type AST = Expr of ...
           | Stmt of ...
           | Comment of ...

  let rec evaluate e =
    match e with
      | Expr # evaluate the ... of an Expr
      | Stmt # ditto
      | Comment # ditto

  class AST {
  public:
    virtual bool evaluate() = 0; // again, forgive the C++ syntax
  };

  class Expr: public AST {
  public:
    virtual bool evaluate()
    {
      // Expr specific evaluate code
    }
  };

  class Stmt: public AST {
  public:
    virtual bool evaluate()
    {
      // Ditto
    }
  };

  class Comment: public AST {
  public:
    virtual bool evaluate()
    {
      // Ditto
    }
  };

Doesn't the discrimination of which branch of a sum type your data represents generally happen at runtime?

What I was getting at is that when we use the algebraic version of AST, we know exhaustively what forms it can take. The definition tells us. In the OO version, this is note true. If we restrict our use to the polymorphic functions, then this is fine. But if we try to inspect the OO version of AST, ask its true type, and then act on it, then we may get into trouble.

A more useful type system would have "union types". They differ from sum types in that you can construct a union type from existing types in an ad-hoc manner. For example, if you have a function that returns a string or nil you can construct a union type "Union of string or nil". To do this with a sum type you'd have to have a common super type of string and nil and no other types.

There are also intersection types, about which I know very little.

I'm not sure you understand sum types. Sum types subsume normal unions; sum types and unions are essentially the same thing and capturing this kind of return type in Haskell or ML is extremely easy. "Supertypes" are completely unrelated and subtyping is not actually needed in languages with sum types (for example Haskell has sum types but doesn't have any real form of subtyping).

For example in Haskell you'd use the `Maybe` type constructor. If your type returns a `Maybe String` that means it either returns a string or nothing.

    public static void test(Foo f) {
        if (f isinstanceof SubA) {
            SubA a = (SubA)f;
            /* ... */
        } else if (f isinstanceof SubB) {
            SubB b = (SubB)f;
        }
    }

Standard ML approaches subtyping in a very different way that renders this kind of approach impossible. It gives you less freedom in that it eliminates downcasting, but that ultimately results in a safer language. Of course Standard ML has proper sum types with all of their compile-time guarantees so there's little reason for it anyway.

   public static void test(MaybeFoo mf) {
      switch(mf.type) {
         case MaybeFoo.NOTHING:
             /* ... */
             break;
         case MaybeFoo.JUST:
             Foo f = mf.getFoo();

             /* ... */
             break;
      }
   }

I would note that when I'm dealing with tagged unions in C, as soon as the operation gets at all complicated (more than maybe 3 lines?) I try to pass it off to another function operating on the internal value which restores safety outside the boilerplate. I agree that support for a true tagged union is ideal, though. (Actually, this might make a good attribute in GCC...)

(I could totally be missing/misunderstanding something, though...)

So that's the blindness. The pre/post conditions around how mf.type relates to its surroundings cannot be expressed first-class in the code. You are required to do risky things like call potentially partial code.

    (a -> r) -> r -> Maybe a -> r

So, I think "bare primitive types lack meaning" is basically correct given that you interpret "meaning" as "meaningful uses/actions".

This is only if you reject the notion of a dynamic type (as most type theorists do, but programmers do not). Otherwise, Python has a rich type system that is checked at run-time. It lacks sum types, but we could easily imagine a dynamically checked realization.

Note that most sum-types are implemented by having a runtime tag on which you can discriminate.

If an "average programmer" here and a type theorist spent 1 minute at the beginning of the conversation normalizing terms then nobody would have any argument here. I don't think anyone tries to claim that "types" and "static types" (or, in the type theorist's parlance, "tags" and "types") are the same... they just try to argue whether or not it's "right" to call one or the other by the name "type".

In particular, many things I can state about static types do not carry over to dynamic types and visa versa.

I believe many conversations get derailed early because among static type researchers, static types are known merely as "types" and dynamic types as "tags" or "classes". There are meaningful ways to compare these two kinds of things but somehow or another they ought to have different words since confusing the concepts will inhibit understanding quite severely.

This seems wrong. You can make a new type that simply contains one of those, right?

In a sense that's what you're doing with a sum type in Haskell, though there you get to produce new types automatically from higher kinded types.

As a starter guide, I highly recommend reading Lawvere's Conceptual Mathematics.