Well, a Functor isn't an HOF, and an HOF needn't be an implementation of Strategy.

I never said a Functor was a HOF, but it's not far wrong. The implementation of the morphism-mapping part of a functor is a higher order function. Show me an example of Strategy that is not essentially a HOF.

All strategy is HOF, not all HOF is strategy: Observe:

  (define (make-counter n)
    (lambda (c)
      (set! n (+ n c))
      n))

This is a HOF, but not an instance of Strategy.

I asked for an example of Strategy that wasn't just a HOF, but I now see you do agree with this. Yes your example is technically a HOF as it returns a function. So I guess your point is that HOF is not specific enough, although one might argue that in general usage it usually does imply function-value arguments.

I mean, that's not the general use I see, so we clearly have different social circles. The thing is, a HOF is a mechanism, the Strategy pattern is an intent. I would in fact argue that there is at least one HOF that takes functions as arguments and isn't an instance of strategy: call/cc.

call/cc does not actually use the passed in function to determine how any action should be done. All it does is provide a capture of the current continuation as an argument. That's it. So it's not a strategy, it's just a HOF.

Well Java didn't add lambdas to write make-counter, it was map and fold that got them envious. In fact, IIUC mathematically make-counter is first-order, counting the nesting of funtion types. In other words, I don't understand why many describe it as a HOF at all. Map and fold are second order. Callcc, ignoring its Scheme implementation as a macro, would be third order. One could argue that the second order argument to callcc is strategy, with the continuation being the strategy.

Call/cc isn't a macro. It's actually a special form, although in CPS-based schemes, it could probably be implemented as a function if there was a lower level continuation primitive.

And semantically, a continuation is pretty much never a strategy.

As for make-counter not being higher-order, that's just not true. A higher-order function takes and/or returns a function. make-counter returns a function: it's higher-order.

No its disputed, for the reasons I gave. But I've already accepted that many regard make-counter to be a HOF. I was careful and said make-counter is not a second-order HOF, which Strategy is.

Callcc can be implemented as a function, no continuation primitives needed. Haskell has examples in its libraries.

Lastly, regarding semantics, I see no difference between Strategy and a second-order HOF. Yes the scope of Strategy is supposed to much more limited, but I don't see value in this. I concede that others might do.

Having thought about this further, I think it is probably only correct of me to talk about the "order" of "functionals" as in functions to scalars in mathematics. The term "rank" is better suited here. Category theory supports the popular definition of higher-order function and gives a different meaning to "order". So make-counter is rank-1 but still higher-order. Apologies for the previous post.