Twilight and Mathematics [mathematics/humiliation]

Greentext Prompt:

>"W-Why am I crying? Why wouldn't I be crying? I dedicated my life to knowledge a-and I know n-nothing compared to you. You- You're an average college-educated human and you know more about physics and math than me. I didn't e-e-even know you could find the exact area under a curve. D-Do you have any idea how many times I've added trapezoids together?"
>"No matter how much I study, I'm stuck with pony advancements. In comparison to human ones, they're w-worthless."

--

"Does Equestria have any mathematicians?"
>"What's a mathematician?"
"You know.. someone who is employed to study mathematics."
>"How can one study mathematics? Math just describes certain patterns in the physical world. I can understand studying magic or studying history, but how can one study mathematics? It's not even a thing."
>No wonder ponies' understanding of math is so primitive. Their conceptual paradigm trivialises it.
"Twilight, mathematics isn't a list of physical, empirical patterns of properties like sine and cosine. Mathematics has nothing to do with the physical world. It is the abstract study of concepts that are in some sense completely unambiguous, a study of the dynamics of rigorously structured thought. Mathematics *is* logic, Twilight, and logic is mathematics."

>"B-but I'm logical, anon. At least, I try to be. But logically speaking, if mathematics is logic, then because I am not mathematical, I am not logical. Am I illogical, Anon?"
"Well, strictly speaking, the argument you just presented was not mathematical. The language of mathematics is one of perfect precision. I claimed that mathematics is logic, but so far we don't even know what either "mathematics" or "logic" is. Therefore none of these sentences is mathematical. On that note, we can't conclude from my claim that "mathematics is logic" that the words "mathematical" and "logical" are interchangeable. Therefore your argument that you be illogical was illogical, and hence your conclusion correct."
>A tear trickles down Twilight's face.
>"Anon, stop! I can't take this anymore! I don't even know what to think anymore. I don't even know how to tell if I'm being logical."
"Actually, it's a consequence of Gödel's Incompleteness Theorems, a result in mathematics, that it is impossible to tell if you're being logical. It is a mathematical fact -- a fact of pure logic -- that no axiomatization of logic can prove its own consistency."
>Twilight remains silent and looks hopefully lost and defeated.

>You hug the purple pony.
"Don't fret, Twilight. Humans had to undergo several paradigm shifts over the course of millennia to arrive at our modern dialectic. In fact, there was an ancient society of humans known as the Ancient Greeks whose understanding of mathematics stagnated for centuries precisely where ponies' seem to be now. For whatever arbitrary reasons, Equestrian society just hasn't undergone the first paradigm shift from classical to analytic geometry that humans underwent centuries ago."
>"So, ponies aren't dumb?"
"Twilight! I'm surprised at you. It's just a difference in culture. That's all.
Besides, most humans don't know any more math than you do."
>Twilight weakly laughed and smiled for the first time since the discussion began.
>"Really?"
"Yes. Only humans who spend years studying it can understand even the basic mathematics of centuries ago, and even then that's just because of our cultural institution. And only the tiny fraction of humans who devote their lives to mathematics as professors and researchers have any hope of understanding modern mathematics, let alone advancing it."
>"*Sniff* Thanks, Anon."
>"Can you teach me mathematics?"

"Sure! Let's start with modern logic. So, a topos is a category C that admits a functorial embedding to the category of presheaves over some small category, where a presheaf over a category is the category whose objects are functors from that category to some fixed abelian category and whose morphisms are natural transformations of functors, such that said functorial embedding admits a finite limit-preserving left-adjoint."
>Twilight's hopeful smile cracks and disintegrates.
"Sheaves are an elementary example of a topos, while a quasi-coherent scheme is an example of a topos to the category of locally ringed spaces. The left adjoints of these structures represented functorally determine them up to isomorphism, by Yoneda's lemma, which I'll explain shortly. Basically, the internal structure of an object is determined completely by its ectomorphisms."
>Twilight begins crying.
"What's wrong, Twilight?"
>"I'm too stupid to understand any of this, Anon! I'm.. I'm just too stupid."

>Your joke wasn't as funny as you thought it would be.
>Purple pony is crying and feels stupid and worthless for the first time in her life.
>Hello erection.
>You put on a disappointed expression, sigh, and say,
"You're right. Maybe I over-estimated ponies. I definitely over-estimated you. You're more like a stupid animal than a sapient being. You don't even wear clothes. I should have known you'd be a dumb animal incapable of understanding mathematics."
>As you say this, you flick Twilight's marehood.
>She was already wet.
"Just like a stupid animal."

