You are just assuming the point you're trying to prove

> There are plenty of smart people - scientists even - who believe in all kinds of deities.

Okay? This is supposed to make me feel - how exactly? I'm not inherently disdainful towards theism or theists, but if I were, I guess your remark would make me like science less, or something?

> We can similarly bring all manner of religious beliefs under the anthropological knife

I'm not really sure we can, actually. At least not in some kind of non-contentious, "objective" sense. I don't really trust individual humans to give an accurate account of why they believe their beliefs, but I trust "anthropology" and "sociology" even less. My distrust for this on an individual scale comes from the fact that many beliefs & memes exist for purposes of social signalling, group identification, etc, and it might not actually be in your interest to know exactly why you believe what you do.

But these auxiliary functions of beliefs, such as signalling etc, seem to me to scale up as you introduce groups and larger-scale activities such as "anthropology" and "sociology". Without some feedback loop keeping them honest, why would I expect anthropologists or sociologists to tell me a true story about why someone believes what they do, any more than that person or anyone else? In aerospace engineering, the feedback loop is that if your design is bad, your jet engine won't work. As a result, I generally trust aerospace engineers about jet engines. But what is there to stop sociologists, anthropologists, etc from just settling on some bullshit that agrees with their preconceived beliefs or flatters their group status and promoting it forever?

But back to math. The history of mathematical ideas is complicated and interesting, but it isn't really that relevant to the question of whether the things those ideas are about are "real", which is equivalent to asking whether mathematical platonism is true or not. The question of platonism comes down to the definition of words like "real" and "exist". It is very easy to equivocate using these words, which is why most discussions about mathematical platonism are so low quality. I think the overall question isn't that meaningful so I'm not really a platonist or an anti-platonist. In most parts of human life, when I say "x exists", I mean that I can reach out and touch x, that it has a mass, temperature, surface texture, etc. In math, when I say "x exists" I just mean that I can talk about x without creating any logical contradictions. The square root of -1 may not exist in the same sense as my laptop here, but it exists in the sense that I can do things with it, such as add, multiply, raise to powers, etc, without reaching a contradiction in my formal system. So the whole "out there" thing doesn't really matter. There doesn't need to be an "out there" in order for me to meaningfully say that the square root of -1 exists.

I think that a lot of philosophy is like this too, when you mentally zoom in really closely on a problem, it often reduces to some kind of equivocation or inconsistent language usage.

Btw I don't really consider anthropology or sociology to be real intellectual disciplines, and I'm pretty on the fence about psychology and economics. I realize that is an unpopular opinion but I've thought about it a lot and I'm pretty certain that it's correct. Aerospace engineering is real because it attaches to some fundamental reality, namely that of the spinning fan blades, the combusting fuel, etc. If you get your engineering wrong, the fan blades won't spin. Likewise, math is attached to systems of axioms. When your do your math wrong, you get a contradiction. Sociology and anthropology don't attach to anything, they're like a closed loop, like theology. If you get your anthropology wrong, nothing really happens.