#a lot of things in math are like this actually...e.g. algebraic topology...

[as-if-and-only-if]

generating functions are really just like "hey, stick that thing in a nice structure and hopefully some of the niceness will rub off on it"

[reductionisms]

gintama mathematics compendium

recent developments in the world of gtama mathematics have convinced me to put together a rough handbook, if you will, of its various disciplines and applications.

the culmination of my gtama-math thinking is a (subjective) understanding that any gtama math is necessarily derivative. gtama mathematicians take gintama's initial unintelligible representation and extract meanings from it into relations we can better understand; in other words, we are looking at the representation of something undescribable and attempting to describe it further. if you like Euclid, this is going from R^n to R^n-1.

the scope of this list is narrow simply because my memory sucks and my knowledge of gtama-math bloggers is small. please send me addendums and updates so i can incorporate them.

My qualifications for this project are that I'm bad at math.

Philosophical Basis

on gintama as a mathematical system (yamameta, 2024)

Set Theory (groups and their relations)

gintoki equivalence class (ibid, 2023)

yorozuya&shinsengumi equivalence class (kraniumet, 2022)

joui 3 set theory (ibid, 2022)

ft. ygh set theory (ibid, 2022)

shoukason equivalence classes (joelletwo, 2023)

Analysis (limits, continuity, sequences, etc)

limit theory (kraniumet, 2023)

zura is schrodinger's wall (ibid, 2022) *geometric analysis

Algebra (concepts and their operations)

shouyou transitive property (yamameta, 2023)

shogun assassination equivalence poem (kraniumet, 2022)

takasugi&gin (me, 2024)

eye equivalence (kraniumet, 2024)

fs castle (ibid, with tags from transjjester, 2023)

cliff (agroupofcrows, 2022)

poles (ibid, 2023)

Euclidean Geometry (parallel lines exist)

types of symmetry (yamameta, 2024)

angle relations (ibid, 2023)

foils/parallels (ibid, 2022)

utsuro samsara geometry (agroupofcrows, 2023)

spiderweb cycles (ibid, 2022)

illustration of the final (ibid, 2023)

cinematic angles (kraniumet, 2022)

shoukason geometry (suchira with regnigt, 2024)

comedy orientations (regnigt, 2023)

Topology (beyond the parallel)

ouroboros framework (yamameta, 2023) *including sequel (application), original poem and its annotation

spheres (ibid, 2023)

spiderweb cycles #2 (agroupofcrows, 2023) *c.f. path connectedness in toplogy

joui 3 homeomorphism (kraniumet, 2022)

joui 4 topology one (ibid, 2024), two (ibid, with my tags, 2024)

Logic (applications)

time math (ibid, 2023)

takasugi math (ibid, 2022)

final triangle math (joelletwo, 2024)

shogun assassination equivalence (triangles) (joelletwo as joelleity, with kraniumet tags, 2023) *tags belong to analysis

shogun assassination equivalence (relational) (agroupofcrows, my tags, 2023)

shouyoutsuro reproductive strategies (joelletwo, 2023)

sakagin existence theory proof (yamameta, 2024)

sacchan/mutsu existence theory proof (ibid, 2024)

Conclusions:

I think gtma math works best when it refuses to describe with generative structures. that is, rather than generating an unrelated, outside structure and forcing gintama into it, we look at gintama and derive structures from it in cooperation with other knowledge. obviously the entire concept of gtama math already seems generative (who in gintama is actually doing math, anyways?), but i think this actually comes down to axioms and proofs, basis and spaces. gintama is already some unknown space; we first acquaint ourselves with the space, consider its pre-existent properties, and then we incorporate outside knowledge-as-description to reduce it into something knowable. this takes a lot of creativity, and is quite literally the procedure of math.

every single person who engages with any text ever is actually doing this in their own heads. we can never know anything, and yet, when we are related to by something from outside ourselves, we are called to incorporate its shadow into who we are. mistakes come when we generate ideas of what outside things should be and force whatever approaches us to fit those ideals (e.g., never-ending complaints about fanon). in a certain sense, though, it's impossible not to do this, which is why the modern mathematical system exists. logic is about appropriate direction; rather than idea in my head->put thing outside me into my idea->this is what the Thing is, math wants to go in the reverse: I can't know the thing outside me->it enters into me nonetheless->from studying it and relating it to other things, i generate ideas. hence axioms, which put the Thing inside your mind while also telling you, what we're talking about, you can never know, it is and isn't real, but you still have to deal with it.

everyone derives, or, in its proper sense, generates, their own frameworks about everything they come into contact with, whether or not they're aware of it. and yet, it takes a shit ton of thought and creativity to shape the framework taking form in your head into something actually expressible in words. the gtama mathematicians cited here are therefore brave warriors of mathematics, literary analysis, philosophy, and general living. pythagoreas has nothing on any of you.

[lipshits-continuous]

Asking both because this is a large hole in my maths knowledge and also because you like this subject:

What actually is a topology? Like I know the oft-repeated coffee mugs and donuts example but how does that actually come about?

The most general way I can describe it is as the study of continuity and properties of spaces which are preserved by continuous maps. However, that's not the whole story and it certainly depends on which part of topology you look at.

General topology, or point set topology, looks at the structures of subsets of a space, e.g. openness, closedness, compactness, connectedness, etc. The actual definition of a topology is to do with open sets, that is, we declare what an open subset of a set is. More formally, let X be a set and let τ be a collection of subsets of X. We say that τ is a topology on X if the following hold:

Both X and ∅ are in τ

The arbitrary union of sets in τ is in τ

The intersection of two sets in τ is in τ

If a set U is in τ, we set U is open in X. From this we may define all sorts of things including a general notion of continuity. These axioms seem very strange at first but they actually arise naturally from considering more familiar spaces, i.e. the real numbers. The notion of open intervals of real numbers and open sets in topology coincide!

One important concept is that of a homeomorphism. A homeomorphism is a bijective continuous map whose inverse is also continuous. Since continuity is defined in terms of open sets, homeomorphisms preserve the structure of the open sets of the topological spaces so in this sense we think of them as the same (more formally, being homeomorphic is an equivalence relation). Informally, we can think of homeomorphisms as maps that mold one space into another such that points "close" to each other in one space remain "close" in the other.

Another main area of topology is algebraic topology which is primarily concerned with classifying topological spaces by describing invariants using abstract algebra. I'm a lot less familiar with this area as I've not learnt anything formal yet (I will be after Christmas and next academic year). By invariants, I mean properties of topological spaces that are unchanged by homeomorphisms (technically it's something more general called homotopy). One of these invariants is the number of holes in a space (there is a whole area dedicated to making rigorous the notion of a hole!).

Using these notions of "the same" is exactly how we consider a coffee mug and a doughnut to be the same. Though the fact that they both have one hole isn't sufficient to show that they are homeomorphic, it does give a good intuition as to why we may consider them to be the same (a counterexample is an annulus that includes it's boundary and an annulus that doesn't, they can't be homeomorphic because the latter is compact but the former isn't, but both have a hole).

As I said, I haven't learnt any algebraic topology formally yet so there might be inaccuracies /lh