Clearly, everyone of you is perfectly aware of the long-standing opposition between the Ogres and the Trolls. This archaic division drives every discussion of mankind, either in the inn of remote villages and in the most sophisticated philosophical pamphlets, since ages. Every war in history can be ascribed to this apparently eternal and incurable hostility.
Its ubiquitous permanence in every aspect of reality makes it impossible to be entirely traced. Basically all the behaviors, patterns and concepts you can think of, line up in a trench or in the other.
Ogres only listen to 432 Hz sounds, and to their derivations. Trolls (provided that their gross ears can appreciate any difference) stick obtusely to the 440 Hz tuning.
Ogres embrace an apollonian living style, while Trolls cannot free themselves from toxic dionysian patterns. This point is so clear that any further word would be wasted.
An Ogre was Leonardo da Vinci, while Trolly was Michelangelo. In fact, they hated one the other.
Ogres/Trolls collocate also in dichotomies like West Coast/East Coast, booty/tits, crips/bloods, to be/to become, to be/not to be, chthonic deities/olympian deities, normal/deviant, day/night, beer/wine, one/dual and many others (the association is so obvious, in all these cases, that it’s left to the reader as a funny exercise).
Once, I heard of a Ogre that claimed to be a Troll only for convincing the Troll in front of him that does exist nothing like the classification into Ogres and Trolls… can you believe it? (Or was it the other way around?)
In any case, for the sake of clearness, let’s state the obvious: Ogres and Trolls are not creatures of the past. They’re all around us.
They’re us.
You are an Ogre… Or a Troll.
You’re not convinced?
Well, then. Let’s make a rapid test, it’s going to take just a second of your time. Something relatable to every human.
So it goes the test: let’s think to the differential of x. Let’s denote it dx.
You are surely comfortable with it, but… did you know that there is someone (maybe close friends of yours, members of your family, people you (used to) admire) that, even if they get the very same results out of computations, even if they call it “differential” like you… nevertheless have in mind a totally different idea of it?
For Trolls, dx is a small amount of whatever quantity the x variable denotes. For Ogres, dx is nothing less than a basis vector of the dual space of the vector space in which x lives (?!?!).
Trolls will employ dx in combination to dy, dz, dm, dt and many other small quantities in order to detect instantaneous changes, like dx/dt. What’s wrong with it, if the very definition of derivative is the limit to 0 of Δx/Δt? Everyone knows that the limit to 0 is satisfied for any quantity less than ε — any small quantity, as they say!
Meanwhile, Ogres apply dx, dy and dz to stuff. No joke! They write in all seriousness and rigor sentences like dx(f), dy(x) and even more ridiculous operations. They claim that dx is an application over x and that the spaces in which they live are not comparable by any means. The Trolls, they just think that if x is 1, dx is something like 10E-10.
What about the chain rule, self-evident to Trolls as dx/dy dy/dt = dx/dt ? Crystal clear, the real problems are others in math.
But Ogres, they cannot even start thinking about the “chain rule” if they don’t define pullbacks and pushforwards, vector bundles and dual spaces… also charts and atlases, in order not to get lost. Lost in their madness, the Trolls think.
This is the abyss of misunderstanding and alienation that Ogres and Trolls can reach.
Maybe, they will agree on the fact that dx measures an euclidean distance from a given point. Maybe they will agree on the fact that if such a distance is too big, everything looses sense. It can happen that they agree on the fact that dx, dy, dz measure such quantities along orthogonal directions. It’s likely that they suspiciously agree on the fact that derivatives are just some tangent stuff to a point… But that’s it. Everything else will be their war.
So, back to the test!
Do you prefer to formalize linearization through its properties, or through the hypothesis under which it applies?
You’re an Ogre in the first case, a Troll in the second! Leave it in the comments! As I told you, just a second of reflection is enough in order to reveal your side!
