Of course a doubling function is not idempotent!
I think the confusion arises because side-effectful functions can be considered as having type
f :: (RealWorld, OtherArgs) -> (RealWorld, OtherOutputs)
and so for a garage door toggle you have something like
t :: RealWorld -> RealWorld
where the new state is the old one with the door opened/closed as appropriate.
Now the idempotence condition becomes:
t(t(world)) == t(world)
but clearly
t(t(doorOpenWorld)
= t(doorClosedWorld)
= doorOpenWorld
!= t(doorOpenWorld)
= doorclosedWorld
so this is where the notion comes from. If you abuse notation and just say a function of no arguments can be idempotent then you'll get confusion like this.
But `GET(GET(x))` doesn't make sense, in general (and if it did, then you would not expect it to be idempotent), so clearly idempotency in this context is meant to mean side-effect free. They should probably just say side-effect free, though, to avoid the confusion.
It's possible to be idempotent without being side-effect free. If you PUT some record, for example, then that operation _will_ have side effects (modifying the record). If you then PUT that same data again, the result will be the same (it's idempotent).
that’s not a side effect. in common usage, a side effect is an extra action, not the desired action itself. if you PUT some record, and some other record or state changes, that’s a side effect.
I suppose logging is a side effect and that makes practically everything technically non-idempotent (though on purpose).
I don't disagree with common usage, but common usage is not what's being discussed here.
It is a side effect in programming usage (not just HTTP).
Something is side-effect-free if and only if the only result of it running is that you get an answer. If you ignore the answer, then you cannot tell you ran the function/method/call/whatever. PUT is not side-effect-free.
That said, side-effect-free-ness is an incomplete paraphrasing of the HTTP spec (RFC7231); you'll notice that the only mentions of the phrase "side effect" are giving examples of legal side effects.