One simple answer to the question “what is purpose?” is “the reference input of a control system”.
On this view, a thermostat acts purposively to maintain a room, a fridge, a shower, etc. at a constant temperature. It senses the actual temperature and adjusts it as necessary to keep it close to the reference temperature. Of course, it is the designer’s purpose that the thermostat should do that, but once the thermostat has been made and installed, the purpose is physically present in the thermostat.
This concept does not occur at all in the letter exchange.
Consider what a person does when they act on a purpose. Some part of the world is not as they want it to be, and they act to bring it to such a state. That is what purpose is: to intend a state of affairs, and act to achieve it.
There is a wider sense of the word “purpose”, where it is considered as a property of things. Maybe Dennett and Rosenberg are talking about that sense? The sense in which we can ask, “what is the purpose of this rock?”, and if it’s just some random rock lying somewhere, the answer would be “none”. And for the question, “what is the purpose of an animal’s heart?”, the answer would be “to pump blood around its body.” The difference between the two once again involves control systems. The heart is part of a control system. It is the actuator that will be made to beat faster and stronger, or slower and weaker, to meet the demand of the rest of the body for oxygen. A random rock is not part of any control system.
I’m inclined to map your idea of “reference input of a control system” onto the concept of homeostasis, homeostatic set points and homeostatic loops. Does that capture what you’re trying to point at?(Assuming it does) I agree that that homeostasis is an interesting puzzle piece here. My guess for why this didn’t come up in the letter exchange is that D/R are trying to resolve a related but slightly different question: the nature and role of an organism’s conscious, internal experience of “purpose”. Purpose and its pursuit have a special role in how human make sense of the world and themselves, in a way non-human animals don’t (though it’s not a binary). The suggested answer to this puzzle is that, basically, the conscious experience of purpose and intent (and the allocation of this conscious experience to other creatures) is useful and thus selected for. Why? They are meaningful patterns in the world. An observer with limited resource who wants to make senes of the world (i.e. an agent that wants to do sample complexity reduction) can abstract along the dimension of “purpose”/”intentionality” to reliably get good predictions about the world. (Except, “abstracting along the dimension of intentionality” isn’t an active choice of the observer, rather than a results of the fact that intentions are a meaningful pattern.) The “intentionality-based” prediction does well at ignoring variables that aren’t very predictive and capturing the ones that are, in the context of a bounded agent.
I would map “homeostasis” onto “control system”, but maybe that’s just a terminological preference.
The internal experience of purpose is a special case of internal experience, explaining which is the Hard Problem of Consciousness, which no-one has a solution for. I don’t see a reason to deny this sort of purpose to animals, except to the extent that one would deny all conscious experience to them. I am quite willing to believe that (for example) cats, dogs, and primates have a level of consciousness that includes purpose.
The evolutionary explanation does not make any predictions. It looks at what is, says “it was selected for”, and confabulates a story about its usefulness. Why do we have five fingers? Because every other number was selected against. Why were they selected against? Because they were less useful. How were they less useful? They must have been, because they were selected against. Even if some content were put into that, it still would not explain the thing that was to be explained: what is purpose? It is like answering the question “how does a car work?” by expatiating upon how useful cars are.