The Only Chapter
Twilight Sparkle rounded the cloud and swung around to the left, heading for the next cloud as quickly as possible. Passing that, she slalomed around it, flying for the finish line. She pushed herself as hard as possible, and flew past the finish markers in what she hoped was record time.
Rainbow Dash was lazily flapping her wings at the end of the race track, and hit the stop button on her watch as soon as Twilight flew past. "Not too shabby, Twilight!"
Twilight slowed down to a stop, and flew back to where Rainbow Dash was hovering. "How'd I do?"
Rainbow playfully hoofed Twilight in the shoulder. "You're making good progress. You shaved three seconds off your personal best."
Twilight grinned. "I'm really getting the hang of these wings now."
"It must be your awesome trainer!" Rainbow puffed out her chest.
Twilight chuckled to herself. For some reason the book of Greek philosophy that she'd read when visiting the human world popped into her head. Particularly the passage by a certain Zeno of Elea. She decided to play a little trick on her friend. "You know, I think I'm nearly as fast as you now."
Rainbow Dash burst into laughter, holding her belly to stop it splitting. After a few minutes she calmed down, and wiped the tears from her eyes. "Good one, egghead!"
A sly smile crept onto Twilight's face. "I'm serious," she said. "Forget these obstacle courses, but on a straight track, if you gave me just a small head start, I could beat you."
Rainbow laughed again, but soon stopped when she saw Twilight's smug expression. "You're really serious?"
Twilight nodded.
Rainbow was stunned — she would never boast about it, but she was one of the fastest fliers in Equestria. "Twilight, you do remember that I've trained with the Wonderbolts and competed at the Equestria Games? I was even crowned Equestria's Best Young Flier a few years ago!" Who was she kidding? Of course she would boast about it!
"But you said yourself, I'm making good progress."
Rainbow rolled her eyes. "Yeah — good for a pony who's only had wings for a few months."
Twilight flew in a little circle around Rainbow Dash. "Well, I still believe I could beat you."
Rainbow was getting tired of this conversation and decided to put an end to it. "What sort of race are we talking about then?"
"A hundred yards," answered Twilight confidently.
Rainbow smiled. She could cover that distance in seconds. The race was in the bag. "And how big a head start do you need?"
Twilight thought for a moment, pretending to do some calculations in her head. "Ten yards."
Rainbow Dash laughed louder than ever. "I thought you'd ask for fifty. There's no way you could beat me with just ten yards' head start," she told her friend, "but if you want to humiliate yourself, feel free to try."
"On the contrary," said Twilight, "I will win, and I can prove it to you."
"Go on then," Rainbow Dash replied, with less confidence than she felt before. She knew she was the superior athlete, but she also knew that Twilight was smart, and wouldn't make such a claim unless she had reason to do so.
Twilight hovered in front of Rainbow Dash and started her explanation. "Suppose that you give me the ten yard head start. Would you say that you could cover those ten yards between us very quickly?"
Rainbow nodded and gestured dismissively. "Very quickly."
"And in that time, how far would I have gone, do you think?"
Rainbow looked at the stop watch, and looked at Twilight, trying to judge her speed over an obstacle-free course. "Perhaps five yards — no more."
"Very well," continued Twilight, "so now there is five yards between us. And you would catch up that distance very quickly?"
"Very quickly indeed!" Rainbow replied with growing confidence.
"And yet, in that time I would have gone a little way further. So that now you must catch that distance up, yes?"
Rainbow's confidence began to falter. "Ye-es," she answered slowly.
"And while you are doing so, I will have gone a little way further again, so you'll need to catch up the new distance," Twilight continued.
Rainbow said nothing.
"And so you see, the whole time, you must be catching up the distance between us. And I — at the same time — will be adding a new distance, however small, for you to catch up again."
Rainbow's jaw dropped. Everything Twilight said seemed to make sense. Once she regained control of her facial muscles, she had no choice but to admit it. "Yes, that must be right."
"And so you can never catch up," Twilight concluded.
Rainbow Dash was speechless. She knew she was faster than Twilight Sparkle, and should be able to easily beat her in a race, even after giving her a small head start. But she couldn't find fault in Twilight's logic.
Twilight flew in another circle around her friend. "So, should we race then?"
Rainbow Dash looked around at the other pegasus ponies training on the flying course. If she lost a race to Twilight Sparkle in front of all these ponies, she'd never hear the end of it. No longer would she have the reputation of the best flier in Ponyville. She sighed defeatedly. "Maybe some other time."
Rainbow would have to step up her daily training regime.
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This is the same paradox as... god I can't remember its name, but the point A to B thing. To get from point A to B, you have to travel first half the distance. Then, to get to B from there, you must first travel half that distance. then half of that, then half of that, infinitely, until the half-distance is infinitely small. In theory you should never be able to get from point A to point B, yet we prove everyday that we can.
