I'll try to keep this as short as possible.

Correspondence theory of truth: Why only one definition of truth? I agree that truth (when applied to judgments about reality are things that correspond to reality. But within other things, like models, or structured languages (mathematics, programming) true is defined as "a statement whose conditions lead to its result/conclusion". Thus 1+1=2 is true in two sense, one that within the structured language of mathematics it is by definition a true statement. It's also true in the sense that "1" is the symbol we use for a singular item, and "2" is the symbol we use for a pair of singular items. The math was created to reflect a relationship that exists in reality (two singular items when considered as a group and categorized are a pair). I would dispute it being "some fact in reality" as the singularity of an item is a fact, two of them can be considered one at a time, or as a pair. It's our minds that want to categorize them as a pair. And we need the structured language of math (which is both a language of descriptors and a model based on rules) to help our brains arrive at conclusions abstractly.

Mathematics doesn’t seem to have any of the second, so it seems to not be a language.

I think the issue here is that math is not just a language. It's also a model with rules of behavior. Think of it as a structured language like programming language. At the most basic level, the 0s and 1s are simply describing the state of the transistor (gate is open or closed) - machine code. But then comes higher level of language where we impose rules based on our needs (decimal allows easier counting and conversion in this instance, base 3 works best in this instance, or this combination of 1s and 0s is equal to the color "white" and can be activated by these other language and rules). It's the language reflecting truth (gate is open) combined with the rules of behavior (this code = "white") that makes math a structured language.

For me it has nothing to do with spookiness. Nor concern over abstract objects being tied to god as that doesn't seem too much of an issue. It's that I'm not convinced math is anything more than a structured language. I've read up on platonism, not convinced.

As far as things like the Banach Tarski Paradox go, it seems more an issue of the structured language than a paradox in reality (such as being able to define a multi-dimensional space in programming that does not correspond to reality, reality isn't at fault, the modeling language is - note that "the theory relies on points with no volume in the ordinary sense" - in other words, its describing something that exists only within the modeling language, not in reality). A point with no volume can exist in reality, but if a "ball" is made up of these, in what sense is it a ball?