Possibly, what I am currently doing can be considered as research in this direction, and I will share about it below.
First, what is metamathematics? I would treat this as a theory in a language which can serve as a metalanguage of the languages used in mathematics. In my article I introduced a language called metalingua intended to serve as one metalanguage common to different languages. Since then, I developed further this language, so that it now has only one symbol of a binary operation. I talk below about this language and its set theoretic interpretation, but first will talk about my understanding of what is meaning (sense) in a natural language which is richer than any artificial language used by mathematicians.
I call meaning of a word the set of all things denoted by it ("referents of the word"). So, a noun denotes objects, a verb denotes actions, an adjective denotes qualities of objects, an adverb denoted qualities of actions, etc. Thus, "meaning of a word" is a set. I treat punctuation signs as operators, i.e. notations of operations, over expressions, and due to my treatment of "meaning" as a set, the punctuation signs denote operations over sets. A text in a natural language is a sequence of words and punctuation signs and, therefore, it denotes a set, which can be calculated proceeding from the meanings of words which are sets.
With this treatment, the next question which appears naturally, is whether there is a small number of operations over sets through which all other operations over sets can be expressed? The answer turned out to be simple - such a binary operation was introduced by Tarski and Givant and is called "adjunction". Currently I am working on axioms of the algebra with this operation and constants to play the role of quantifiers. You might want to look into my questions - this this. and this
We can also discuss about this in more detail if you drop a message to my email indicated in my profile.