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Monads aren't really about math, just like OO isn't really about objects in the real life, it's just inspired by them. Learning more about object in real life doesn't help much when doing OO, just like learning more about math doesn't help much when using monads in programming.

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I agree that in principle one ought to be able to do without thinking explicitly about logic, algebra, etc... but is that how it really goes in practice?

It's not, and I think that's why some people take a long time "getting" monads. Lots of tutorials online spend a lot of time talking about category theory to introduce monads. You can see the opposite view in things like this: http://dev.stephendiehl.com/hask/#eightfold-path-to-monad-sa....

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