When is a Religious Belief Delusional?

[Angrboda]

What do you mean by a mathematical object?  That's a term of art, and it's not clear what your question is.

I say that triangularity would exist in potential even if the physical universe had never come into being to manifest objects that sentient organisms would recognize as triangular.  Is my belief in triangularity wrong or delusional?

While how triangularity works is indeed a mystery, we have little doubt that triangularity in general exists, even if we don't understand it. It's an interesting question, but rather beside the point here. If by a mathematical "object" then you're referring to things like triangularity, or mathematical notions such as number, or mathematical objects like the Mandelbrot set, then you haven't aquitted your analogy. We don't think that any of these things "exist" in the same sense that we think God exists. I'm not arguing that there might not be some substance to Platonic notions as opposed to say, constructivism or whatever, just on its own merits, the ideas of forms and universals don't have the same ontological status as God does. And beyond that, we have everyday experience with triangularity, and can reason out transparently why such things as geometric theorems work. Why the Mandelbrot set exists is a mystery, but it could easily be accounted for via Ramsey theory, as could other "mathematical objects." In general, most mathematicians and (probably) most philosophers don't view mathematical objects or Plato's form or Universals as falling into the same category as things like God. The reasons for that are probably threefold. One, it tends to demonstrate it's existence pragmatically, in ways that things like the sensus divinatis do not. Two, our reasoning about these things is transparently related to reasoning in general, and related in the specific by things like the multiplicity of geometries depending on axioms. And three, we don't hold these things to be true in the same sense that many/most Christians hold God to be true -- there's nothing tentative or conceptual about God belief.

And that last point brings into focus the other side of the question, and that is the behavior of theists in relation to their beliefs. The confidence enjoyed by most theists is not abetted by the evidence for such beliefs, though I admit there is some, and depending, could consider it persuasive; theists aren't operating on the basis of evidence, their behavior is motivated by conviction which is only retroactively justified by appeals to evidence and argument. Which points to another area in which theist behavior is similar to delusion -- theists spend an inordinate amount of time rationalizing their beliefs; something I know I spent a lot of time doing when I suffered organic delusions. In general, the relationship between reasons for believing and the confidence and surety of belief are skewed tremendously in the case of religion. There are some obvious reasons for that, having to do with the substance of the beliefs, but I doubt that explains all of it. And even if it is the substance of the belief that is skewing the behavior, that skewing is still problematic, even if there is an innocent cause. Anyway, as noted, if this is all you meant by believing in the existence of mathematical objects, then your analogy is flawed and inapplicable. I have no doubt there probably are people who take the ideas of Platonism and mathematical objects to extremes (religion aside). I'm sure there are people who think that these ideas and forms exist in, essentially, a separate universe, mysteriously influencing this one, in a case that parallels that of belief in God. However, those are the exception, rather than the norm, whereas in religion, thinking such things is just par for the course, and believed with irrepressible certainty. And as noted, some of it is due to the nature of the beliefs, but there is still a fundamental difference in how such a Platonist acts with respect to their beliefs, and how a typical religious person does.