I like paradoxes....
5662670 That one is called the dichotomy paradox, and they're each variations on the same idea, yes. Zeno actually wrote about both of them. Aristotle considered them to be the same thing as each other.
Hah, I enjoyed that.
Congratulations, this story is number 250000 on FIMFiction.
Upvoted for 250k GET, also good story
No, seriously, good story.
It's not a paradox because an infinite sum doesn't have to be infinite. 1/2 + 1/4 + 1/16 + 1/32 + 1/64 + ... all the way to 1/infinity is equal to 1.
So, Rainbow was replaced by a changeling!
5662850 As I said in the notes at the bottom, the idea is no longer considered paradoxical. Back 2500 years ago it had them stumped though. They didn't have the same tools to deal with infinite series that we do.
But really, the problem is not summing an infinite series. The problem is paying attention to the infinite series. It's artificial to consider smaller and smaller units of distance when all we care about is who goes 100 yards first.
5662670
The theory you are summarizing is one of Zeno's surviving Paradoxes, which is in fact the title of this story. Zeno wrote the seven paradoxes in order to argue against Aristotle's physical theory of motion. The modern simplification of his argument is what you have described, namely that in order to travel a measured length, an object must travel a smaller length, and so on ad infinitum. However, each of the seven paradoxes argues essentially the same conclusion. This version is more reminiscent of the swift Achilles and the slow turtle; but I would prefer to deal with Zeno's paradox of the arrow:
Premise 1: Everything that occupies an equal space is at rest.
Premise 2: Everything in locomotion occupies an equal space at any moment.
Conclusion: Everything in locomotion is at rest at any moment.
This appears to be a paradox, because the conclusion seems to be self-contradictory. The problem with this argument, however, is that at least one if not both of the premises fail to hold true. Looking at premise 1, I just don't see how it can be certainly true; since it's logical negation, "something occupies an equal space that is not at rest," seems very plausible. I think the main reason that people give credence to this premise is that they are fond of viewing time as composed of discreet units; but this, I think, is merely a invention of human design. We see time discreetly, because we want so badly to measure it. As mere observation will tell, this premise in fact goes against the whole of human experience, which views time non-discreetly. As Heraclitus put it, time flows. Thus I think that premise 1 begs the question.
Premise 2, on the other hand, may be seen as meaningless. As one might see in the work of the quantum physicist, Hendrik Lorentz, according to the Minkowskian interpretation of Einstein's fundamental equations, depending on the reference frame of an observer, as objects accelerate and maintain velocities that approach the speed of light, they shorten. Thus the same object which moves at different speeds in fact does not occupy an equal space. Premise 2 therefore is at best false, and at worse an unspecific assertion, which does not hold a meaningful truth value.
Story # 250,000
Has 1,000 words
That's a lot of zeroes.
(Twilight, stop using your brain to bully poor Rainbow Dash!)
Rainbow Dash isn't the smartest pony in town, but she wouldn't give this up without a fight. I could say the same thing between me and an Olympic sprinter, but he'd still blow past me in a heartbeat, no matter what I say.
I wont read this, but I love the concept. I'm a philosophy major and this just makes me smile each time I read the title.
I think my brain exploded from overthinking this too much.
5666086 I'm trying to think of other Socratic dialogues to ponify. Many of the ancient Greek dialogues are pretty long, and would be difficult to condense down into a bite-sized one-shot story like this. But I am interested in doing others, probably with other foils for Twilight. (I can't have her picking on Rainbow Dash constantly.)
I can't actually see Dash giving any credence to the philosophical conundrum proposed by Twilight. She's far to action oriented a mare to back down because she can't find the logical solution to something she knows to be true. The only way I can see this little prank playing out is for Dash to simply say "Let's do it!" and having the race. Which would, of course, end in a resounding win for RD. As well as weeks of teasing rights.
Now if this little puzzle was presented to Pinkie... That would be an interesting story!
Well, the actual Paradox is that a finite system can be infinite as well.
It was resolved as we understood that you can divide a static system an infinitely number of times.
Another paradox, for you.
Based on Xeno's, it was crafted by Lewis Carrol.
Essentially, it takes the idea of the Syllogism (If A=B and B=C, then A=C), given as properly logical.
Achilles puts that forth.
The Tortise puts forth that it's only true if you accept the idea of the syllogism as also true. Therefore, if A=B and B=C, and Syllogism1, then A=C.
However, you can also reject the veracity of that, and keep going down the proverbial rabbit hole.
(If A=B, B=C, and we accept that the Syllogism is true, and we accept that we accept that, and we accept that we accept that..... then A=C)
A great one for breaking Twilight
5668320 Yes, I'm aware of the Lewis Carroll one. I'd planned on doing that one first with Twilight and Spike, but I couldn't think of a way to keep it interesting, so did this one instead. I'm considering covering the use–mention distinction in the style of the white knight from Through the Looking-Glass.
