"Sum types" (aka "Rust enums") are something that I've really come to love for representing the idea "There are 3 possible cases here, here's the data we need in each case, and please remember that these cases are strictly mutually exclusive."

Making this super easy and short to express makes a lot of designs clearer.

There's also an interesting tradeoff here between sum types and abstract interfaces:

- Abstract interfaces: "We have an unknown number of cases/subclasses, but they each support this fixed set of methods." The set of implementations is open, but the set of operations is harder to update.

- Sum types (aka Rust enums): "We know all the cases, but there may be many different methods." The set of cases is harder to extend, but the set of methods manipulating them is completely open.

A good example of the latter pattern is actually LLVM! LLVMs' intermediate representation is a essentially a small assembly language with a fixed set of operations (and an escape hatch). But because the set of basic instructions is small and known, it's possible to write dozens and dozens of optimization passes that each manipulate that small set of instructions.

And a surprising number of programs work well with an LLVM-like structure.

So in a language like Rust, which supports both abstract interfaces (via traits) and sum types (via enums), the tradeoff is whether you think you have many different types of values that you'll manipulate in a limited number of ways, or a small number of types of values that you'll manipulate in many different ways.

(Oh, and the metaphor of "algebraic types" goes deeper than you might think; there are some super interesting data structures based on taking the "derivatives" of simple algebraic types.)

> Algebraic Types are Just Elementary School Algebra

My math prof did say the exact same while torturing students with question about proofs about their arcane set of arbitrary numbers and if they can be considered a field or ring or a group or everything at the same time.

Sure, just some + and *...

And sure, for a programmer it is mostly about which operations are defined on the type. But with just a few tweaks here and there you can transform a tool into a torture device...

Jokes aside, I think this is a good explanation about the concepts and parallels.

...and a monad is just a monoid in the category of endofunctors of some fixed category

Just kidding but algebraic types are a great abstraction paradigm,