What happens if I add water and... oil? Well, first of all, let's specify that if I add water to water, I still get water; and if I add oil to oil, I still get oil. Now, I could have water on top of the oil, oil on top of the water, water in the oil, and oil in the water. Let's collapse this so that now we have a collection of combinations that are water, oil, water and oil mixed, water on top of oil, and oil on top of water. If we can agree that we can "add" things in this collection together and get other things in the collection, then we have created an abstract algebra. The math is just a formalization that gives us extra power to reason about this collection of things.
In a more computer sciencey case, what might be the outcome of "adding" two Twitter users? Does one follow the other? Do they both follow each other? Is there something else that happens? Monad was a buzzword for a while because it made questions like this solvable with some goodies like easy parallelization. What if 200 million accounts suddenly needed to be added together? It sure would be nice to have an idempotent, parallelizable way to do so. Monads were one way of doing that, and abstract algebra gives us mathematical methods to rigorously approach problems like this (and many that aren't like this one).
It might be worth it to check out algebird
It's not unlike say Haskell, which IMO is pretty straightforward once you have a feel for all the terminology.
Suppose that S is a set and • is some binary operation S × S → S, then S with • is a monoid if it satisfies the following two axioms:
Associativity For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds. Identity element There exists an element e in S such that for every element a in S, the equations e • a = a • e = a hold.
In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique.[1] A monoid in which each element has an inverse is a group.
That's what I was getting at. I took a few classes in AA and its sibling topology, including at graduate level. Most of the time was spent defining things, because there's so many terms with very specific definitions. Only after maybe 2/3rds of a course do you really start getting into deeper material. In my brain, reading definitions was always a process like: read through, substitute definitions for jargon terms, read through, substitute... iterate until you hit a fixed point.
That's standard for math, but AA is particularly well known for its colorful choice of names. Some people enjoy it, but I was always more attracted to the analysis-family areas of math.
I've always found the other name for category theory amusing though: https://en.wikipedia.org/wiki/Abstract_nonsense
It's even worse with category theory -- how one is supposed to understand why functors are useful before encountering them in a natural context is beyond me.
Alright? Yeah, why wouldn't it be alright?