at the most inopportune times. You have chosen well.”

And then the room fell still. I was expecting Evenstar to lead the project, but for some reason, he had decided to put me in charge instead. “Well?” he asked as he took a few steps from the chalkboard with Amethyst quickly following. “Where shall we begin?”

“What?!” I nearly shouted. “I don’t have a clue!”

Evenstar snickered. “You’ve got an idea, that’s all you need to start. The better question is: Where do you want to end up?”

I caught myself before I responded with my ill-conceived answer and pondered the question for a moment. Where do I want to end up? The question was one that I never really considered before now. I had always put large vague goals before my life, ones that I had only start to cross, and even then I never really knew when I did. I wanted to study magic, but I had been gradually dropped into the subject, not suddenly woken up one day as a magician.

Either way, I now had to figure out what I could do with this concept and prove it, with the help of the others of course. The more I thought about my responsibility, the more worried I became. “Uh… I have no idea where to start. I mean, sure I’ve got a concept of what I can see in my head, but I don’t know what to do with it.”

“Have you ever heard of mining?” Evenstar asked as he gestured toward a hunk of raw ore lying on one of the bookshelves.

“Well of course. Why?”

“I’ve used this analogy in my classes when discussing research. I think it’s quite relevant now. A hunk of ore alone is worth nothing. It would take hundreds of hours to extract the gold laced inside of it, but there’s hundreds of bits worth in that rock. Your concept is like that ore. It’s not very valuable in a useful sense, but once you refine it, mathematically explain it, prove it… then its value becomes apparent.”

I nodded as my mind slowly wrapped itself around the process ahead. It wasn’t going to be easy, nor was it going to be fast, but in the end, we were going to get our effort’s worth.

“So…” Evenstar prodded. “Where do we begin?”

“Well,” I said as I turned toward the board, “I guess we should start with what we need. It looks like this whole thing has potential, but so far all we know is that the process can show that one over X at infinity is 0.” I paused for a moment as I stared at the graph. “So what if we went the other way?”

“There you go,” Evenstar said as he watched me with his powerful gaze. “Now you’re getting it. Explore the concept, feel it out. The more you play with it, the more it will tell you about itself.”

I watched, in my mind’s eye, as X approached zero. As it did, the value of the function itself rapidly increased to incalculably high numbers. It seemed to approach infinity. The more I explored the idea, the more I felt I understood it. There was something about it that let me peer into the world of the infinite, to see its secrets, and that was a source of unending excitement and wonder.

“So, we can conceive this motion toward zero,” Evenstar said, trying to draw my thoughts together, “But what else can it do? Does your concept do anything mediocre?”

“What do you mean?”

“Well, we need a place to start. Here,” he offered as he tapped on the board. “Here’s a simple function, Y equals X. What can your concept do?”

We deliberated for nearly a week on how to approach the idea. Evenstar and I put our heads together to figure out the equations, and Amethyst told us when we were dead wrong whenever we incorrectly surmised a solution. The math turned out to be much more finicky than I had thought, but the concepts therein were so tantalizingly accurate that we refused to give up. It had taken months for us to come to such a strange way of describing math, and yet we were still so far from understanding its true use.

As we neared the end of the week, we had filled most of the boards with different conditions that Evenstar and I had put together to try and break the idea. Some of them shattered it completely, offering answers of both positive and negative values for infinity. Some others were mathematically sound in the most juvenile way. It was these quirks that really drove me to understand the fundamental issues with the mathematics that we were trying to solve. There was something about the typical procedures that never felt quite right, and blazing a new trail to unlock the secrets of infinity was something that drove me to work harder than I ever had before.

Despite our efforts, it took quite a while before we finally started making headway on the concept. We had at least given it a name, the limit. By now, the three of us seemed to have a fairly decent grasp on the way this limit worked. Our efforts to prove it wrong had become rather predictable.

In short, the limit of a function was like adding a bit of magic to a standard function. Instead of just plugging in a value for X to solve, you guessed closer and closer to that value of X until you figured out where it was heading. Just coming up with that description alone took all of our heads put together, but it eventually wrought a strong foundation that we could use to build our proof. That method of bringing the function closer to the desired value worked if you could use your imagination to fill in the gaps. It even seemed to work on functions specifically designed to have no answer at a certain value for X. Since we could bring our imagination into the mix, our understanding of mathematics was beginning to grow.

Evenstar, Amethyst, and I worked feverishly to understand why such a strange concept would work so reliably. It wasn’t until that one fateful day that Evenstar had a breakthrough. I remember him racing across the room to the board with Amethyst hot on his tail. He had figured something out in his office and he was ecstatic to find out how he had possibly proved the limit.

“Alright, I’ve got something so crazy it just might work,” Evenstar repeated to himself as he rifled through the papers on a table. “Amethyst! You might need two pieces of chalk, because I’m going to talk fast.”

A moment later, and Amethyst had readied herself by the board. She and I both stood with baited breath as Evenstar composed himself.

“We can use this limit method for finding the slope on typical functions, right?”

We nodded as Amethyst began to write.

“And we’re trying to figure out the infinitely small portion beside the final X value, right?”

Again we nodded.

“Then in that case, what if we subtract an arbitrary value from our X value and take its absolute! By definition, we should be able to take the answer that we know and find it’s displacement on the Y axis as a function of the function’s absolute value when subtracted from the final limit!”

Neither Amethyst or I had followed him through that train of thought, and his conclusion left us profoundly confused.

“It finally makes sense!” Evenstar shouted moments later before galloping around the room.

“Alright,” Amethyst whispered through the side of her mouth. “If you take him down, I can tie him up and we can get to the bottom of this.”

Comments

[Booster Spice]

Welcome to Normalisation.

Instead of trying to find the value made up of the huge value and the small adition. subtrract the previously known huge value, and whatevers left, must be the truth. The change in the function at infinity. Or infinitely small as the case may be.

Or you can shoot some more arrows at that tortoise and keep trying to hit it.

[Kevin Lee]

They've finally gotten outside of High School math class: They've realized that ∞ isn't just a theoretical concept that can never be reached, but is a real number with a real value.
Finding the slope of a line 1/X as X->0 IS ∞ and as X->∞ IS 0. The next step is to realize the relationship that idea has where it comes to Trig.

"Slope" is actually the definition of TAN(ϴ), which is also the relation of SIN(ϴ)/COS(ϴ), and immediately leads back to the cyclic nature of the cartesian coordinate system, cementing the implied relationship of +∞ = -∞ as a real value, in addition to all the other Trig identities. At the same time, Trig will make their efforts to understand, construct and expand Calculus much easier.

BTW, this will be a good movie to reference:

[MudakTheMultiplier]

Chapter XXXVII
Epsilon
Or:
In which Tackleswirl the Heavy and Douchethyst the Magical hog-tie their mentor.