I've often criticized what analytic philosophers of religion have said about the classical doctrine of divine simplicity, mainly because analytic philosophers of religion have said many stupid things about it, but I don't think I've really laid out the basics of the doctrine here. Since, in the perpetual cycles of argument, arguing against divine simplicity seems to be coming back into fashion, it might be a good idea to do so. The doctrine is actually not all that difficult; it is, one might even say, if you will not hate me for the pun, relatively simple.
I. Triplex Via
It's useful, although not strictly necessary to start with a question: How do we know anything about God at all? Since we don't have first-personal knowledge of God, we know God through His effects. When we know something through its effects, the inference that constitutes that knowledge has certain features.
First, we obviously are identifying it in a specific way, namely, as cause of those effects.
Second, we obviously are identifying it as something distinct from those effects.
Third, we obviously are not committing ourselves to saying that its being a cause of those effects completely exhausts everything about it; instead, in causal inference we are saying that the cause is at least able to cause those effects. While we can sometimes eventually come to the conclusion that there is nothing more to something than its being a cause of certain effects, this is usually very difficult to establish, and is not a feature of the original inference by which we identified it.
The inference having these three features does not change in the case of God. For instance, if the effect we start with is Sacred Scripture, we are concluding to God as its original source, as distinct from Scripture itself, and as being at least adequate to cause that effect. If the effect we start with is the cosmos, we are concluding to God as its original source, as distinct from the cosmos, and as being at least adequate to be the source of the cosmos.
This is the foundation of what is historically known as the triplex via. Our knowledge of God, insofar as it is tied to effects (which certainly covers most of our knowledge of God, at least), necessarily has these features, which are historically usually called causation, remotion, and eminence, although sometimes precise terms vary. Triplex via is often mischaracterized; it's important to grasp that these are not successive, they don't have any particular order, and they can't actually be isolated from each other. They are just features that always go with causal inferences about God.
As Saadia Gaon famously noted, when we look at the world and the things in it and ask if we find evidence that they are effects, we do: they change, they are composite, they are limited in duration, etc. From this we infer that they have a source, and although there can be complications with issues like infinite regresses, an ultimate source (causation); this ultimate source, to be an ultimate source of the changing, composite, limited, etc. effects, must be unchanging, noncomposite, unlimited, etc. (remotion), but God, as ultimate source, is supereminent and therefore the divine nature is not exhausted by this discovery.
The doctrine of divine simplicity is just the doctrine that God cannot be composite because He is not an effect. To be composite is to be composed of parts; being composed of parts implies being composed by a composer; God is not composed by any composer; therefore God is not composite. Or, to put it in yet other terms, composition is a mark of being created; God is not created; therefore God is not marked by composition. That's it. That's the entire doctrine. Some people want rhetorically to make out that the doctrine is a bugbear of incomprehensibility, but in fact you can go anywhere in the world and find people, of all kinds of educational backgrounds and all kinds of religious backgrounds, who will helpfully explain to you that God is not made up of anything because God is not made at all. This is an entirely correct and entirely adequate presentation of the doctrine of simplicity; nothing more technical is required.
II. Composition and Noncomposition
The doctrine of simplicity doesn't require anything very technical. There are reasons, however, why you might want to be more technical. It makes intuitive sense to say that, since God was not composed by anything, He doesn't have components; everything we know that has components is composed by some cause or set of causes. But you might have questions about what 'component' covers. There will obviously be lots of obvious things, but there might be some cases where you aren't sure, and therefore need a more precise account of what makes something a component. The second reason you might need to get more technical is that there's inevitably that guy who will say, "But I think God can be composed of parts without being composed out of parts. God is composed of components without anything that composes God." And you'd need something more specific to say to that guy. You need a specific account of composition.
It is here where we start getting some technical disputes. For instance, the occasional dispute between Palamists and Thomists over divine simplicity is a very specific case of a more general dispute about whether Platonism or Aristotelianism provides the best account of composition. Nonetheless even so there are commonalities among disputants. The most general account of composition is that it requires something that is both actual and passively potential; the relation of part to whole is a particular case of the relation of potential to actual. Thus a common line of argument, particularly popular among Thomists, who have a very deeply elaborated account of potentiality and actuality, is to argue for divine simplicity by arguing that God must be purely actual because there is nothing with respect to which he could be passively potential. This is complicated a bit by the fact that we use 'potential' to describe two things, only one of which, passive potential, is relevant here, and by the specific details of the causal inference for God's existence that you are using (for Thomists, that would usually be the First Way or the Second Way). But on the basis of an account of composition in terms of potentiality and actuality, Thomists will argue that the potential of part to whole can only be ultimately actualized by something actual that is distinct from itself; from which it directly follows that if God has parts, something else puts God's parts together and keeps God's parts together. If this argument is granted, it is impossible for God to have components and not be composed by something else. This is only one possible line of argument, but it gives the idea.
