Matthew Baddorf has an interesting article arguing that there is an inconsistency between the doctrines that God is beautiful and that God is simple. The essential argument is:
(1) God is beautiful.
(2) If God is beautiful, God's beauty arises from some structure.
(3) If God's beauty arises from some structure, God has proper parts.
(4) Therefore God has parts (and is thus not simple).
Both (2) and (3) are certainly false, but it's worth considering Baddorf's argument for them.
With respect to (2), it's obviously being assumed that beauty is something that depends on something -- i.e., that beauty has to 'arise', which is doubtful for a case like divine beauty. But setting that aside, Baddorf gives as his reason for (2) a position that he calls Structuralism (which I'll call S1), which he defines as:
If an object is beautiful, it has a kind of unity, proportion, harmony, or some similar relation of various elements of a whole.
It's important to distinguish S1 from another claim, which I'll call S2:
If an object has a relevant kind of unity, proportion, harmony, or some similar relation of various elements of a whole, it is beautiful.
S2 is an extremely common historical thesis. It is much less clear than Baddorf suggests that S1 is quite so common, and indeed, there are highly popular historical accounts of beauty in which S2 is true but S1 is false. Indeed, the single most influential account of beauty in the history of philosophy, the Neoplatonist account, is such a case, as Baddorf himself has to recognize. In any case, Baddorf's argument for S1 does not adequately distinguish it from S2; the arguments for S1 could easily be seen as arguments for S2, instead, such as the arguments that many things in our experience that we find beautiful have 'harmony relations' of this kind, and some of the historical predecessors for S1 are arguably not actually committed to anything stronger than S2 (e.g., Aristotle, who does not actually have more than a few comments on the subject). In any case, Baddorf does consider Neoplatonist rejections of S1, and in particular, some of their counterexamples: pure color patches and thoughts.
Baddorf recognizes that pure color patches may be pleasing, but wants to say that calling them beautiful confuses beauty with pleasingness. This is not consistent with the way we ordinarily talk about beauty, since people regularly will not only talk about things like a beautiful shade of blue but even talk about the beauty of a single dimension of a color's appearance, like saying that a color has a beautiful deepness or lightness or vividness; nor do they seem to be talking about merely finding the experience pleasing. But Baddorf, I think, would suggest that in such cases they are merely assigning the beauty of a larger whole to the color. (He wants to say that any counterexample with color patches would have to be of experiences involving nothing but that color, to rule out that possibility. It's not at all clear why anyone would think this, but let's assume it for the moment.) Consider a pure field of blue. Can it be beautiful? Baddorf wants to say no. I, and the Neoplatonists, and, indeed, most people who have ever talked about the beauty of colors, want to say that yes, it can be so. What is more, I think it's clear that it could be the case even given S1; a pure field of blue has complete unity, proportion, order, harmony with itself -- and Baddorf has not ruled out that some of the relevant harmony relations are things an object could have with itself. He will go on to say this, and to say that we do not experience color patches as having harmony relation, but this actually makes no sense: we do, in fact, experience pure color patches as one and harmonious. That's what makes them pure.
Baddorf's fallback position is that even if we assumed that there was some way that the pure field of color was beautiful, it would have to be a minimal beauty. Again, I, the Neoplatonists, and most people who have ever talked about the beauty of colors, deny: pure, intense, vivid color is the height of beauty in colors. Yes, there are greater beauties involving colors -- a beautiful painting has, perhaps, a greater beauty than a beautiful color. But this is just because colors themselves are not the most beautiful things. It does not show that pure color cannot have the highest beauty as regards color itself. And the Neoplatonist can turn Baddorf's error-diagnosis against him: Baddorf is making the error of assuming that the highest beauty with respect to color (i.e., the highest beauty of a whole involving color) is the highest beauty of color itself. But this is not true, the Neoplatonist will (rightly) say. If we just look at color itself, its highest beauty lies in its purity; but there are things that exhibit higher beauty, and Baddorf is attributing the beauty of wholes that include beautiful colors to the beautiful colors themselves. Indeed, this necessarily follows: if a color on its own can't be beautiful, 'beautiful' has to be an extrinsic denomination when applied to colors.
