See the introductory post of this sequence to make any sense of the following.
The many sciences that deal with telic questions, from linguistics to anatomy to genetics to sociology, have at least two basic tools in common:
Since reductionism is the science of adding up parts to get a whole, it is natural to expect telism to behave a bit like subtraction.
If I want to know the role of a part, I can remove it to see how the whole changes:
* linguistics: remove a word from a sentence and see if it still makes sense
* genetics: knock out a gene and see if the organism survives
* neuroscience: remove parts of the brain to see what cognitive functions are affected
This suggests that there is at least a mapping between the role of a part and the way that the whole fails without it. This notion of failure indicates that we consider the whole itself to be under selection (to make sense, to survive, to think). We thus try to explain the part by showing how selection on the whole translates into selection on the part.
If I want to know if two objects have the same role, I can substitute an object for another and see if the whole does *not* change:
* anatomy: replacing a heart by a mechanical pump, without dying
* linguistics: replacing a noun by a pronoun, or by a phrase, without losing meaning or grammaticality.
This allows us to identify objects that have the same function in the context of a given whole, even when they have very different natures. The same nature can hold different functions, and the same function can be achieved by different natures. We can use this to build a many-to-many relationship of natures and functions.
In other posts, I have grouped together a bunch of concepts like function, role, goal, purpose and meaning, claiming they are all more-or-less related to telos. I wish to give them more precise technical definitions that also clarify distinctions.
Here, function is to be understood as it is used in biology or sociology, and probably quite unlike its common mathematical sense (which is deeply embedded in set theory, which itself is probably not very adequate for this kind of questions).
Traditionally, functional explanations, from social norms to anatomy, are timeless: we consider a system in working order, and explain some part of that system by its role in the whole (e.g. the body "needs" a brain, a heart, etc. to fill various roles). There is no need for dynamical models at all - knowing the temporal sequence of failures after I remove a body's heart is the domain of forensic science, not anatomy.
More accurately, function is how external* selection on the part derives from external selection on the whole: the function of the heart (being a pump to keep us alive) is shorthand for "selection on the heart [to behave like a pump] derives from selection on the organism [to stay alive]".
(*Here "external" is generally thought in terms of spatial boundary conditions.)
I will note that, in large part because of evolution, a historical dimension has been added to the concept of function in biology. I deem this usage inconvenient, as I think there are many instances where we can and should separate purely functional and purely historical explanations (for instance, etymology is a thoroughly historical science with very little in the way of functional explanations, and Chomskian grammar is a thoroughly functional science without a trace of historical explanations).
Purpose could be for temporal boundary conditions (initial and final states) what function is for spatial boundary conditions (what lies outside a subsystem at any given time).
In other words, purpose may be how selection on a short time interval (the present) is deduced from selection on a longer time interval (including some boundary condition such as "being still alive in a thousand years").
Despite this temporal dimension, I believe this is a way of explaining phenomena that is entirely distinct from the historical method (a point I will certainly belabor in a future post). If nothing else, it tends to explain the present by thinking about the future, rather than by thinking about the past. And more starkly perhaps, it invokes selection and necessity, whereas accidents and chance are the province of historical explanations.
Goals are related to purpose, but specific to discussions of agents and intentionality.
Generally, I will avoid going too deep into these kind of distinctions, because I think that a massive number of low-hanging fruits can be harvested at the level of basic telic questions before going into autotelism, i.e. generating one own's purposes or selective forces.
I am talking here about meaning in the linguistic sense (meaning of a word, whereas meaning of life typically pertains to function or purpose).
Meaning as reference to something external, or possibly as an at-least tripartite compound (linguistic symbol, mental concept, physical referent) is a more sophisticated instance of context and selection, but it is selection nonetheless. Many basic judgments of function in linguistics are actually judgments of meaning - see above: is a sentence grammatical or meaningful when I remove or substitute a word for another?
To conclude, telos is my grab-all word for all these objects. It is a bit provocative, as teleology is the bogeyman of many a biologist, though most are, at least unwittingly, daily users of teleonomy.
I do not care for these distinctions: my interest is in types of explanations, not in underlying material implementation and mechanisms. Whether a system is designed or emergent is a major question from a reductionist perspective. Either way, functional or purposive explanations (as defined above) are not only valid, but perhaps necessary in many places where roadblocks to reductionism appear quite daunting -- in many of what we currently call "complex systems".