I can see the position that math is something entirely created by our minds, and I'm kind of attracted to that idea, but it doesn't explain why such complexity / so many theorems can be derived (and proven--there's no other possibility) from such a simple set of axioms or procedures, nor does it explain why math is so ridiculously effective at modeling and predicting the universe.

Another thing to consider is that math is actually a particular application of logic itself--I think it's even been proven that all of math can be reduced to logic, so the question then becomes, is logic something invented by our brains, or something that exists "objectively" in the universe?

You might say it exists objectively out there, but consider this: if there were actually a logical inconsistency in the universe, you couldn't possibly perceive it. For example, consider the logically inconsistent example of a square circle. You can never possibly be shown such a thing, or if you were, you wouldn't know it because what could you ever actually, conceivably categorize/label as being a square circle with a sound mind? What would it possibly look like? There is no reasonable answer to that.

So that would seem to imply that logic is a way our minds frame things, because if it were an incidental state of the universe then there would be the possibility of the universe being some other way, which we've just shown there isn't. (The universe would have to have the possibility of being some other way *as perceivable and knowable by us*, as it makes no sense to say something is the case which is not perceivable or deducible from some sort of perception *even in principle*, because we are epistemological beings who know things by finding them out, so the very meaning of the 'existence' or 'state' of things is tied in with the perception of them.)

But then, I guess logic could be a *non-incidental* state of the universe? Like the universe is logical because it *had* to be? That seems odd, though, because it seems to me that the only things that can be *necessarily* true are tautologies, and tautologies actually contain no information about the ontology of things..

Perhaps not coincidentally, some philosophers (e.g. by Frege) believe that all of math is tautology (I guess excepting for its fundamental axioms), and I wouldn't be surprised if some philosophers believe the same thing about logic in general.

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I should add that any purported logical inconsistency in the objective universe that can be rationally perceived could be effectively reframed as a mere change in something from one state to another over time, or as a difference between what was observed and what our models tell us should be observed, even if it's an unexpected and inexplicable change over time or anomaly in observation, as our models of the universe are of course fallible and incomplete.

they are necessarily incomplete, as the universe is unlimited and so is the scope of things that what we're observing could possibly be affected by, and as far as the 'laws' that determine how it works go, what determines why the laws are the way they are, and what determines what determines why the laws are the way they are, etc. etc.? maybe it's infinite all the way down, too, and hence endlessly open to possible surprises.