88 lines
5.6 KiB
Markdown
Executable file
88 lines
5.6 KiB
Markdown
Executable file
---
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title: Dysfunctional Programming
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date: 2025-11-24
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modified: 2025-11-24
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draft: true
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summary: Group Theory Adjacent Nonsense for people who slept through high school (me)
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---
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*Group Theory Adjacent Nonsense for people who slept through high school (me)*
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> A Cringe set is Just a Co-Based set
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>
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> \- Unknown
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So anyway, what the FUCK is a Monad
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- A monoid in the category of endofunctors
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- a burrito
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- a co-comonad
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Do any of these mean anything? No, so lets start at the beginning
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# Numbers
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- In the beginning, there was the empty set
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- and then Zermelo and Fraenkel said "Let there be a power set of the empty set"
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- ...which is a line of exploration that will get us nowhere productive for the scope of this work, and is probably something I'll talk about in the future.
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# Magmas
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- The easiest Algebraic Structure to explain is The **Magma**
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- **Definition**: A magma is a set \(M\) with an operation \(\cdot\) that sends \(a,b\in M \to a\cdot b \in M\)
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- the tl;dr is, you've got a set of stuff, and you can shove any two of those together to make a third thing, as long as it's still in the set, we call this property **closure**
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- you're probably used to \(\cdot\) meaning multiplication, but that's a fake word that doesn't mean anything
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- Aside: '\(\cdot\)' is hard to read on its own, hereafter it will be referred to as multiplication or as <strike>bottom surgery</strike> 'the operation')
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- The main example of this is the Free Magma, which is basically just a binary tree
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- In this case, 'multiplying' two binary trees just means concatenating them
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- Why is this called the Free Magma? I honestly have no clue, Free means *something* here but I haven't figured out what yet. This will not be the last time it comes up.
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# Monoids
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- The next step we'll on the Algebraic Structure Skill Tree is the Semigroup, but that's less important to focus on here so I'll fold it into the Monoid Section
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- When you add Associativity to a Magma, you get a **Semigroup**
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- Associativity just means that \(a\cdot(b\cdot c) = (a\cdot b) \cdot c\)
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- A more concrete example of this is natural number addition, \(2+(3+4) = (2+3)+4\)
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- The proof of this is trivial and left as an exercise to the reader
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- When you take a Semigroup and add an **Identity** to it, you get a **Monoid**
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- An identity is just something that you can combine with something and get that same something back out
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- The easiest way to think of this is that we need something that acts like '1' when you 'multiply' by it
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- If we go back to the example of adding numbers together, '0' acts as the identity since \(0+x = x\)
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- Yes, this means that the '1' element of \((\mathbb{N},+)\) is '0'
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- No, I'm not sorry for coming up with that absolutely dumb ass example
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- The Free Monoid is strings with concatenation as the operation
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- Told you 'Free' would come up again
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- the Identity element here is '', or the empty string
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- This is closed since adding two strings together always results in a string
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- This is also associative because Trust Me Bro
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- Another example of a Monoid is RGBA colors
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- The identity here is `rgba(0,0,0,0)` since adding clear to any color doesn't change it
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- Colors are closed under saturating addition since you can add two of them and get a color back*
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- *Actually I'm sure there's some sort of floating-point nonsense that makes this not true but IEEE 754 can go suck a log
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- actually you know what, lets pretend that #RRGGBBAA is a real notation that real people use, since it'll let us replace define multiplication as 8-bit saturating addition which is a much more well defined thing to do to colors
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- RGBA Color addition is associative because I said so
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- Absolutely unrelated, but I'm pretty sure OKLab isn't closed under the set of displayable colors, which feels cursed but is probably reasonable tbh
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# Categories
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- The Magma -> Monoid skill tree (and the several parts of it that I left out), are a scuffed refresher on Group Theory
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- Up next is Category Theory, which they did not cover in high school; As such I will be spewing a significant amount of hot air out of my ass.
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# Functors
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> A category is a collection of "objects" that are linked by "arrows"
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>
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> \- [Wikipedia - Category](https://en.wikipedia.org/wiki/Category_(mathematics))
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- ...FUCK, I was going to say that, unfortunately Wikipedia's definition is pretty spot on here.
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- Okay but seriously
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> In mathematics, abstract nonsense is a nonderogatory term used mainly to describe category theory and homological algebra
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>
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> \- Paraphrased from [Wikipedia - Abstract nonsense](https://en.wikipedia.org/wiki/Abstract_nonsense)
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- Okay forget finding a way to explain it to an audience of (<small>uhhh whatever my audience is, I kinda try to write these at a level high school me could follow along with and not get bored, but I don't actually know what kind of masochist is reading this</small>), I need to figure out how to explain this to me first
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- I'm back from Wikipedia, and I've returned neither smarter nor wiser, as such we proceed in our endeavors with the rallying cry: "Fuck it, we ball"
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# Endofunctors
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# Monads
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- Examples include: `Future<T>`, `List<T>`, `Option<T>`, `Result<T,E>` also logging
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- TODO: find someone who can explain how/why logging and file i/o are monadic
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----
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# Questions Mira hasn't answered yet and should really get around to before publishing this
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- Does free just mean the operation is concatenation?
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- https://en.wikipedia.org/wiki/Free_object
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- it's supposed to be the 'most generic'
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- How does using monads deal with side effects, eg Logging, File IO
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- Do they encapsulate state somehow? how does that work exactly?
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- what exactly does `Co-` entail
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