Notice that I have said nothing at all about distinction or identity. This is because the doctrine of simplicity was not historically formulated in term of distinction and identity. It was discovered very long ago that trying to make sense of composition entirely in terms of sameness and difference ran into serious difficulties; this was one of the motivations, in fact, for trying to account for composition in terms of potentiality and actuality. The reversion to sameness and difference, or identity and distinction, is a relatively recent one, although it has some anticipations in the Cartesian account of divine simplicity. It is sometimes said, for instance, that the doctrine of simplicity is the position that all of God's attributes are identical with each other. This is not how the doctrine was historically formulated. If God is noncomposite, it follows that God's attributes are not distinguished as component parts composing God; but it's another question entirely whether there might be other distinctions between them.
People will sometimes refer to the Augustinian "Everything in God is God", but Augustine at the time was arguing against a particular variant of Arianism, and so his point is actually that there is no such thing as being partially God; if the Second Person has one divine attribute, He has them all. Augustine's analogy is illuminating, since he appeals to the unity of virtues thesis. That is, in a virtuous character, having the fullness of any virtue requires having all of the virtues; to have justice in a complete form, you need to have fortitude, to have fortitude in a complete form, you need to have prudence, and so forth. Thus virtuous character is a simple multiplicity and a multiple simplicity. The difference is that you don't need every virtue to have some virtue in some way; whereas to have any divine attribute in any way, you have to have all the divine attributes fully. If you are God in a way, you are God; the divine attributes are not divisible. There is no way to be partly God, so if the Son is equal to the Father in any divine attribute, He is equal to the Father in every divine attribute. This is certainly a claim about divine simplicity; it can indeed be put in terms of sameness and difference; but the reason is that Augustine is arguing something very specific here.
III. The Doctrine of Incorporeality as Training for the Doctrine of Simplicity
While it's not the only kind of composition, the most obvious kind of composition is corporeal or physical composition. Saying that God is incorporeal, and thus has no physical parts, is already to attribute at least a relative simplicity to God; you are saying that, since God has no bodily components at all, God is noncomposite at least thus far. In fact, historically, incorporeality has been the hard step. Augustine, for instance, who was no intellectual slouch, struggled to get past the idea that God had physical parts, and one of the reasons why Platonism was so important to the early history of Christianity is that it actually provided arguments that something could exist and be incorporeal. Thinking of incorporeal existences like pure spirits is not particularly natural to human beings, who often think with their sensation-based imaginations; and the inability to imagine something existing without having any physical parts is always the first and most basic stumblingblock people have when it comes to the doctrine of divine simplicity. Therefore, while it won't deal with every issue, it's always a good idea to take any claim made about divine simplicity and see how it would affect the doctrine of divine incorporeality.
In Christianity, the doctrine of divine incorporeality is closely associated with the claim that God is spirit (John 4:24). Since incorporeality is a relative simplicity, the arguments made for simplicity all apply here. For instance, it has been argued that being corporeal is a mark of an effect, so that the cause of all bodies cannot be corporeal; likewise, it has been argued that all bodies are potential, and therefore God, being purely actual, cannot be corporeal. Likewise, it has been argued that corporeality is associated with defect, which God cannot have. And so forth.
Once one grants that God is not corporeal, however, it follows that God also cannot be anything that would be sufficiently analogous to being corporeal, in the sense that very similar reasons would apply. Thus if God is not corporeal, it is very likely, just considering that on its own, that there are other kinds of composition that are not physical composition but are sufficiently like it that reasons for saying God has no physical composition apply to them, as well. That is, if God is relatively simple in the sense of not physically composite, God is likely relatively simple in other ways. For instance, human beings are psychologically composite; this is different from our physical composition, but it is clear that at least a lot of our psychological composition is due to our physical composition. For instance, we have separate sensory parts whose information needs to be integrated. If God has no physical composition, then it is clear he can have no psychological composition that depends on physical composition in this way. Thus, despite incorporeality being a narrower and more limited attribute than simplicity in the full and proper sense, recognizing God as incorporeal is already moving away from taking God to be composite, and quite forcefully.
When we start with incorporeality and move along even just by analogy, then, we have at least a reasonable argument for divine simplicity as the natural limit of this line of thought. And we can use the basic arguments for simplicity at each step. Thus Aquinas argues for God having no physical composition, having no hylomorphic composition, having no composition of nature and subject, having no composition of essence and existence; and by the time that we hit the last one, we're at such a basic level that we can simply take God to be simple and have no composition at all. Perhaps someone wants to get off this train before then; if so, (1) they need a principled reason to hold that arguments analogous to those that apply for incorporeality can't hold against the kind of composition they want to admit; and (2) if you're on the train at any point, you already hold qualified divine simplicity, so we're no longer arguing over whether God is simple but over the limitations of that claim.
IV. Issues with Common Objections
Ah, the objections. There is no way to go through them all. But it is worth pointing out a few things that objectors often forget.
(a) Simplicity is attributed to God by triplex via, which means that it does not set limits to what else may be said of God, as long as it does not introduce composition into God.
(b) Since things are attributed to God relative to His effects, and only relative to His effects, anything attributed to God, in the way it is attributed to God, has to be done so through some causal pathway, which will constrain the senses in which you can take the attribute.
(c) While simplicity removes any distinction that is a distinction between components, it tells us nothing at all about any other kind of distinction.
Thus the only objection against divine simplicity that is even worth paying attention to is one that starts with an effect and argues that God must be composite to have that effect, using an explicit account of composition that makes clear why we must think of this as composition rather than either as a higher unity or a noncompositional distinction.
