A general description of systems has the problem of either becoming too content with being a mere description of the element of these systems as to become an inventory of phenomena, or to be an analysis so abstract as to not contain any reference to the specific nature these systems are of, and therefore making the quality of systemifiability (of being able to be concieved to be elements of a system) a mere analytic category, a term without meaning beyond methodology; the two extremes being the understanding of a system of elements as these elements themselves (a strong form of reductionism to matter) or as the relation of these elements only without their nature, as the formal characteristics of a mathematical graph (this being a strong form of reductionism to _form_; a rarer form of reductionsm to be sure, but still not fundametally different). If we want to be more clear about the relation a system has to its elements or points of reference, or actors which the actions of the system are based upon (such as people and things in social systems, thoughts and desires in subjective systems, objects of perceptions or drives in psychological systems, cells in biological systems, and foundamental particles in physico-chemical systems and machines), then we need to understand what that reference is in general, or how a system in some sense can be its actions (or that what happens between these actors/elements), without loosing the actors that create them; or in which way the self-constitution of a system can create functional substitutes for their elements, that don't in fact replace them, but represent them in a sense in a then closed system of actions and reflections, which openness is then reintroduced as a general kind of mapping of these actions to other things, and create a transferability of that representation to the general question which elements each action can represent.
To illustrate this question, take an example from mathematics, namely that of the definition of a category. A category can be defined in two distinctly different ways: in one we first say, there are elements of a category, and then define a class of morphisms between any pair of elements A and B, written as f : A -> B (as indicated by this notation, mathematicians often use this to describe functions, but these f can be different things, like connections in a graph, numbers or visual elements), and then say that for morphisms f : A -> B and g ; B -> C we have a combination fg : A -> C, and that every element has an identity 1_A so that 1_Af = f1_B = f. However, there is different way to define this same thing. Suppose we have any class of objects, which sometimes can be multiplied and sometimes not; if we require, that if we have xy and yz we also have (xy)z and x(yz), and that they have to be equal, and that every x has units A and B, so that Ax and xB exist (where a unit is any X, so that xX = x and Xy = y for any x and y where the product exist), then it is also a category; namely, it is the union of all the morphisms in a category, with the objects being represented by these units, which are our identities 1_A; that is, we have the objects A, B represented by the identity operations 1_A, 1_B instead.
Obviously, general social, biological systems etc. don't have their elements represented by "identity operations" in any obvious way, but this example is an interesting jump off point to think about the general kind of representability of the elements of a system within it, purely by its operations. The question here is: is any object of any system representable by its action form? Is there, in other words, such a thing as a social, subjective, psychological, biological or even physical Yoneda-Lemma? And what would then be the "elements" of such a representation form of the actors of a specific system?
But let's back up a bit. This is all very speculative, and not that useful for a general discussion; althought the idea of isomorphisms between statements of mathematical categories and real systems is interesting, real systems aren't conceptual, so there won't be a proof of anything like that, at most a description of possibility Instead then, let's begin how these elements of the system are described to begin with, starting with the most basal system of the hierarchy, that of physical/chemical systems.
The elements of these systems are the elementary particles, or, to be more general (also including classical physics and not just quantum theories), bodes that have qualities such as mass, energy, momentum, electric charge, spin and so on. These phsical properties are what define an object in a physical system. If I were to swap out one object with a different one with the same qualities, there would be no difference for the system; for example, it doesn't matter much which material the string of a pendulum is made of, as long as its stress endurance is roughly the same and it weighs the same, as in the system of a pendulum, only the stress on the wire and gravity are what impacts the rest of the system. If I were also to put a magnet next to the pendulum, it would start to matter if the string was of cotton or metal, and if metal, what kind (say iron in comarison to aluminium fibres); but not matter what I do, if all the physical qualities involved in the interaction of physical laws where to be the same, then it truely would not matter if I swapped them out, and it would be impossible to prove if i in fact did. This is the same for a chemical reaction: if all qualities are the same, it is impossible to measure a difference; ultimately, this comes down to the kind of molecule used, and their relations to each other (crystal vs. asymmetric solid vs liquid etc.); ultimately, the sentiment expressed here leads to the replacability of elementary particles, of not being able to tell two electrons, photons or up- or down-quarks apart. But that is only one version of this general idea; if in fact we would not be able to go down to the level of elementary particles, then other qualities would be this last stage of interchangeability, be it the sum formula of a substance together with its pH value, its behavior under pressure and temperature etc. It does not matter what level our technology or understanding of nature is; as long as two elements behave the same under all known laws of nature, which is the same as to say that they have the same qualities, we cannot tell them apart.
In this very basic sense the elements themselves _aren't a part of the physical systems, only their properties are_. Really, a physical object is an X that has mass and velocity and charge etc., not any specific such X. It is a predicate, rather than an object that falls under it. But the object still exists, right? Otherwise we were to say that no physical object exists, simple because we cannot tell objects apart that seem identical to us, but that seems obviously absurd. And indeed, I am not saying that; I am merely saying that the object, that exists, is not part of the physical _system_, instead only its properties being a part of it and that fact _that it exists_. In this sense, systems consist not of entities, but of existences of entities, regardless of their underlying identity, that fall under a specific sense of identity-relative-to-the-system. Any system of anything as such is not an ontical, but an ontological phenomenon.
