Can we distinguish between an object's having trivial and having nontrivial parts?

Question

This distinction, if possible, would lend itself to a reformulation of divine simplicity, as the claim that God has only one nontrivial part, Itself, even though It might be "divided into" trivial parts (when properties are taken for parts). But so we might also say of pure shapes that their constituent lines are trivial parts of the shape, or infinitesimal sections of an infinitesimal patch of color might be trivial parts of the patch, etc. At least, I would hope to be able to apply such a distinction in such ways.

Are there any mereologists who have drawn such a distinction? Or does it maybe collapse into the proper/improper distinction? The SEP article on mereology says:

Taking reflexivity (and antisymmetry) as constitutive of the meaning of ‘part’ simply amounts to regarding identity as a limit (improper) case of parthood. A stronger relation, whereby nothing counts as part of itself, can obviously be defined in terms of the weaker one, hence there is no loss of generality (see Section 2.2 below). Vice versa, one could frame a mereological theory by taking proper parthood as a primitive instead. As already Lejewski (1957) noted, this is merely a question of choosing a suitable primitive, so nothing substantive follows from it.

Searching for the phrase "trivial part" with "nontrivial part" is giving pretty much just elaborate mathematical examples, which is fine, but I'm wondering about a metaphysical interpretation.

Answer 1

Great question. Learned a new sub-field (mereology).

The SEP article you reference give the core axioms of "partness":

(1) Everything is part of itself.
(2) Any part of any part of a thing is itself part of that thing.
(3) Two distinct things cannot be part of each other

Taking these as largely uncontroversial (which SEP article suggests is the case) we can see that for some "whole" thing (A), A is trivially a part of A.

Further down the article, there is a discussion of Supplementation, which uses the idea of a Proper Part (PP) predicate as an extension of the Part (P) predicate:

Proper Parthood
PPxy := Pxy ∧ ¬x=y

We also need the concept of Overlap:

Overlap
Oxy := ∃z(Pzx ∧ Pzy)

Supplementation is defied as:

Supplementation
PPxy → ∃z(Pzy ∧ ¬Ozx)

Which the article goes on to say

[Supplementation is the only concept] that appears to provide a full formulation of the idea that a whole cannot be decomposed into a single proper part.

This leads to the concept of an "atom":

Mereologically, an atom (or “simple”) is an entity with no proper parts, regardless of whether it is point-like or has spatial (and/or temporal) extension:

Atom
Ax := ¬∃yPPyx.

Going back to your question, we can say that a trivial part is simply an improper part and we need to show that God is simple iff God has no proper parts, or equivalently God is an atom.

To guide intuition, we could examine a more mundane object like a red ball (rB)

Is "red" (r) a proper part of rB? Is r even a part of rB?

Per the article:

mereology assumes no ontological restriction on the field of ‘part’. In principle, the relata can be as different as material bodies, events, geometric entities, or spatio-temporal regions, as in (1)–(8), as well as abstract entities such as properties, propositions, types, or kinds

Therefore, we can consider red a property, and hence a part, of the ball.

Is it a proper part? Per supplementation, to be a proper part there needs to be a remainder if we remove it. Will removing red from the red ball leave a remainder? The answer appears to be yes, we will just have an colorless ball ("not red") but all the other parts of the ball will remain unscathed as they do not overlap with redness.

We can do this because "red" is not constitutive of the entire ball.

In contrast, consider if clay is a proper part of a sculpture. The clay is co-extensive with the sculpture, so taking away the clay would remove the entire sculpture, and so clay is not a proper part of the sculpture. However, "hardness" is a proper part, because we will still have the clay (albeit limp).

So, on to God. The question here seems to be if God is constituted by all his properties, so God iff a set of properties obtain. The most important property is what Aquinas called "esse", most accurately translated as "being". Removing "being" would indeed remove God, so "being" cannot be a proper part of God.

But being is not the only property of God. We have the usual ones like omniscience, omnipotence (with caveats, per Aquinas), benevolence, etc. Are these proper parts of God? Omniscience and omnipotence seem to be orthogonal qualities, insofar as one pertains to action and one to knowledge. An omniscient but not-omnipotent God (e.g., deism) would still be very powerful, but not all-powerful: there would be limits on what God could do apart from logically contradictory actions.

On this reading, God possesses proper parts and hence is not an atom, and hence is not simple.

A counterargument to this would be to say that God is necessarily omnipotent, omniscient, benevolent etc. In this case, we are not arguing as we did with the red ball, but are more arguing from definitions. A mundane example would be an Algebraic Field, which is defined by the field axioms. If we remove, say, associativity, we no longer have a field, similarly to how if we remove clay, we no longer have a sculpture.

In this view, God is simple, since all of God's properties hold necessarily, and therefore cannot be thought of as proper parts. Therefore, God possesses no proper parts, making him an atom, and hence simple.

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BTW: You are not alone in this interest. Looks like someone at Fordham University investigated this as part of their thesis (behind paywall outside of a preview): https://research.library.fordham.edu/dissertations/AAI10810626/#:~:text=Divine%20simplicity%20is%20central%20to,distinction%20between%20existence%20and%20essence.