There have basically been two objections to polytheism over the years: one, that it is impossible, and the other, that it is too complicated.

I. Impossible!

The impossibility objections traditionally come from more classical quarters, as the very idea of there being two or more Gods has seemed contradictory in light of a more metaphysical view of divinity. You’ll hear things like that it would be contradictory for two beings to both be omnipotent, or infinite, or the Good, or simple, and so forth, since then each would constitute an exception to the other. There are many, many ways to put these objections, and thinkers as diverse as Aquinas and Spinoza have advocated them.

My considered response has been that the impossibility objections to polytheism are like the rock-too-heavy-to-lift objection to omnipotence: they miscalculate the target concept’s range and so end up including impossible or nonsense scenarios where none should have been permitted. Just as the supposed rock is nonsense rather than a logical possibility in need of consideration, forcing Gods to be co-instantiations of something greater or prior, and so to not be “Gods” after all, is not a state of affairs that polytheists need to even try to account for.

We could conceive of Gods as compounds of essences and existences (or whatever), and so as impossibly co-instantiating things like omnipotence and such. But this would be hard news to break to the God of classical theism!

In light of the absurdities that impossibility objections to polytheism raise, and the fact that the God of classical theism isn’t even thought of in composite terms like this, it would seem that far from compelling polytheists to conceive of Gods as compounds of essences and existences, and so as impossible, reason compels us to conceive of Gods otherwise, such as pure existences.

II. Too many!

Simplicity considerations are more likely to come from contemporary quarters, where theoretical simplicity is a more prevalent consideration. You’ll hear things like that Occam’s razor renders polytheism unnecessary, or extravagant, or that a “committee” of finite gods involves untidy limitations not present in the simpler hypothesis of a singular omni-God (to phrase things Swinburnely).

I’ve likewise written a lot about this problem over the years, so let me just rank (in dramatized fashion) my estimations of the dialectical strengths of different responses.

F. The tu quoque/parity response: polytheism suffers because of its plurality? Sounds like a pretty good reason to doubt Trinitarianism as well.

D. I agree that we shouldn’t posit things unnecessarily, but it’s necessary, or called for, to posit each God.

C. You’re right: proposed limitations call for explanation. Why are we proposing the limitation of exactly 1 again?

B. Category mistake. If we’re talking about pure existences, we’re not even talking about how many of an essence to posit…

A. Insofar as we talk like this, the amount of pure existences is intrinsically, necessarily indeterminate; meaning it’s not 0 (atheism) or 1 (monotheism).