However, what does it really mean to say the object is not part of the system? Importantly, it does not mean the object does not exist. Obviously they need to exist, as the system as such only would function if the objects do, otherwise the system also would not exist. In fact, this is maybe the most peculiar function of a system: that it's existence is dependent on things that only need to exist in general, independently of what these things, that it needs to exist, really are. But, crucially, that is only true for this _form_ of a system. Systems are in this sense forms: they are either real forms (that is, relations of objects, that form relations of the form specified), or ideas describing the commonalities of all arangements of that form. As material forms they possess these two sides, as it has a long history of describing it in the scholastical picture of the material form, as potential and realized. But the question is then in what way systems are more than there form of realization.
I think the best way is here to differentiate a system from its form, by looking at what it is besides its form. The obvious Other of the form would be the matter of the system, that is what it is made of. However, as we said, the system actually is not made of its elements directly. Rather, the matter of a system is its interactions, the relations between parts; and the form of the system is then the shape these relations take over all. In this sense, the system of a pendulum is not made of the shape, or the idea of pendulum-ness - that would be the form of the system - but of the physical properties, forces, impulses etc. that make up the interaction of the ball, the rope, the base, gravity to earth etc. These interactions themselves compose the system, not the elements, but also are not the whole system; the system is only made of these interactions, but is these interactions and their specific shape, the way they interact with each other, have positive or negative reinforcements on each other etc. And that whole system is then dependent, as a necessary antecedent of it, on elements which are only formally defined by these interactions.
However, there is still something very abstract about this, in which the elements seem to disappear in the interaction apparatus. The reason for that is, that we might use the term "system" here in two different ways, once referring to the whole situation, and in a different meaning referring specifically to what is added to it beyond the elements - what is more than the sum of the parts, if you want - which happens also to be the only thing specified by the interactions, or by any possible observation that is on the same level as that system. Therefore, from that standpoint of observation, this interaction system is really all there is, but that is only in the sense of observability; it needs to assume the elements are there, but since we can't know what they are in that form of interaction/observation, we can only know of them as of theoretical entities, which is why they themselves are not part of the system, only the fact that they exist. (To illustrate this: the only way physical objects ever appear in social systems is by social observation by the ways society interacts with them - their cost, or their value for scientific discovery (science is a social system after all) - molecules by themselves, without being observed and talked about by a human, are not part of that social system, as much as they may have materially composed the building materials of the ancient Romans for example, as these elements, if not interacted with, are not observable by them, and therefore cannot produce a real difference in the system. The same is true of private thoughts in relation to society, bodily systems the mind does not have senses for etc.)
Both the whole-situation picture and the interaction picture are sensible for dealing with systems, and both exist clearly in some way. When I choose the interaction picture then, I don't do it because I would assume the whole situation does not exist, but because the specific reason I call something a "system" is to clarify that I am talking about interaction. For example, if I talk about the "economic system", I would refer to the economic system of payments and lending of money, maybe also of bartering; but when I also want to refer to the people and institutions partaking in that, I would rather call it the "economy", without referring specifically to systems theory, as systems theory has been, from my knowledge, more closely associated with the interaction picture than with any other mode of observation. Nonetheless, the whole-situation picture exists, and to any system therefore there exist what you could call its underlying sub-system or its environment, that would consist of a specific realization of a system and its elements in that situation.
Looking back at the elements then, there is in fact, as I noted in the example, a way that they can come from the environment into the system: by representation. Much as the identity functions in a pure mathematical category, these representations are not the objects they represent, and don't map much of their structure, as can be easily seen by the environmental discourse in society, which is much empoverished as compared to the biological diversity that it tries to preserve. However, it also adds to it a kind of formal element. It is an interesting program for future research, to study the difference of these kinds of representations to the mathematical, pure mappings, of something like the Yoneda-Lemma, and to ask ourselves: by how much do systems, that try to represent their environment, miss the actual object in their operation - how much can they reconstruct? And what do they add as a formal element of reconstruction? Naturally, I am mostly interested in this for the case of subjective systems, for which we can reformulate this question in a way I asked elsewhere too: What is the relation of thinking to the outside, material objects that is has - how can it represent them? And more poiniently, since the knowledge of these material objects (if believed to be truly different or only imaginary, aside) only comes to us from psychological perception: How can I represent my own psychological life in thought? What do I add to it in categories of my way of thinking? - And here we do not try to find a specific "immediate", a bare sensation, rather, we try to scour the edges of our system, the subjective system of me and the other headmates, of that what represents phenomena outside our own imagination, and maybe the sources of it. In this sense, activities like trauma work or search for the history of gender identity, natural plurality etc. as a relation between subjectivity and the psyche are not just practical work, completely seperated from theoretical exercises; they are in fact lived theory, a form of research into the environment relations of the system of the mind.