Another Neoplatonist counterexample is thoughts. Baddorf wants to say that these are in fact structured. Even if that were so, what he actually needs to argue is that it is specifically about their structure we are talking when we call them 'beautiful', which is not an argument we get. But it's hard to argue that what Neoplatonists have in mind when talking about thoughts -- things like unified insights -- are always structured in the way Baddorf needs. What he needs for (2) to go with (3) is structure that has definite proper parts (otherwise, the argument fails through equivocation). But while our descriptions of these things have proper parts (namely, parts of the descriptions), what exactly are the proper parts of an understanding of the Pythagorean theorem as a whole? What are the proper parts of the insight that the hypotenuse of no right triangle can ever be strictly measured by the sides? What are the proper parts of the apprehension that there are no square circles? The Neoplatonist will deny there are any. And rightly so. Our descriptions of these things have proper parts (i.e., phrases and words), but these are not part of the things being described; and trying to break down an insight about a whole into its mutually exclusive parts seems to involve a category mistake.
This is related to a problem with the role that Structuralism plays in the argument. S1 says that something's being beautiful has a harmony relation of elements in a whole. It does not require that these parts be proper parts, and one of the harmony relations mentioned is unity. Now every whole is (by definition) an improper part of itself; and every modality of a thing is (by definition) not a proper part; and every whole is (by definition) one and proportional with itself. So if one assumed a whole with no proper parts, it could still have improper parts or modalities that are related by the relation of unity. Indeed, it would have to be so. S1 does not give us a tight enough account of structure to guarantee that 'structure' in (2) will actually be univocal with the kind of structure discussed in (3).
So how do we get (3)? The primary argument for (3) is the following dilemma, which assumes the truth of (2). Either the structure of divine beauty involves non-divine proper parts, or not. If it does, this is inconsistent both with simplicity and divine aseity. If it does not, how could there be such structure without proper parts?
There is no dilemma here, though; proper parts are not the only things that we take to contribute to structure. Wholes without proper parts can have improper parts (indeed, by definition all wholes are improper parts of themselves); modalities of a whole can be related to each other by relations like unity and are not proper parts; wholes can have rational relations to themselves (like self-consistency) that have structure; and if we reject Intrinsicness, we have the possibility of a whole necessarily having rational relations to other things that are beautiful. Mereology, modal logic, and indeed lots of other areas study these kinds of things as structures of some kind. If you accept S1, you seem to be be committed to something like these being enough if you accept a lot of attributions of beauty in mathematics and logic. And, of course, if we reject (2), as a Neoplatonist would, then the dilemma has no force to begin with.
And indeed, Baddorf's entire argument is suspicious from the get-go. If attributions of beauty are based on unity, harmony, and the like, noncomposite things (i.e., simple things) are more unified than composite things and guaranteed to be necessarily harmonious in ways that composite things cannot be guaranteed to be. This is the whole point of the Neoplatonists: things are beautiful in proportion as they are unified and harmonious, and noncomposite things are necessarily more unified and harmonious than composite ones. What Baddorf wants to hold is that both variety and unity are equally necessary to beauty-attributions. But the Neoplatonist counterexamples are precisely put forward to argue that in fact this is not true: beautiful faces, beautiful colors, beautiful actions, beautiful mathematical theorems, beautiful logical arguments, beautiful ideas, all are very different, but what brings them together as beautiful things is that they have unity; and in any given order it seems that the greater the unity, the greater the beauty of the thing in that order; and as we move from sensible things to intelligibles, it seems that beauty and unity continue to go together even as it becomes less and less reasonable to talk about proper parts. What can exhibit unity, harmony, and the like more perfectly than what is without division or composition? And on the other side, in practice we often attribute uglinesses to lack of unity; we often say that something is ugly because it is too busy or not sufficiently unified or incoherent. There's also a reason Neoplatonism has always been such a major player when it comes to accounts of beauty; whatever you might think of other aspects of Neoplatonism, it describes beauty in ways that people very often find resonant with their actual experience of it. We should be wary of arguments appealing to plausibility against a position for which we can easily point to the fact that many people through history and across different cultures have actually found it plausible.