Well, if you are actually doing the real hardcore FP™, and not just the lame pretentious PF, then yes higher-order functions will indeed very much make heavy use of closures. Did you miss the part where I gave haskell as an example? And note that I didn't say that closures and higher-order functions are the same thing.
> Higher-order functions that use closures make use of closures. Higher-order functions that don't use closures do not
How are these tautologies even an argument? This does not say anything meaningful, like saying wet water is wet. Don't worry, I'm not even trying to keep a straight face while reading what all you have said so far.
> I don't think there is even a purpose to higher-order functions without closures.
Heh, I did qualify my statement with "I don't think" since I didn't really give that much though on that one.
But okay, I admit that statement is dumb and invalid since the usual map, filter, reduce functions are good examples where higher-order functions is not a closure. But more often than not, you really do need to use closures to do anything beyond simple cases like map(array, x => x*x).
My overall point still holds. I'm still in a very good position to challenge your dogmatic beliefs that: heavy-usage of closures is pseudofunctional and unfunctional.