We’ve now had two episodes dedicated to exploring the relationship between algebra and the Swift type system. Let’s do a quick recap:

In the first episode we showed that addition and multiplication that we all know from algebra have a very real and direct correspondence to enums and structs in Swift. Specifically, a struct is like the product of all the types inside, and an enum is like the sum of all the types inside. This had a very real world application in allowing us to refactor data types so that values that we know should not be possible to exist are provably unrepresentable by the compiler.

Then, in the second episode, we showed that exponentials that we all know from algebra, that is take b to the a power, directly corresponds to functions from a type A into a type B. This was amazing because we could then take all the properties we know about exponentials and transport them over to the world of types. This allowed us to understand how making small changes to the inputs and outputs of a function affect the overall complexity of the function.

Well, today we are exploring the next piece of bridging algebra to Swift types. We are going to explore how generics and recursive data types fit into algebra. Turns out, like most things we do on this show, there is a very beautiful interpretation of these things in the algebra world. And we get to use a whole body of knowledge from algebra in order to better understand the Swift type system.

Let’s get started by looking at generics!