Yep, in case you didn’t know, there is now a Rational Catholicism YouTube Channel where posts here are made into videos!
Socrates, the father of philosophy, mentor of Plato—whom himself went on to become the mentor of Aristotle, who greatly influenced St Thomas Aquinas—never authored any works; however, many of his ideas and discussions were recorded by Plato in his “Socratic dialogues.” One of these recordings, entitled “Euthyphro,” recounts a discussion between Socrates and, the titular character, Euthyphro on what it means for something to be pious or good. Socrates seeks to learn the definition of that which is pious through a series of questions posed toward Euthyphro. Initially, it is concluded that that which is pious is that which is pleasing to the gods, but Socrates is quick to point out that as the gods quarrel, they, therefore, differ in what pleases them and in what they consider to be pious or holy. The discussion continues, yet they return to defining that which is pious as that which is pleasing to the gods. Needless to say, Socrates was not satisfied with this final conclusion.
Scholasticism—the philosophical school largely founded upon the works of St Thomas Aquinas, key school of theology within the western Church—quickly makes note of, and corrects, the error which leads to this unsatisfying non answer. In a similar fashion to the moral argument in making the case for the existence of God, this “pious argument,” if one wishes to call it that, argues against polytheism. The pious argument, in the searching for error in Socrates and Euthyphro’s attempt to answer the question, finds it unable to be answered within a polytheistic context—for unless all the gods are identical, and, therefore, one and the same god, there will be contradiction amongst them, as their natures will differ, resulting in them being pleased by different things. Therefore, should Socrates’ assumption that “piousness” exists and needs to be defined be true, in consideration that his proceeding logic is sound (which it is), there, therefore, must be only one God.
Rational Catholicism 
You say many Gods would have different natures. But, what does it mean for a God — especially in light of divine simplicity — to have a “nature”?
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The nature of any given thing is its definition, as per Aristotle. This again knocking another hole in polytheism, as divinity can be defined (as per divine simplicity) as being itself, which admits of no differentiation, which is necessary if two things are to share a nature.
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A lot of things are being taken for granted here that would need to be argued for, especially when in discussion with a polytheist. For example, the idea that to be divine is to be “being itself” is rejected by the Neoplatonists (cf. e.g. prop. 115 of Proclus’ Elements of Theology); and Thomists and Neoplatonists alike will not only deny that we can define that which is divine in any Aristotelian fashion (cf. e.g. S.T. I.13.1); but, more specifically, that one can be divinely simple and have a nature to be defined in terms of (cf. e.g. S.T. I.3.3.; prop. 127 Elem. Theo.). Thanks for clarifying though.
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The Thomistic ontological view of God as pure being is not a necessary presupposition for the argument (I used it as an example of the further problems which can arise from polytheism, since I myself am a thomist, and because you asked the question specifically in reference to divine simplicity), as no matter how one defines the natures of the gods, they will still need to differ in order that they be truly distinct deities, which again results in the problems pointed out in the original article. In the case of the Neoplatonists, the One cannot be divided, and hence in the argument is functionally identical to the notion of pure being. And yes, I’m aware that God is unable to be properly defined under the Thomistic view (hence why I noted the “as per,” which is a qualified definition, rather than saying “can be defined in the proper sense as”), as again I myself am a thomist.
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It is a category error — from which not even Aquinas is immune — to expect one “God” to be differentiated from another by parts (such as natures, properties, powers, perfections etc.). The Gods are not wholes, they are each divinely simple, and so no God has parts at all, let alone any that other Gods do not. Divine individuality is thus not a function or result of having some unique set of parts. It’s not a function or result of anything. Each God just is his or herself, and for that reason, is not the same as any other God.
The One in Neoplatonism is not any one deity in particular (as Plato concludes from the First Hypothesis of the Parmenides: “The One neither is, nor is one”), it is any God considered precisely insofar as she is herself. The One is not a “thing” that many Gods would divide, it’s just a way of talking about any God whosoever; a collective singular like “man” in the phrase ‘all that is man’.
