Can an 'abstract object' be a collection of constituent parts?

Question

When I ask this, the use of collection or set is not necessarily 'mathematical', so if in this case I mean a collection of ideas that encapsulate it, 'make up' the idea in the same way the various components 'make up' the computer I'm typing this on.

I'm aware the idea of a number as a 'set' mathematically (ZFC etc) but I feel that there is more to a number than being this kind of 'mathematical' object of set which itself could maybe be separated into a collection of ideas of what it is to be a 'set' in a mathematical sense.

Most physical objects can be readily dissected into constituent parts, can abstract objects be the same?

Answer 1

Yes, abstract objects can be composites.

Take several prototypical examples of abstract objects:

US Supreme Court: This is a collection of individuals, who are imbued with a degree of authority and powers. Changing out individuals, or even changing the NUMBER of USSC justices, or a change in the amount of authority they have, still leaves a USSC. Like all such composite objects, both physical and abstract, it is not a reducible object, as the changes at the reduced level still leave the composite item in existence.

British Constitution: This is a good example to refute efforts to claim abstract objects are actually physical, as the British Constitution has no physical realization. It consists of inference from a variety of documents, earlier decisions, and a jurisdictional tradition. This is composite, as well as abstract.

Selfhood: This is Hume's classic example of a composite (he used the term bundle) object. Hume's example of changing thoughts and experiences has mostly been rejected, as selfhood today is associated with more stable aspects of a person -- their memories, personality, and inclinations. But these more stable phenomenon are STILL variable, and multiple, hence selfhood is still, as per Hume, both a "bundle" composite, and if one thinks only in terms of reductive rather than emergent identity, then one would then not be the same "self" over time.

Memes -- Most memes are composite memes -- for instance the meme of a musical earworm can be decomposed to notes, timing, timbre, etc. The study of memes has revealed extreme difficulty in identifying actually NON-decomposable memes -- it seems likely that NO memes are actually the fully reduced components of thought -- we may only be able to think in composites.

Virtues, such as Love, Loyalty, or Patriotism: These are classic concepts that need an essay rather than a single line or word definition to explain what they are a nd consist of. Most of our ideas need an encyclopedia entry rather than a dictionary entry to explain, and that is because they are complex composites.

Answer 2

At the risk of missing your point and perhaps stating the obvious...

If an abstract object is that which cannot be physically (concretely) instantiated; if it is that which exists only in the imagination of those who contemplate it, then it is limited only by the limitations a person wishes to place upon it, and/or by the powers of a person's imagination.

Even a basic number such as the number 1, which is subject to shared understandings, can be reimagined to be constituted by a potentially infinite range of possible parts ('potentially infinite', because the imagination is likely unable to comprehensively conceive of an actual infinite or to actually imagine an infinite range of things).

A common imagining of the constituent parts of the number 1 might be the decimal fractions of the number 1, as these fractions or parts possess a utility we routinely employ.

There is nothing however to prevent a person imbuing the number 1 with any number of constituent parts which diverge from common usage (such as little pixel fairies, entire universes, abstract ideas, poetry...a combination of these), or to imagine a number 1 which has no constituent parts whatsoever. The number 1 can even be understood to mean something entirely different to what most people consider 1 to represent; including the number 2.6756 for instance, or an emotion, an instruction, a person, a rule, or a god.

Abstract objects are useful in part because they are malleable; because they can be adapted to suit a person's or a community's needs. If constituent parts of an abstract object prove of value to us, then those parts come into existence, and they don't have to conform to anyone else's notion of what those parts (if any) should be, unless the value they hold is shared and/or if their value must be translatable by the members of a group or by artificial systems.

In relation to your notion of a "collection of ideas that encapsulate it, 'make up' the idea"; ideas/parts that may constitute the number 1 might include concepts like 'quantity', 'relation' and 'sequence'. These in turn might have constituent parts, even if those parts are merely the arrangements of letters which allow us to articulate them, and to compare them with one another and combine them with one another.

Are there abstract concepts which can't be reduced? It seems unlikely if all abstract concepts can be constituted by the languages we use to express them, and if these languages can then be constituted by other properties, much in the same way the number 1 can be. This leads to a kind of circular constituency which could theoretically be repeated endlessly.

Answer 3

If I'm to take what you're asking as "Can abstract objects be broken down into parts the same way physical ones do?" Then my answer would be yes, in general.

In the physical sense of describing something as a set of its constituents, this is even somewhat unclear and abstract. Take a chair for example. This can be described as a set of atoms, which can be described as a set of particles, which can be described as a set of quarks and smaller particles, of which can be described as wave interferences in certain fields, of which...

It really seems never-ending. Your middle school teacher might try to tell you atoms are the building blocks of everything, but what they don't say is that even the building blocks have building blocks. This sort of recursive reasoning isn't even necessarily "completed" and is the primary subject of study in modern physics. With that said, it's hard to even claim physical objects can be broken down into constituent parts, as there is no defined foundation at the bottom of it all.

With that said, your question isn't necessarily asking of defining constituents with infinite recursion. So instead I'll take it to mean if at least one step of "breaking down" can be done, of which is clearly true for any macroscopic physical phenomena.

The difference then arises in that, with physical phenomena, there are atoms which we can defined to be the constituent parts of any object, but in abstract spaces this isn't necessarily true. There may not be some well-defined "building block" that can always be relied on as a potential constituent of a "larger" object in this space. This is better understood in understanding the vagueness of what it means to have an abstract space. Truthfully, I could define a space such that there is an object called "abstract-emon", but not bother to define any properties or constituents of it nor any other objects within the space that my abstract-emon could interact with. In this case it is impossible to defined the object in terms of its constituents. However, this is not typically the case. To define a space in this way is rather trivial and, frankly, useless. Typically, abstract spaces arise as a form of study, and in doing so, have well-defined logic and objects. In this sense, you may think of properties and objects as things that have been reasoned to, and if there are not well-defined constituents within the space, it's perfectly okay to define the reasons that led to these objects as the constituents in themselves. Therefore, any "proper" or "interesting" abstract space necessarily has the property that a given object can be defined in terms of its constituents. Once again, I want to reiterate this is not necessarily infinitely true. In abstract spaces especially, the notion of a "building block" that is recursively defined is very unlikely.

So yes, in most cases, and certainly all "useful" cases, it is possible to define an abstract object in terms of its constituents.