Thank you EricC for the additional nuance and clarifying examples. In what follows I will attempt to lay out what I (mis)understand, ask questions that I have, and further develop some thoughts. I look forward to corrections and additions.
Before getting to the main content of this post, there are some points of confusion for me between the thermostat model and the gull model worth identifying. In the thermostat model, function (wanting the room to be 80 degrees) is founded upon setting the goal of the thermostat system (keeping a metal switch bent) via the dial. On the other hand in the gull model, the function is scoped much tighter to an evolutionary teleology, ie. the function is defined in terms of what is good for the species. That evolutionary theorists find it meaningful to distinguish goals from function, I plan to consider them as different sorts of objects under this investigation. It may be that some construal of the set of goals is isomorphic to some construal of the set of evolutionary functions. Because these concepts are given as different, I wish to preserve the possibility that relevant categories exist where they are non-isomorphic[⇅][₾]. Taking steps in this direction, I think it may be useful to distinguish the structural role that goals and functions play relative to one another from the ways that they differ in their objective content. Goals are tightly-scoped to individual action, in that, the behavior is directed toward the satisfaction of a goal[†]. Goals, seen in this way, come equipped with an associated mathematical function from the collection of an individual's behaviors to an individual's satisfaction. In other words, a behavior is goal-directed when it can be valued in terms of personal satisfaction, satisfying an individual's need: goalEval :: Behavior -> PersonalSatisfaction Functions[⍾], on the other hand, are better interpreted in terms of the species, in that, functional behaviors support the fitness of the species. Evolutionary functions come equipped with a function from the collection of an individual's behaviors to a valuation in terms of the species. functionEval :: Behavior -> SpeciesSatisfaction I consider the above descriptions of goal-directed and functional behavior to be distinctions in the content of the objects themselves, differences of type at the very least. Additionally, there may well be differences wrt the way evolutionary functions relate to goal-directedness, structural difference. Given the thermostat model, it seems that the founded-ness of function upon goal (as alluded to above) suggests that it might be worth characterizing variation in function as contravariant variation in goal. Goals and functions are contingent on the behaviors of the individual. While goals are directly observed in terms of *what the organism does* (rubbing against an egg satisfies an itch, clearing away shell or fixing the angle of a piece of metal), the function is discovered indirectly. There is quite a bit more to think through here than I have managed to do, but I feel that this observation may be as important to understanding the goal-function relation as the content of the objects themselves. In fact, it can easily account for why we would interpret a goal as being distinct from function, even if the categories of each were in fact the same. Mostly for my own notes, further reading on the importance of this structural distinction can be found here [⁂]. [⇅] Here, by category, I am referring to a mathematical category that is to be understood as preserving some *phenomenological* notion or other via the notion of isomorphism. Size is the notion preserved by isomorphism in the case of sets. Dimension is the preserved notion in the case of vector spaces. Symmetry is the preserved notion in the case of groups. Closeness is the preserved notion in the case of topological spaces... [₾] For instance and allowing for some hand waving, in treating Goal and Function as the same we may rush to assume the same open set structure on both. Now, a small perturbation of the goals may satisfy the laying hen (by satisfying her itch), but doing so may fail to preserve the hen's laying-evolutionary function to regulate the temperature of her eggs. What would begin as a continuity preserving isomorphism would fail to preserve the topological structure necessary to our theory. [†] For the sake of style and clarity I will attempt to speak more directly than some readers may feel I have the authority to do. Please understand that I understand that I have no expertise here :) [⍾] By goals I understand us to mean something like instrumental goals and by functions something like evolutionary function. Because of the obnoxious overlap with other commonly used referents, I will try to distinguish typographically or explicitly when otherwise ambiguous. [⁂] https://altexploit.files.wordpress.com/2017/06/1992-categories-of-space-and-quantity.pdf
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