Maybe someday RD will be more of an egghead and figure out that infinities are summable. Eh, probably not.
Nifty.
Next time on My Little Pony, Twilight explains background ponies using Hempel's Raven Paradox...
It actually depends on where the limits go to. If it's at or beyond the designated endpoint, we have a winner-or at least a tie. If it's before that, though...the logic is unsound, because you can't hear a sound after your ears have just been blown out.
Wait, the boom actually takes its toll on the eyes, not the ears...but then is it really a boom? That's another fillysoflycal question for you all!
5671187
Of course it isn't. The original version of the paradox is with a tortoise and Achilles, the greatest athlete in Greek mythology. Naturally the tortoise could never beat Achilles, even with a massive head-start, but the way the tortoise describes how the race would go convinces Achilles that he could never overtake the tortoise.
Hopefully I made it clear that Twilight doesn't actually believe that. She's just messing with poor Dashie's head.
Edit: OK, you've added a graph to your comment. The way Twilight's logic makes it appear that Rainbow could never overtake her is basically by examining progressively smaller chunks of that little triangle in the bottom left of the graph.
Poor Dashie.
That was petty funny, mindgames ftw!
Oh, Rainbow, silly rainbow.
Twilight's proposed paradox speaking made you forget theory is much different than physical placement.
5940119 In theory, everything that works in theory should work in practice. In practice, it doesn't.
5940135
Exactly! and here I thought I was doing all of those physics labs wrong.
5940135
Great little quip, by the way.
What? How is this a paradox or a mind game, as everyone's saying? It seems more like Rainbow Dash didn't consider that she could eliminate the head-start faster than Twilight could elongate it, to the point where they'd be in the same position at some point before the finish line where Rainbow would then overtake her... I mean once Twilight's head start reaches a number close enough to 0, it might as well be 0...
Oh, I just realized the whole point to the paradox was that they don't consider that there isn't an infinite measurable distance can be divided.
You should do a sequel where Twilight tries her logic out on Pinkie. One broken universe later...
6010640 Need to come up with some philosophical problem that can be expressed via the use of cupcakes.
6069971 Thanks. I wrote this one pretty quickly — it took an hour or two rather than my usual weeks and weeks — but it turned out not to be so bad.
It's one of my few stories that doesn't actually ship any ponies. Weird, right?
6069982 I liked it. It's very very cute. Good job.
Ah, Zeno's Paradox has two parts. She recited the first. The Second disproves all possible motion by saying to move to a point you must move half the distance, the a fourth, then an eighth, and so on.
And from that moment on, Rainbow never let anyone have a head start.
Dash Really needs to get her head out of the clouds if she wants to beat Twilight in a mind game.
......
The inevitable sequel has Rainbow Dash accept the conditions to the race and then beat Twilight even with her head start. Rainbow Dash then turns out to actually be a hardcore math nerd who already knows of Zeno's Paradox as well as the proven solution, and sends Twilight into a conniption as RD explains the solution in excruciating mathematical detail
7886520
5671234
VitalSpark, please make a sequel, or at the very least a follow-up chapter.
It feels like RD got jipped, and I really don't like when characters take advantage of anther's lack of intelligence for petty reasons...
7991854
Just published a sequel.
Heh.
Honestly, I would have thought Rainbow would be more of a Diogenes kind-of thinker.
Boy, there's mathematics flying fast and furious and I haven't even read it yet. You had to have adjusted it to 1000 words exactly after seeing the story number.
5662670
There's been some good takedowns of it.
8023582
I quite often aim for specific numbers of words. Six of my stories have exactly 1000 words. Another five stories have multiples of 1000. Quite a lot of others are multiples of 500, 250, or 100. Two have word counts which represent the year of somebody's birth. One has 4096 words (which is 2 to the power of 12). There's a 4444 word story, a 2121 word story, and a couple of word counts are a reference to a friend's favourite number.
The Zeno paradox is solved by math: s= a((1 - r^n)/(1 - r))
Enjoy
8023626
This must have been the perfect OCD wet dream aside from like, a perfect 30000, then~
8023968
Yes, I was pleased with the number.
Also, a friend is writing a story with 12 chapters, with lengths 500 words, 1000 words, 1500 words, 2000 words, etc. And a favourite book of mine, The Luminaries, has its chapter lengths as a reverse Fibonacci sequence (as well as having a load of other geeky patterns in the story).
5662682 HA! IT TOOK TWO YEARS, BUT I NOW HAVE PROOF THAT RAINBOW DASH WOULD WIN!
Basically if we were to take a limit of the summation it would converge on a non-infinite number.
I DON'T CARE IF IT'S WRONG! I'M TIRED! WOOHOO! YEAH!
Zzz...