In the world that I imagine could exist, we'd do away with algebra 2 and pre-calculus in high school, which are a waste of 2 years, and instead do something like algebra -> geometry -> calc1 -> calc2 -> linear algebra -> abstract algebra, with linear algebra being concrete things like adding arrows and solving systems, and abstract algebra introducing basics of monoids, groups, vector spaces, and homomorphisms. It's sort of unfortunate that even the basic ideas of algebraic thinking (i.e. structures and structural transformations) are pretty much not even hinted at to anyone but math majors, and yet we spend years of school on something called "algebra". So even technical people can't see the point of structural modeling.

Why? Serious question, but what's the use of monoids? I encountered the term years ago, when I had an ill-fated ambition to make sense of monads. I've let it go and made peace with the world. But outside that narrow context, I've never even heard the term "monoid". What are people using it for in the real world?

Roughly, something is a monoid exactly when a parallel reduce type of algorithm can be used. The associativity lets you break it into sub-problems, and the unit lets you insert padding where necessary to get same-sized blocks for parallel processors. It's also a useful concept to know for library design. e.g. when there's a "combine" or "reduce" operation on some data type, it should occur to you that your users will probably want a neutral "do-nothing" element and that your operation should give you a monoid. APIs without one are usually annoying to work with and require extra if statements/special casing.

More generically, named concepts like this give you a way to compress knowledge, which makes it easier to learn new things in the future. You get comfortable with the idea of a monoid, and when you meet a new one in the future, you immediately have an intuitive ground to build on to understand how your new thing behaves.

Thanks. Parallelization of T[] => T operations makes a lot of sense. Monoids seem to introduce exactly the necessary constraint to allow some kinds of operation re-ordering needed for various perf optimizations like parallelization. I get it!

Taking that one step further, a monoid homomorphism is a function that transforms one monoid into another while preserving that essential structure (homo-morph: same-shape), so that map-then-reduce is the same as reduce-then-map. Being able to swap the two might be a useful optimization if one is more expensive than the other, for example. e.g. `e^(a+b) = e^a*e^b` turns addition into multiplication. Fourier transforms turn convolution (expensive, complicated) into multiplication (cheap, simple), if you know what that is.

In some other contexts, it's useful to talk about transforming your problem while preserving its essential structure. e.g. in engineering a Fourier transform is a common isomorphism (invertible homomorphism) which lets you transform your problem into an easier one in the frequency domain, solve it, and then pull that solution back into the normal domain. But to understand what's going on with preserving structures, you need to first understand what structures are even present in your problems in the first place, and what it means to preserve them.

This stuff isn't strictly necessary to understand to get real work done, but without it, you get lots of engineers that feel like the techniques they learn in e.g. a differential equations class are essentially random magic tricks with no scaffold for them to organize the ideas.

Another useful purpose of these concepts is to have the vocabulary to ask questions: A semigroup is a monoid without a unit. Given a semigroup, can you somehow add a unit to make a monoid without breaking the existing multiplication? A group is a monoid where the multiplication has inverses/division (So if your unit is called 1, then for any x, there's a "1/x" where x/x = 1). Can you take a monoid and somehow add inverses to make it into a group? etc. In a programming context, these are generic questions about how to make better APIs (e.g. see [0]). It also turns out that groups exactly capture the notion of symmetry, so they're useful for things like geometry, physics, and chemistry. If the symmetries of the laws of physics include shifts, rotations, Lorentz boosts, and adding certain terms to the electromagnetic potential, can I somehow understand those things individually, and then somehow understand the "group of the universe" as being made out of those pieces (plus some others) put together? Can we catalogue all of the possible symmetries of molecules (which can tell you something about the the states they can be in and corresponding energy levels), ideally in terms of some comprehensible set of building blocks? etc.

[0] https://izbicki.me/blog/gausian-distributions-are-monoids

It's a type of nice structure. Lists with concat, strings with append, etc. "Friendly chunkability", if you like. For instance, map reduce is a monoid homomorphism - when I see 'monoid homomorphism' in the wild, I think 'parallelizable' etc. It's a handy concept.

As the article explains, fold functions are a simpler way to think about monoids. So in recursive pure functional programming, many important iterative processes (pipelines, accumulators, machine states...) can be expressed as an application of fold-l or fold-r.

The value of monoids is being a single concept that generalizes the ideas of several different things like arrays/lists/strings, integral numbers, or paralelizable computation.

Pretty much every time you see people start talking about monoids/monads/endofucntors/etc... they are trying to get their compiler to automatically vectorize their program.