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Neither Aquinas nor I believe that divinity can be differentiated by parts. The nature of a thing, after all, is not a part (at least not when defined in the Thomistic or Aristotelian sense), but rather the whole of a thing, as the whole of a thing is its substance, which, when taken in regards to how it causes a thing to act or be a certain way, is called a nature. Thus it is clear that the nature of a thing is not a part of it, but the thing itself in its entirety. Now, under Thomism, no thing can be said to have a perfectly simple nature (or be perfectly simple), and hence be a first cause and divine, unless it is being itself, which is unable to be differentiated in any way from another thing except by means of privation, in which case it would then not be being itself, and hence composite and not divine. This because if something were to be in any capacity in which another is not, then that which is not could not then be being itself, which is in all and every way, rather than in a certain way (that is, having a privation). The multiplicity of perfect simplicity is a blatant impossibility in Thomism. Under Thomism, something cannot be itself unless its nature is either being itself or some privation of it.
Regarding the One in Neoplatonism, what you said is simply not true. The One is an intellect, so to speak, which formulates the ideas of the forms and consequently brings about the existence of all things; neolatonist thus regard it as divine. They regarded it as absolutely and entirely transcendent, and hence can not be properly said to have any sort of traits which exist in dichotomy (eg. one or many; is or is not) which are properly attributed to lesser existing things. I recommend a read through the Standford Encyclopedia’s article on it, or even just Wikipedia’s if you lack the time for something as in-depth as Standford’s. In addition, Plato’s Parmenides is far too contested in interpretation to use as forceful evidence. It would be better to draw from his Republic, which too was composed in his middle period, and in which Plato clearly refers to the One as a singular deity when expounding on it.
All of this, however, is superfluous to the point that things which differ in nature (that is, are different things) are pleased by different things, as what pleases each thing is determined by the nature (that is, the definition) of it.
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There’s a lot to address here — from minor points of interest such as what “nature” refers to in Thomism, to more substantive issues like how the Neoplatonists understood The One, especially in light of their interpretations of The Parmenides, or how The Republic is to be interpreted — but it all strikes me as circuitous to our exchange in light of your statement that being itself is “unable to be differentiated in any way from another thing except by means of privation”.
This statement tells me you’ve not yet grappled with polytheism. True, being itself cannot be differentiated from another except by privation. But, polytheism would not involve the differentiation of being itself from what is not being itself: it would involve the differentiation of one ‘being itself’ from another ‘being itself’. This differentiation would itself be secondary to and resultant upon the individuality of each God as a way of being in all and every way.
I’d recommend reading Edward Butler’s work on polycentricity. Although he rejects the Scholastic notion that to be divine is to be being itself, I believe the concept is transferable to a Scholastic worldview, and would in any case greatly illuminate what polytheism involves. Many of his articles are available on https://henadology.wordpress.com/philosophy/ but I’d particularly recommend “Polytheism and Individuality in the Henadic Manifold” and “The Gods and Being in Proclus.”
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I’m aware that you’re arguing that there could be multiple sheer beings. But as under Thomism each thing is distinguished from another insofar as it can be said to be in a way in which another is not, or to not be in a way in which another is, that which is sheer being cannot be distinguished from something else which is sheer being, because sheer being by definition is in every way, and thus neither can be in a way in which the other is not, making them then substantially indistinguishable from each other and under Thomism one and the same thing. It’s unsound to suppose that each god could be different yet still sheer being on the grounds that each “differentiation would itself be secondary to and resultant upon the individuality of each God as a way of being in all and every way,” because if each god was “a way of being in all and every way,” each would then lack the act the other’s “way of being in all and every way,” and hence neither would be being itself, because the aforementioned is a privation found in each.
I have grappled with Polytheism before, and it simply doesn’t hold up unless one applies the appellation of “god” to things which are not sheer being, but merely immaterial, such as angels, but of course Thomism is strict in its definition of God, and refers to only sheer being as such.
That being said, I will give your recommendations a look.
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