For a formal language $L \subseteq \Sigma^*$ over an alphabet $\Sigma$. From https://proofwiki.org/wiki/Definition:Syntax
The syntax of a formal language is its structure, and is specified by a formal grammar of the formal language.
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning
How do you define the "structure" of a formal language, and therefore define the syntax of a formal language? Ideally define them formally or mathematically?
The syntax of a formal language should be defined formally, because it pertains to the formal language which is defined formally as a subset of $\Sigma^*$, right?
Raphael once said that the syntax of a formal language is the language itself. Is it true?
Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal language that constitute the well-formed formulas of a formal system.
We shouldn't equate the two concepts: the syntax of a formal language, and a formal grammar of a formal language, should we?
Note that a formal language can have several formal grammars that all generate the language. When a formal language has multiple different formal grammars that generate it, is the syntax different for different grammar?
Also a formal language may not have a formal grammar that can generate it, and still syntax should make sense to such a formal language?
For a non-formal language $L'$ over an alphabet $\Sigma$,
What are the definitions of its syntax and its semantics, ideally mathematically?
When defining its syntax, how do you distinguish what pertains to the formal part of the language, and what is not?
Can its semantics be defined as a mapping $L' \to S$ where $S$ is some other set?
Maybe, for example, when $L'$ is a programming language?
Maybe, for example, when $L'$ is a natural language, such as English?
Thanks and regards!