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I think there's an important difference between saying Haskell uses monads to do IO and we use rings to do arithmetic. In Haskell, understanding monads helps you do lots of tasks more effectively, since so many data types are monads: IO, Maybe, lists, ...

In math, there are other uses for rings, but if you're using them, it helps to know a little ring theory. Similarly, if you want to use any of the other Haskell monads, it helps to know how monads work.

You say "I think there's an important difference", but then, you end up saying precisely the same things about monads and rings. What was the difference?

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