The reason to treat a programming language as context-free is that the context-free grammar tells how to parse the language. If you considered just the subset of C consisting of programs of length no more than $2^{32}$ that would be regular, but the regular expression would likely consist of millions of individual cases, and wouldn't be helpful for parsing the programs. You might not even be able to fit the compiler in memory...
For a simpler example, consider a context free grammar for arithmetical expressions with natural numbers, addition and multiplication.
- E -> {sequence of digits 0-9}
- E -> ( E + E )
- E -> ( E * E )
If you only look at expressions with 500 or fewer symbols, that fragment of the language is regular, but it is much more difficult to describe as a regular language than as a context-free one. Plus, the parse tree for the context-free grammar gives a direct way to evaluate the expressions, while the parse tree for that smaller fragment as a regular language is not likely to help evaluate the expression.
So, here are a few directions you may wish to explore for formalizing natural language, which is a broad topic. From looking a bit, this question does not appear to be an exact duplicate of an earlier question, but there may be some other questions such as this one that will help.
One is formalized systems that resemble natural language.
This includes the Mizar system which is a piece of software that validates proofs written in a syntax that's like a cross between a programming language and mathematical prose. There is a Proof Assistants Stack Exchange with more information on Mizar and other proof assistants.
One direction might be studying metalogic.
Metalogic uses natural language to avoid circularity, but its use of language is different from natural language in other settings. In particular, metalogical if is the material conditional, usually.
One thing to check out might be the Open Logic Project which has a few free and open source textbooks on logic. Boxes and Diamonds, the book on modal logic, covers some approaches to formalizing the notions of possibility and necessity (and some other things like time and deontic status). It includes a lot of examples of explicit metalogical analysis using Kripke frames.
One direction might be cataloging the difficulties we would run into if we attempted to formalize natural language.
Aside from the difficulties with quantifiers mentioned in the comments, such as in the famous example someone loves everyone. The connectives themselves like and, or, not, and if are tough to formalize. A compelling account of all of their usages is elusive.
The Connectives by Lloyd Humberstone, has examples taken from natural language of different kinds of phenomena that defy a straightforward encoding in a logical system. This book is quite big and full of technical details about different logics, but the introductory sections on the chapters about specific connectives have good examples of natural language use.
Best Answer
Your question is not completely clear.
This is the mathematical definition of Formal language :
Having said that, what are you meaning with "a pure mathematical concept/name of 'formal languages' " ?