As far as I know formalism is the view that mathematics is just a manipulation of strings according to established manipulation rules, which can be thought of as a game having its rules. In set theory there are the primitive notions of a set and set membership. A platonic view might be that sets denote collections and the element symbol "$\in$" denotes that the symbol on the left represents an object that belongs to the collection that is referred to by the symbol on the right, for example $x \in X$ means that the object denoted by $x$ is contained in the collection/set called $X$. I assume that a formalist would avoid words such as "contained" and "collection", as those seem to be based on the platonic interpretation. A formalist might just say that $x \in X$ is true, without further mentioning what "$\in$" means and that we can write this sequence of strings down since we agreed on it being a well formed formula before.
In some logic texts interpretations are not really mentioned. It is said that the string $tall(a)$ for example means that $a$ is tall. Hence this is immediately interpreted as the property that it should naturally stand for. This, however, should strictly speaking be an interpretation of the string, which might be natural in this case, but nevertheless could be different, right? As an example: I could technically also interpret this as anything I want, even as "a is small", couldn't I? Now, the notion of $tall(a)$ is chosen naturally to represent "a is tall", but nothing would guarantee that "$\in$" would denote membership of a collection. This could only be one interpretation of the symbol (which would be fairly natural), right? Technically I should be able to interpret it as whatever. Hence my questions:
I think I am confused by what an interpretation actually is and at what level it takes place. But I feel like anything beyond the notation of strings should be an interpretation in some sense.
$Q0)$ Is anything I wrote above incorrect? I appreciate any comments on what I wrote, since I am really not sure whether what I think is correct.
$Q1$) Is anything beyond the sequence of strings an interpretation? Meaning, as soon as I write anything other than a well formed formula, I am writing down an interpretation of the formula? An example would be given by what I wrote above on "$x \in X$".
$Q2)$ Is this the usual notion of an interpretation? I have read that there is a notion of interpretation in logic, but I do not really understand it, which is why I am asking.
