In this context, the mathematical dual of something is the consumer of the something. So "tagged unions are the mathematical dual of structs" means that a tagge

In this context, the mathematical dual of something is the consumer of the something. So "tagged unions are the mathematical dual of structs" means that a tagged union consumer is isomorphic to a struct. If you have a tagged union consisting of the elements Foo(int), Bar(char) and Baz(int,string) then to consume such a thing you have to specify what to do with a Foo, what to do with a Bar and what to do with a Baz. You can specify this by giving a struct with a function that consumes Foos, a function that consumes Bars and a function that consumes Bazs. In ML syntax we have: type Qux = Foo of int | Bar of char | Baz of int * string type QuxDual = Consumer of (int -> result) * (char -> result) * (char * string -> result) Now if you have a QuxDual, you can consume a Qux by picking the appropriate function depending on what kind of Qux you get, and then calling that function on the data inside that Qux. You can see that because of these `result` types duality is not particularly nice in languages based on functions. It's nicer in languages based on processes or continuations, because they have a direct notion of a consumer of something. A consumer of an int is an `int -> result` function but in languages based on processes we can directly take a process that takes an int as an input or a continuation C(int). So in those kind of languages we don't have extra result types hanging around.