>Twilight has a fetish for being made to feel stupid.
>Who would have thought? Incredible, really, and almost funny.
"Ok, listen. I'll break this down as simply as I can. A category is a class of objects endowed with a set of morphisms between each pair of objects and an associative binary function from composable morphisms to morphisms. In the category of categories, the morphisms are functors, which are structure-preserving maps between categories."
>Twilight looks at you dejectedly.
"This is as simple as it gets, Twilight! Come on!"
>Twilight is despondent.
"Get me a ruler and go stand in front of the chalkboard."

>Twilight fetches a ruler and stands meekly in front of the chalkboard.
"Do you want to learn, or not?"
>"I- I want to learn, please. Please, I'm not a stupid pony."
"You're not stupid for a pony, but you *are* a stupid pony."
>Twilight winces at her stupidity.
>You spank her with the ruler, eliciting a relinquished sob.
"Every time you say something stupid, I spank you. Clearly, you're incapable of actually understanding mathematics, but as an animal maybe you can be trained to not appear so goddamn stupid."
>She looks humiliated.
"Now, first question. Are you stupid?"
>She looks at you with pathetic eyes, and meekly says, "I'm a s-stupid pony." She emphasises the word "stupid."
>You place your hand on her back, and say,
"Yes, you are. I'm glad you finally realize this."
>Her marehood is soaking.

"I'm going to teach you Yoneda's lemma. A corollary will be that if a functor is naturally isomorphic to Hom(A,·) for some object A, then A is unique up to isomorphism. This will allow us to define objects by specifying a Set-valued functor. Do you understand?"
>"Y-yes."
>You whip her marehood with the ruler.
"No, you don't understand. You're a stupid pony."
>She whimpers.
"So, let's formally state the lemma. Given a functor F: C --> Set, Yoneda's lemma states that there is a natural bijection between Hom(A,·) and F(A). It is natural in the sense that induced hom-map C x Set --> Set is a bifunctor. Now, need it be the case that this bifunctor have a left adjoint?"
>Twilight remains silent.
>Spank.
"You're as dumb as a sack of bricks. Need this bifunctor have a left adjoint?"
>"Y-yes?"
>You whip her hind with the ruler again, hard.
"No! Of course not! Consider any category that isn't co-complete! Are you sure you're not retarded?"
>Twilight whimpers and shudders.
>"Well, are you?"
A drop of liquid hits the floor between her hind legs.

>You sigh.
"Let's try a different subject: algebraic geometry. Maybe you won't be such a worthless dunce at it."
>Twilight begins shivering psychosomatically.
"The fundamental object of algebraic geometry is schemes, but before we define a scheme we need to define the concept of a sheaf, which will generalize such concepts as smooth manifolds and sections of vector bundles, while at the same time providing geometric intuition for such algebraic notions as localization of modules and their analogues in abstract additive categories."
>Twilight is shivering. She knows you'll ask her something soon and she'll be too stupid to even understand your question.
"Regard a topology T as a category whose morphisms are subset-inclusions. A sheaf is simply a contravariant functor on T that respects co-equalizers. I want to make sure we're on the same page. Describe to me how a symplectic manifold can be regarded as a sheaf of holomorphic functions."
>Twilight's hind legs shake. She looks stressed. Her marehood is dripping furiously.
>"I-- I don't know! I just don't know."
>She begins sobbing profusely and collapses.
>"I'm sorry I'm so stupid. Such an idiotic animal pony. I'm sorry I'm sorry I'm sorry, stupid stupid stupid."
"Mentally retarded," you add.
>"I'm a mentally retarded stupid retarded dumb pony!"
>You gently pet her, and soothingly add,
"You are a stupid animal."

"Well, Twilight, I give up. It was already obvious you're too stupid to understand mathematics, but turns out you're so stupid you can't even be trained to appear even a little bit less idiotic. There is no hope of pretence here, Twilight Sparkle. Your whole life has been a lie, and you are mentally retarded. Do you understand?"
>Her nose is runny from crying. She sniffs and nods her head. A piece of snot dribbles down to her mouth.
"That is completely disgusting, Twilight. You're a retarded animal. Don't even pretend that you understand what that means. Now go clean up your humiliating cunt and "think" about what you are."
>You were sure to emphasize that "think" was in quotes.
>Twilight is shivering, covering her face in shame. Her cunt is still dripping. She doesn't get up.
>You get up and leave her there, stupid and humiliated.