I am a long time OO programmer and a functional programming newbie. From my little exposure algebraic data types only look like a special case of inheritance to me where you only have one level hierarchy and the super class cannot be extended outside the module.

You are describing closed sum types, the most common form of algebraic data types, as seen in F# and Haskell. Basically, everyone agrees that they are a useful feature to have in the type system, primarily because pattern matching makes it easy to dissect them by shape as well as by content and also because they permit exhaustiveness and redundancy checking.

However, there are other forms of algebraic datatypes. An important limitation of the conventional form is that they are closed, meaning that a previously-defined closed sum type cannot be extended with new type constructors (part of a more general problem known as "the expression problem"). OCaml's polymorphic variants allow both open and closed sum types and, in particular, the inference of sum types. In contrast, Haskell and F# cannot infer sum types. Polymorphic variants solve the expression problem and they are extremely useful. In fact, some languages are built entirely on extensible algebraic data types rather than closed sum types.

In the extreme, you also have languages like Mathematica where "everything is an expression". Thus the only type in the type system forms a trivial "singleton" algebra. This is "extensible" in the sense that it is infinite and, again, it culminates in a completely different style of programming.

So my (potentially dumb) question is: If ADTs are just that, a special case of inheritance (again this assumption may be wrong; please correct me in that case), then why does inheritance gets all the criticism and ADTs get all the praise?

I believe you are referring specifically to implementation inheritance (i.e. overriding functionality from a parent class) as opposed to interface inheritance (i.e. implementing a consistent interface). This is an important distinction. Implementation inheritance is often hated whereas interface inheritance is often loved (e.g. in F# which has a limited form of ADTs).

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